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INT. J. RADIAT. BIOL., 1984, VOL. 46, NO. 5, 569-585
Adding two components of radiosensitization by oxygen
TIKVAH ALPER
Birkholt, Sarisbury Green, Hampshire S03 6AL, UK.
(Received 12 March 1984; revision received 23 May 1984;accepted 25 May 1984)
It has been shown, or inferred, in various contexts that radiosensitization of cellsby oxygen is the sum of two (or more) components. If the component sensitivitiesconform with the Alper and Howard-Flanders equation their sum cannot alsoconform, but, in practice, even the most meticulous experimental techniques willfail to reveal lack of conformity unless one of the component K values is at leastnine times the other. Thus, despite the many results that have demonstratedconformity with the equation, the existence of at least two components may wellbe a general phenomenon. The killing of cells by radiation is attributable to asummation of lesions in different structures; different K values for the contribut-ing components are therefore to be expected, since neither oxygen nor itscompetitors are likely to be present in uniform concentration in all elements of thecell nucleus. Provided the components have intrinsic values of o.e.r. greater thanone, their addition results in sensitivity that increases monotonically with Po2,approaching asymptotically to the overall o.e.r. which is a weighted average of thecomponent o.e.r.s. In a curve plotted with Po2 on a linear scale a point of inflectioncan occur only if one component o.e.r. has a value less than one (i.e. oxygen isprotective for that component), and then only if relationships between the otherparameters satisfy certain conditions. In cases in which points of inflection in thesensitivity curve have been observed these are unlikely to be accounted for by theaddition of two components. The analysis of the consequences of adding twocomponents of oxygen sensitization could apply also to chemical sensitization ofhypoxic cells.
Indexing terms: K values, oxygen.
1. IntroductionThe proposal that radiosensitization of cells by oxygen is attributable to more
than one 'component' has been made in various contexts. Using techniques forobserving very fast reactions, Shenoy et al. (1975) and Michael et al. (1978) observedtwo resolvable rates of reaction of damaged intracellular sites with oxygen; but those'components' would not be separately observable in steady state conditions. Powerset al. (1960) discriminated between 'immediate' and 'latent' effects of oxygen onirradiated dry bacterial spores. Ewing and Powers (1980) described three, possiblyfour, 'classes of oxygen-dependent damage' and they, and Powers (1982) postulatedthat different processes occur at low and high partial pressures of oxygen (Po2). Theproposals of Powers and his colleagues regarding different 'components' of theoverall effect of oxygen differ from our own, since we ascribe these to reactions ofoxygen at the sites of energy deposition in cellular structures having differentchemical compositions and environments, and therefore differing intrinsic oxygenenhancement ratios (o.e.r.s) (Alper and Gillies 1958, Alper 1959, 1961,1963, Obiohaet al. 1984). According to that hypothesis, the overall o.e.r. observed for cell killing isa weighted average. Despite the conceptual differences in the approaches of Powers
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and ourselves, both demand an analysis of the result of adding together componentsof radiosensitivity which differ in their mode of change with Po2. The need for suchan analysis was pointed, in particular, by the experiments of Obioha et al. (1984).These were done to determine the dependence on Po2 of the induction of a specificmembrane-related lesion that was expressed as lethal only when the irradiatedbacteria were exposed to penicillin (Gillies et al. 1979). Occurrence of the lesioncould be observed only as a component of the overall effect of radiation plustreatment by pencillin, which of course included the killing that would have beenseen in the absence of that treatment. Thus in that case the overall result was due totwo components that were clearly resolvable.
The analysis which follows is to be regarded as applying to radiation-inducedcell killing and its enhancement by oxygen. The term 'sensitivity' will be definedas the slope of an exponential survival curve, symbolized by S in the equation f(surviving fraction) = exp (-SD), where D is the dose; or as the slope of theexponential tail to a shouldered survival curve, expressed byf = n exp (- SD) in thehigh dose region, where n is the extrapolation number. For simplicity, the analysiswill deal with the summation of two components of sensitivity that actindependently.
2. O.e.r. when total sensitivity has two componentsLet the overall sensitivity be NS under anoxia and let
NS=NS1 +NS 2 (1)
and let the intrinsic o.e.r.s for the two components be m1 and m2. Let the overallo.e.r. be m. Then, irrespective of how sensitivity changes with Po 2, we have, at veryhigh Po2,
ml NS1 + m2 NS2mNS= mlNS1 +m 2 NS2 . . NS +=NS2 (2)NS1 '+FNS 2
3. Hyperbolic oxygen equation as the basis for the analysisWhen cell killing has been investigated for its dependence on Po2 at the time of
irradiation, several investigators (e.g. Bryant 1973, Cullen and Lansley 1974,Chapman et al. 1974, Ling et al. 1981) have reported results conforming with theequation proposed by Alper and Howard-Flanders (1956):
S mP+K-- -r (3)
NS P+K
where K is a constant and m gives the value of S/NS at very high Po2, i.e. m is theo.e.r., P being the Po2.
Differentiating r with respect to P, we have
dr K(m -1) (4)-_= (4)dP (P+K)2
Thus, if r is plotted as a function of P, the maximum slope, where P-*O, is given by
dro-M- (5)(dr) m--1k dP }1 o K (5)
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Adding two components
3.1. Addition of two sensitivity components, both conforming with eqn. (3)Equation (3) may be rewritten
S-NS P(m-1) (6)NS P+K
Assuming that NS = NSl + N5 2, with proper parameters for the components given bymI1, K1, and m2, K2 , and that S= S1 +S 2 for all values of P, then
S1 -NS1 _ P(m 1 -1) S 2 -NS 2 P(m 2 - 1)
NSl P+K 1 ' N$2 P+K 2
S-NS=(Sl+S2 )-(NS1+NS2)=P NS (ml - 1) NS2(m2 - 1 } (7)P+K P+K 2
The terms NS 1(mi - 1) and NS2(m2 - 1) are constants, so, for convenience, they willbe represented by the symbols a and f. Equation (7) may thus be written
S NSP{P 7 + P1} (8)
Equation (8) will express a hyperbolic relationship between S/NS and P only ifK1 =K2 . However, if K 1 #-K2 , the departure from a true hyperbolic relationship islikely to be difficult to detect unless one of the K values is at least nine times the other(see §6.3.3) or unless one of the component o.e.r.s is less than one (see next section).
3.2. Point of inflection in K-curve with Po2 on a linear scaleUsing Chinese hamster V79 cells, Millar et al. (1979) reported no change in
radiosensitivity over the range 103 to 5 x 103 parts per million of oxygen in theequilibrating gas mixture: their results are plotted with oxygen content on a linearscale as abscissa in fig 1. Millar et al. (1979) attributed this result to the existence oftwo 'components' of the oxygen effect. The horizontal section of the curve suggests
103 5.103
10 1.5.10' 2.10'parts per million of oxygen
Figure 1. Results of Millar et al. (1979) plotted with 02 content of equilibrating gas on alinear scale.
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that the first derivative, dS/dP, may have averaged a value of zero over that range. Onthe assumption of two components, both of which conform with eqn. (3), and whichcombine to give the overall result, we have, from eqn. (8)
dS a(P + K)-aP (P + K 2)- 3P caKl + K 2
dP= (P+K )2 + (P+K2 )2 (P+lK1 ) 2 + (P+K2)2 (9)
dS/dP could become zero only if ac or /3 were negative, i.e. if ml or 2 were less thanone.
The results of Tallentire et al. (1972) for sensitivity of B. megaterium spores, as afunction of oxygen concentration in the suspending medium, manifested a well-marked point of inflection, and the authors suggested that their results 'could... be areflection of events associated with two distinct (oxygen-dependent) processes'. Fora point of inflection in the curve relating relative sensitivity to Po2, the secondderivative of S must become zero at some value of P. Differentiating eqn. (9), wehave
d2S _ 2aK1 2PK 2
dp2-- (P+K 1 )3 (P+K2 )3
As with the first derivative, the second can have the value zero only if the o.e.r. forone of the components is less than one. Thus the curve relating overall sensitivity toP0 2 cannot demonstrate a point of inflection, if it is the resultant of two componentsconforming with eqn. (3), unless one of the components has m less than one.
While that is a necessary condition, it is not sufficient; the magnitude of all theother parameters (NS1, NS2, K1 and K2 ) must be such as to enable the first or secondderivatives to become zero at some value of P. It is therefore not suggested here thatthe results cited above should be interpreted as demonstrating, in those particularexperiments, the presence of a component with m< 1: some additional evidencewould be required. Indeed, in the case of the V79 cells used by Millar et al. (1979),there is evidence to the contrary. If their particular experimental conditions hadresulted in the presence of such a component, it could be expected that the overallvalue of m that they observed would be significantly less than has been reported byother investigators using the same cell line (eqn. (2)). But the o.e.r. noted by Millaret al. (1979) of 3.1 was about the same as that observed by Cullen (1976) or Walker(pers. comm.) who found no point of inflection in their K-curves.
3.3. Slope at origin of S-NS as given by eqn. (8)
(dS) m - 1 m2 -1m2 -1 ~~~~~~(11)(dS) S NS 1 -1 +NS2 K2 (11)(dP)p_ 0 N K1 K
which is the sum of the two components at the origin.
4. Overall effect known to be due to two componentsThe experiments of Obioha et al. (1984) were designed for the construction of a
K-curve pertaining to the induction by radiation of a specific penicillin lesion in thebacterial membrane (Gillies et al. 1979). The overall sensitivity, with o.e.r. 4'8, hadtwo identifiable components: the ordinary, 'control', sensitivity, with o.e.r. 33, andthat attributable to induction of the penicillin-sensitive lesion. The latter wasmeasurable only indirectly, as a component of overall sensitivity. Both 'control' andoverall sensitivities conformed with eqn. (3), within experimental error. If it is
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Adding two components
assumed that this was true also for induction of the penicillin sensitive lesion, theresults may be analysed in terms of the following general treatment.
The overall effect (parameters NS, mo0 , K 0) and its components all conform witheqn. (3), within experimental error. Thus
pNS(mO - 1)=pNSl(ml-1) +pNS2(m2-1)
P+K o P+K P+K 2
From eqns. (5) and (11),
(mo-1 '~ [m- + Nr 2-1\NS K )=NSl(MK )± ( Kmj)
But, from eqn. (2),
NSmO=NSml +NS 2 m 2
NS(mO-1)=NSI(ml-l)+NS2(m2-1)=ac+ (12)
c~+/3 _ ,8 /i.e. + # (13)
K0 K 1 K 2
Let K 1 =K0 +Al, where A1 is positive.From eqn. (13),
=_ + _ _ /Ko Ko+A1 K2
Oe { Al = i Ko- K2{Ko+A } { K 2 }
K 2 <Ko, or K 2 =Ko -A 2 , where A2 is positive.Equation (13) may then be simplified to the form
a KjA2 (14)
/ K 2A1
or
K1 calK2 /3A 2 (15)K2 -- ~ 2
Equations (14) or (15) embody known parameters in the case under consideration,for example o.e.r.s and anoxic sensitivities will be measurable both for the overalleffect and for one component. Parameters relating to the second component, orrelationships between them and the measured parameters, should then be calculablefrom eqns. (14) or (15).
4.1. Application of eqns. (13)-(15)The above analysis may be applied to the results of Obioha et al. (1984), with the
symbols they used corresponding with those used above as follows: their K. (killingof bacteria by radiation alone) = K1 ; their K0 (including additional killing by post-irradiation penicillin treatment) = K0 ; their Kp (for induction of penicillin-sensitivelesion) = K2 .
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Within experimental error, they found K0 and K1 to be equal, so, from eqn. (13),K2 should have had the same value. Separate estimates of Ko and K1 gave the latteras the greater, so K2 should, if anything, have been less than K0 . However, theauthors made an independent approximate estimate of K2 based on the values of 'D0 'for the penicillin-sensitive lesion, by calculating reciprocals of the differences inreciprocals for measured values of Do in the two sets of conditions for which thosevalues were directly measurable. Values of 'Do ' calculated in that way for inductionof the penicillin-sensitive lesion were clearly subject to considerable error. Whenthose values were used to derive K2, its value, 59 mmHg, was considerably greaterthan K0 , which would mean that the value of K1 would have to be less than K 0.Equation (15) can be used to calculate what the 'true' values of Ko and K1 would haveto be to give K2 = 5-9 mmHg, since 'a' and '' are known. Calculation shows that ifK0 had a 'true' value near the upper of the 95 per cent confidence limits on theestimate for K0 and K1 combined, K1 would have to have a value near to the lower ofthe confidence limits; and vice versa. This calculation supports the conclusion ofObioha et al. (1984) that the independent estimate of K2 represents an upper limit.
5. Relationship of estimated 'K value' to those of two (initially undetected)components
There are various methods for estimating K from a set of results apparentlyconforming with eqn. (3) within experimental error. For example, K may beregarded as equal to that Po2 for which SNS=('m'+1)/2; i.e. for whichS-NS=NS('m'-1)/2 . Since eqn. (3) may be written in the form
('r - 1')/('m-r') = P/'K' (16)
'K' would be given by that value of P for which 'r -1' = 'm - r'. A more accurate wayof estimating K by the use of eqn. (16) is to measure the slope of the straight linewhich should, if eqn. (3) applies, fit all the experimental points corresponding withthe left hand side of eqn. (16). But, theoretically, that will not be the case if the overallsensitivity has two components. The properties of the function ('r- ')/('m-r') insuch a case are considered in §6.2. (See §§2 and 3 for definitions of m and r.)
When there are two components eqn. (7) gives the true value of S- NS. Therefore'K' will be given by the value of P for which
NSm l N S l +
m2 NS 2 _1
S { + a| NS (eqn. (2)) (17)P +K P+K 2 2 2
where
a =NSl(Ml-1), =NS2(m2-1)
Equation (17), when simplified, becomes
p2 _ p (a f (Ki-K2)-K1K2=O (18)
and 'K' is given by the positive value of P which satisfies eqn. (18).The equation has a unique solution only if a = fl, in which case the value estimated
for 'K' will be equal to 1/(K1 K 2). If c7: Af, the estimated value of 'K' will depend onthe relationship of a to fl as well as of K t K2 . Some computed examples are set out
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Adding two components
in the Table. As these show, 'K' will be greater than /(K 1 K 2) if the component withthe larger K value also has the larger value of NS(m- 1), and vice versa.
6. Detection of the existence of two components of oxygen sensitization
6.1. Lack of agreement between observed values of S-NS for P>>'K' and valuespredicted by using 'K' in eqn. (6)
Equation (6) may be written
FS=NS PMlNS m2N2 1
S-NS=1NS P+ KFI -N P p+SK' K (eqn. (2))______{pmlNSmN- 1}
i.e. S-NS=Pp ,l (19)P+ 'K'
Thus values of S-NS at high Po2 will be predicted on the basis of eqn. (19), whereasthose actually observed will conform with eqn. (8). Detection of lack of agreementbetween predicted and observed values will depend on the accuracy of theobservations and on the magnitude of the ratio between predictions andobservations.
Let (S - NS)predicted Q
(S- NS)observed
The value of Q depends on the relationships between K 1 and K2, and ; and itvaries with P. No unique solution can be given if a # /3; however, Q has a maximumvalue when a = P, so if we regard those parameters as equal, when computing Q, weshall be dealing with the maximum possible ratio and therefore considering the bestchance of discovering a discrepancy between predicted and observed values ofS-NS.
Value of apparent 'K' for change with Po 2 in two-component sensitivity
K1/K 2 4 9 16
NSL(m-Il ) ~'K' 'K'/(KK 2 ) 'K' 'K'/J(KK2 ) 'K' 'K'/V(K1 K 2)NS 2 (m2- 1)
05 156 078 194 065 222 05507 175 088 238 0-79 288 07209 192 096 280 093 362 0911'0 200 100 3-00 100 400 1001.1 2'07 107 3'20 1-07 437 1091'4 2'27 113 3'74 1-25 544 1'362-0 2'56 1-28 4'62 154 722 181
t 'K' expressed as multiple of lower K value.
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When a=f#, 'K'=1/(K1K2), so, from eqns. (8) and (19),
Q=_ _ + '3P- 2 P+ K {P+K+ P+K2 }
Q- 2(P+K1 )(P+K 2)(2P + K + K2)(P + /KK 2) (20)
Let K 1 =n2K 2, i.e. x(K 1 K 2 )=nK2Q has its maximum value when P = K2 ./n (n + ,/n + 1)Inserting that value for P in eqn. (20), and putting K1 =n 2K 2, we get
Q 2(n+1) (21)
(see Appendix)Computed values of Q as a function of n are shown in fig. 2.The accuracy with which a 'K-curve' can be constructed depends on various
factors. There are potential pitfalls, like unsuspected oxygen contamination, orunsuspected oxygen depletion by metabolic or radiolytic action (Alper 1979,Chapter 6). Evans (1969) demonstrated that suspensions of cells in bubbling vesselsmight fail to be equilibrated with oxygen present in the gas in low concentration,even when radiation dose rates did not appear very high and gas flow rates appearedadequate. Silvester et al. (1982) showed that, if cells were irradiated on a surface,doses to cells in an atmosphere of nitrogen (commonly used to displace oxygen)would be less than in oxygen, because of the reduced yield of photoelectrons.Provided that attention is paid to technical points like these, in experiments in vitro,values of Q of 11 and more should be significant and should suggest that eqn. (3)does not describe the observations; the discrepancy may be due to the existence oftwo (or more) components of oxygen sensitization. As can be seen from fig. 2,Q reaches the value 1 1 only when K 1 = 16K 2, in the most favourable case, whenNS(ml-1)=NS 2 (m2 -1), i.e. NS('m'-l)=2 NSl(m-1). For any value of n theseparation between prediction and theoretical (ideal) observation may be visuallypresented by the pair of graphs of the functions 2/(P+nK2) and[1/(P+K2 )+I/(P+n2 K 2 )] against P, P being measured in units of K2 .Those functions are respectively the predicted and observed values of
K1 as multiple of K2
Figure 2. Maximum ratio, [(S-NS) predicted: (S-NS) observed], when there are twocomponents of sensitization and the prediction is based on 'K' measured at low Po2.
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Adding two components 577
2(S-NS)/NS (mI -1), and are valid for any value of 'm'. Figures 3 and 4 show pairsof curves for n=3 (K1 =9K2 , Q=1-07) and n=4 (K 1 =16K2 , Q=1-1). Even themost meticulous experiments are unlikely to yield results that could be clearlyresolved from the upper curve in figure 3.
It is concluded that K values of two components should differ by a factor of atleast 15 or more, if their existence is to be detected by virtue of the failure of results athigh Po2 to conform with those expected on the basis of a value of 'K' establishedfrom observations at much lower Po2
6.1.1. Lack of agreement between observed and expected values of S- NS for P <'K'The previous section is applicable to methods for estimating K that have
commonly been used by experimenters constructing 'K-curves'. Such estimates areordinarily based on observations of r at low values of P, ranging from about 0-2 'K' toa Po 2 about two or three times 'K'. However, another method, used, for example, byLing et al. (1981), is to make observations over the whole range of P, from as low aspossible up to 100 per cent 02, and to assume that the observations conform witheqn. (3). 'K' is then established by getting the best fit of the equation to theobservations. If there are, in fact, two components of sensitivity with differing K,then the Q of eqn. (20) will be less than 1 for values of P less than 'K', i.e. for
2.0
E 1.5
LIz
COI
I .5LI)
C0j
Figure 3. (S-NS)
0 10 20 30 40 50 60 70 80 90 -100
P in units of lower K valuepredicted (upper curve) and observed (lower curve) when components
have K values in the ratio 9:1.
2.0
E 1.5
LrIz
I 1.0
zI .5
C0J0
0 10 20 30 40 50 60 70 80 90 100
P in units of lower K valueFigure 4. As figure 3; components with K values in the ratio 16:1.
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P</K1K2 in the simplified case in which =fP. Observations will lie above thefitted curve for P>/(K1K 2) and below the curve for P<V/(K1K 2).
The ratio Q decreases with decreasing P: no minimum occurs for P>O0 (seeAppendix). Whether Q can become sufficiently less than 1 for 'significance' in thedeparture from the fitted curve must depend even more critically on the experi-mental conditions than in the case considered in the previous section. For example, ifthe K values of two components were in the ratio 16: 1, Q would attain the value 0'9with P 05 'K'. Values of Q even less than this, corresponding with lower values ofP, could, in theory, demonstrate lack of true conformity of the observations witheqn. (3), and thereby suggest that there were two components of sensitivity. But itmust be borne in mind that, at low P, Q is the ratio of two values of (S- NS) that arevery small, and is therefore subject to considerable error. Furthermore, it might bedifficult to be sure that the appearance of points below the fitted curve, for P<<'K',was not attributable in part to failure in equilibration of the cells with oxygen in lowconcentration, a point emphasized by Ling et al. (1981).
6.2. Equation (16) in terms of parameters for two componentsIf results conform with eqn. (3), the plot of (r-1)/(m-r) versus P will give a
straight line through the origin, with slope 1/K. Departure from linearity mightindicate the existence of two components.
If each of the postulated components conforms with eqn. (3) each with its propervalue of m and K, the observed quantity 'r' (= SINS) will be given by
S1 +S2 ere S1 mP+K1 S2 m 2 P+K 2
NS1 + NS2 NS1 P+K 1 NS2 P+K 2
The function ('r- 1')/('m-r'), which will be designated F(P), may then be given interms of the parameters of the components. The value of 'm' is given by
MlNS+ m2NS2 (eqn. (2))
NS
Substituting for 'r' and 'm' in terms of the variables and constants pertaining to thecomponents, and simplifying, we obtain
F(P)' P[ x(P+K2)+ #(P+KO K (22)A L (P+K2)+ K2(P+K1) ]
where XC=NS(m - 1), and fl=NS2 (m2 - 1) (see Appendix).
6.3. Slope of ('r- l')/('m-r') or F(P)Failure of F(P) to be linear with P would be demonstrated by different slopes, i.e.
different values of the first derivative of F(P), for small and large values of P.It may be shown (see Appendix) that where P-*0
(dF) O K 2 +iK (23)dP) p-o KK2(1+ fl)
and if P>>K, and K2
(dF) c+fl (24
(dP) P(large) K, + flK 2
T. Alper578
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Adding two components
The difference between the first derivatives for P-O and P (large) is given by
cK 2 + K1 _ a + _ c(K -K 2 ) 2 (25)K 1K 2 (ci + f) caK 1 + K 2 K 1K 2 (a + Of)(aK 1 + /K 2 )
This is a positive quantity, provided c and /3 are both positive, i.e. provided m1 andm2 are both > 1. Thus if oxygen sensitization is due to two components, the slope ofthe curve for ('r- 1')/('m-r'), where 'r' and 'm' are observed quantities, will be lessfor very large values of P than at the origin.
6.3.1. Detection that ('r-1 ')/('m-r') is not a linear function of PThis depends on the accuracy with which slopes can be measured when results of
this type of experiment are plotted in that form. The results of Cullen (1976) are aguide, since she determined the 95 per cent confidence limits of her estimates ofslopes of the function (r- 1)/(m-r), and the data on which those estimates werebased were probably as accurate as the exigencies of this type of investigation willallow. Cullen's results suggest that it should be possible to regard as significantlydifferent two slopes in the ratio 5 :4.
6.3.2. Demonstration that. equality between a and /3 would give the minimum requiredratio between two K values
The ratio of slopes of the function F(P) = ('r-1 ')/('m- r') at very small and verylarge P is given by
(dF) /(dF) (K 2+K 1 ) / (a+)dP) p-0 (dP)P(large) K 1K2 (a+3) (aKl+ +K 2 )Vsay
which simplifies to
V= 1+ afl (K1 -K 2 )2 (26)(O+ q /) 2K 1K 2
aB/(a + /3)2 has its maximum value when c = fl, in which case the ratio of K valuesrequired for detection of non-linearity of F will be the minimum.
6.3.3. Minimum required ratio of one K value of anotherThe greatest and least slopes of F(P) are respectively at P--O and P very large; but
it would not be realistic to expect the slope at P-0 to be measurable, or the change toa lesser slope even to be detectable. As is shown in figure 5, the plot departs from thatinitial slope at values of P so low that the difference between 'r' and 1 would scarcelybe measurable. The most accurately determined values of F will be those which giveF(P) not greatly different from 1, i.e. for P in the region of 'K'. Where
= #, the slope of F(P) at P='K' is given by
(dF) 4(27)(27)
(dP)p=,K,K2 K 1 +K 2 +21(K1 K2 )(see Appendix)
By the reasoning given above, recognition that F(P) is not linear with P should beachieved if
4 5/2\= - (28)
K 1 +K 2 +2/(KK2 ) 4 K1 +K 2
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K.- 9K-
P in units of lower K value
Figure 5. Thick line: plot of ('r- I ')/('m-r') when sensitivity has two components with Kvalues in the ratio 9:1. Initial slope shown. Crosses: imaginary experimental pointsscattered closely about the true curve. Thin line: regression line through theexperimental points, constrained to pass through the origin in the expectation that theresults conform with eqn. (3).
Eqn. (28) is solved by K1 =9K2 (or vice versa). A plot of F(P) for K 1 = 9K 2, = , isgiven in figure 5. Inserted in the figure are some imaginary experimental points. It iseasy to see that, even with that relationship between two component K values,experimental errors of acceptable magnitude could result in the conclusion that theplot was indeed of linear form. Rather large deviations of experimental points from 'astraight line through the origin', for P>> 'K', are likely to be disregarded because theerrors in calculating 'm -r' would be large.
Detection of the non-linearity of F(P) would clearly require observations over awide range of values of P, in which case the experimenter might find it moreconvenient to plot log ('r-l')/('m-r') against log P (figure 6). With that plot,
I10
IL
.4
2
.1
'I
'I~~~~~~~~~~~~~~
i .1 A 1 *4 A JU 0.n A3 stun-1 Z 4 I z 4 IV ZU 4U IUU
P in units of lower K value
Figure 6. Thick line: computed for accurate delineation of ('r- ')/('m-r') on a log-log plot,if sensitivity has two components with K values in the ratio 9:1. Crosses: imaginaryexperimental points. Dotted line: 'best' straight line fitted to the experimental pointsand constrained to have a slope of , in the expectation that the results conform witheqn. (3).
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Adding two components
observations conforming with eqn. (3) should yield a line with slope 1 (Alper 1976).The computed 'line' defining F(P) for K 1 = 9K 2 is not straight; but, if it were definedby very accurately determined points, it would probably be seen as a straight line notof slope 1. But, as shown in figure 6, the scatter of experimental points about the truedescription of the function might be such that the expectation of a straight line withslope 1 could result in the fitting of such a line to the observations, particularly sincethe most heavily weighted points would be those near F(P)= 1.
It may be concluded that the use of a 'linear' transformation of eqn. (3) might beof slight benefit for detecting non-conformity of the observations with the equation(cf. Powers 1982, but see also Alper 1983); but, in the analyses of most sets of results,two component K values would have to be different by a ratio of more than nine toone before the existence of the two components would be suspected.
7. DiscussionThe theoretical derivation of equation 3 (Howard-Flanders 1958, Alper 1979)
implicates concentrations of oxygen and of molecules competing with oxygen (andpossibly with other 'damage-fixing' agents) for reaction with radiation-inducedradicals in cell targets. The theory is based on consideration of metionic reactionsafter energy deposition in a single site. But there is inferential evidence that morethan one such site may be involved (Alper 1961, 1979, Shenoy et al. 1975, Michaelet al. 1978). If that is correct, there can be a single K-curve, with unique parameters,only if the concentrations of reacting molecules are equal at all target sites.
The role of non-protein-SH (NPSH) as a competitor with oxygen, and thereforeas a determinant of K, was demonstrated in experiments with bacteria (Michael andHarrop 1979) and with mammalian cells (Cullen et al. 1980). Those experimenterschanged the NPSH levels by chemical treatments; but Cullen et al. showed also thatintracellular NPSH, and concomitantly K, changed with the growth phase ofmammalian cells, observations that have been extended to several cell lines (Walkerand Cullen, personal communications). Changes in K correlated with intracellularNPSH can be substantial: in a mutant of E. coli that is deficient in ability tosynthesize glutathione, Harrop et al. (personal communication) found that intra-cellular NPSH was about one-ninth of that present in the wild-type strain, and the Kvalue about one-twelfth. It must be borne in mind, however, that determinations ofintracellular NPSH refer to the cells as a whole, and they may be only a crude guideto concentrations at critical sites within the cell. It is plausible that these may varywidely, by analogy with oxygen, the concentration of which in different parts of thecell is extremely non-uniform, whatever the Po2 (Benson et al. 1980).
If components of sensitivity differ only with respect to intrinsic o.e.r., or if theo.e.r. pertaining to one of the components is so near to one that the overall 'K'pertains almost entirely to the component with comparatively large m, then theexistence of the two components could not be detected from the shape of the K-curve: some other feature would be required, such as, for example, changes in mcorresponding with treatments that affect one component more than the other (e.g.Alper 1961). But if components differ with respect to intrinsic K values, perhapsbecause of differing concentrations of reactive species in the immediate vicinity ofthe sites of energy deposition, then, as the analysis shows, this would be difficult todetect from an experimental K-curve without substantial differences in the K values.
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The analysis could therefore be regarded as providing support for the contentionof Powers (1982) that the existence of more than one component of oxygensensitization is a general radiobiological phenomenon. As Powers suggested, theexpectation of conformity of results with eqn. (3) may in some instances have beenresponsible for obscuring features that could betray the possibility that componentsof sensitivity with different K were involved. An example of the detection of such apossibility is afforded by work of Walker and Cullen (personal communication) whofound that 'K' for V79 cells, established by observations at rather low values of Po2,was much too small to account for the increase in sensitivity seen when the oxygencontent of the equilibrating gas was increased from 21 (air) to 100 per cent. Theminimum ratio of K values of putative two components that could account for thediscrepancy was about 12. However, in some cases in which conformity with eqn. (3)has been reported, it has not been tested; in others, it has been tested by the use of oneor other linear transformation of the equation, and observations at several values ofPo2 considerably greater than 'K' have not been made. Some transformations, e.g.that used by Alper et al. (1967), are not really applicable to observations at values ofPo2 >> K, because the errors involved become very large. Figures 5 and 6 may suggesthow statistical tests of the validity of fitting a single set of parameters to the K curvemight be developed.
If there are undetected components of sensitivity with differing K, the measured'K' will, as shown above, be of the order of their geometric mean, one with K muchless than is regarded as 'usual', and another with much higher K. Powers (1982) tookthe view that oxygen acts by different processes at low and high Po2 . It is not hereenvisaged that there is any change in the mechanism of oxygen sensitization, as Po2increases: but it is true that the increase in sensitivity will be governed mainly by thelower (or lowest) of the component Kvalues at Po2 near to zero; and by the higher (orhighest) K when the Po2 is large. The latter consideration might account for someunexpectedly severe late effects on normal tissues, in the Medical Research Council'sclinical trial of hyperbaric oxygen radiotherapy (Watson et al. 1978). The rationalefor that treatment involves the assumption that most normal tissues are sufficientlywell oxygenated not to suffer a significant increase in sensitivity when theconcentration of inspired oxygen is increased by a factor of 15-an assumptionapparently justified by the 'usual' range of K values for mammalian cells:3-8 mmHg, say. But if those represent geometric means, the highest component Kmight well be high enough so that oxygen-breathing at 3 atmospheres could result inan effective increase in dose of 20 per cent or more to normal cells at risk (see Alper1979, Chapter 17).
With certain 'hypoxic cell radiosensitizers' the increase in sensitivity withconcentration has been reported to conform with eqn. (3) (Fielden et al. 1978,Hendry and Sutton 1984). As with oxygen, such apparent conformity might likewiseconceal the existence of two or more components, each with its own properparameters, and the foregoing analysis would apply.
AcknowledgmentsAt the outset of this work I found a discussion with Professor D. B. A. Epstein, of
Warwick University, to be very helpful. I am very grateful to Drs Beulah Cullen andHilary Walker for helpful discussion and for showing me, and allowing me to quote,some of their results before publication.
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Appendix8.1. Value of Pfor which ratio of predicted to observed values of S-NS is a maximum
Putting K1 =n2 K2 ,
(P+ n2 K)(P + K)Q=2 (2P+ (n2 + 1)K)(P+nK)
and Q has its maximum value when (dQ)/(dP)= 0
The numerator of (dQ)/(dP) is:
{2P+K(n2 + 1)}{2P 2 +PK(n + 1)2 +nK2(n2 + 1)} minus
{4P+K(n+ 1)2}{P 2 +PK(n2 + 1) + n 2K 2}
Collecting the coefficients,coefficient of p2 is: 2K(n + 1)2 + 2K(n2 +1) - 4K(n 2 + 1)-K(n+ 1)2
i.e. -K(n-1)2coefficient of P is: 2nK 2(n2 + 1) +K 2 (n2 + 1)(n + 1)2 -4n 2 K 2 -_K2 (n+ 1)2(n+ 1)
i.e. 2nK2 (n-1)2
constant term is: nK3 (n2 + 1)2 -n 2K 3 (n+ 1)2i.e. nK 3 (n-1)2 (n 2 +n+1)
The value of P for which Q is a maximum is given by
p 2 _-2nKP-nK2 (n2 +n+1)=Oi.e. P=nK+(n2 K 2 +nK 2(n 2 +n+ 1))1/2
=Kn 1 2 (n+n 1/ 2 +1)
The value of P for which Q is a minimum, i.e. P= -Kn1/2 (n-n 1
l2+ 1), does notoccur for P>0.
8.2. Expression ('r- 1')/('m- r') in terms of two components of sensitivity
, I 5+S2 ,= (S1 -NS) + (S2 - NS2)
NS NS
p [xl(P+Kl)+ fl/(P + K 2 )
L N-S '
'm-r'=mlNS +m 2 NS 2 S1 +S 2m-r=NS NS
NSl(ml - 1) +NS2(m2 - 1)-(S1 -NS)- (2 NS 2 )
NS
c +fi-P/(P+K) -Pfi/(P+ K 2 )
NS
'r-l' p[ al(P+K,)+/(P+K2) K'm-r ' La +f-Pp/(P+K )-Pf(P+K2)1
=Pr a(P+K2) + f(P+ K 1)
L(a + f)(P + K,)(P + K2 ) - Pot(P + K2) - Pf(P + K)]
F(P) = P c(P+K 2)+(P+K) 1L.K,(P+K2 )+ fPK 2(P+K,) J
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8.3. Slope, i.e. differential coefficient of F(P)F(P) =C/D, say.
(dF) C (d. C)
(dP) D (dP\ DJ)
(dF) (C)
(dP)po (D)po
cK 2 + PK 1
K K 2 (a + /)
dF' a(P+K2 )+f/(P+K1 )
dP XK(P+K2 )+ K 2(P+Kl)
+p ( +f){oaK(P+K 2) +fK 2 (P+K 1 )}-{ac(P+K2)+fi(P+K){(aKL +iK2)
L[~~ {cKo(P+K2)+#K 2 (P+K 1 )}2
dP P,,,K2 P(K 1 + K 2) P2 (oK 1 + #K 2)2
a+,B(aK 1 + iK 2 )
For c=f, i.e. 'K' =K 1 K 2
2P2 + P(K 1 +K 2)
P(K 1 + K 2) + 2K1 K 2
Numerator of (dF) = 4P2 (K1 + K 2 ) + P{(K1 + K 2)2 + 8K1 K 2} + 2K 1 K 2(K1 + K 2 )(dP)
-2P 2 (K1 + K 2 ) -P(K 1 + K 2) 2
= 2P 2 (K1 + K2 ) + 8PKlK2 + 2K l K 2 (Kl + K 2 )
(dF) 4 K 1 K 2 (K1 + K2 ) + 2(K 1K2 )3 /2 1
(dP)P=¥KlK2 L[V(K 1K 2){K 1 +K 2 +22(K1 K 2)}12
4
{K1 +K 2 + 2(K 1 K2 )}
ReferencesALPER, T., 1959, Int. J. Radiat. Biol., 1,414; 1961, Ibid., 3,369; 1963, Phys. med. Biol., 8,365;
1976, Int. J. Radiat. Biol., 30, 389; 1979, Cellular Radiobiology (Cambridge: CambridgeUniversity Press); 1983, Int. J. Radiat. Biol., 44, 313.
ALPER, T., and GILLIES, N. E., 1958, Nature, Lond., 181, 961.ALPER, T., and HOWARD-FLANDERS, P., 1956, Nature, Lond., 178, 978.ALPER, T., MOORE, J. L., and SMITH, P., 1967, Radiat. Res., 32, 780.BENSON, D. M., KNOPP, J. A., and LONGMUIR, . S., 1980, Biochim. Biophys. Acta, 591,187.BRYANT, P. E., 1973, Int. J. Radiat. Biol., 23, 217.CHAPMAN, J. D., DUGLE, D. L., REUVERS, A. P., MEEKER, A. P., and BORSA, J., 1974, Int. J.
Radiat. Biol., 26, 383.CULLEN, B. M., 1976, Ph.D. Thesis, University of London.CULLEN, B. M., and LANSLEY, I., 1974, Int. J. Radiat. Biol., 26, 579.
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Mel
bour
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/10/
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rson
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se o
nly.
Adding two components 585
CULLEN, B. M., MICHALOWSKI, A., WALKER, H. C., and R/vtsz, L., 1980, Int. J. Radiat. Biol.,38, 525.
EVANS, N. T. S., 1969, Radiat. Effects, 1, 19.EWING, D., and POWERS, E. L., 1979, Radiation Biology in Cancer Research, edited by R. E.
Meyn and H. R. Withers (New York: Raven Press), p. 143.'IELDEN, E. M., SAPORA, O., and LOVEROCK, P. S., 1978, Int. J. Radiat. Biol., 33, 41.GILLIES, N. E., OBIOHA, F. I., and RATNAJOTHI, N. H., 1979, Int. J. Radiat. Biol., 36, 587.HENDRY, J. H., and SUTTON, M. L., 1984, Br. J. Radiol., 57, 507.HOWARD-FLANDERS, P., 1958, Adv. Biol. Med. Phys., 6, 553.LING, C. C., MICHAELS, H. B., GERWECK, L. E., EPP, E. R., and PETERSON, E. C., 1981,
Radiat. Res., 86, 325.MICHAEL, B. D., and HARROP, H. A., 1979, Radiation Sensitizers, edited by L. W. Brady (New
York: Masson), p. 14.MICHAEL, B. D., HARROP, H. A. MAUGHAN, R. L., and PATEL, K. B., 1978, Br. J. Cancer, 37,
Suppl. III, 29.MILLAR, B. C., FIELDEN, E. M., and STEELE, I. J., 1979, Int. J. Radiat. Biol., 36, 177.OBIOHA, F. I., GILLIES, N. E., CULLEN, B. M., WALKER, H. C., and ALPER, T., 1984, Int. J.
Radiat. Biol., 45, 427.POWERS, E. L., 1982, Int. J. Radiat. Biol., 42, 629.POWERS, E. L., WEBB, R. B., and EHRET, C. F., 1960, Radiat. Res., Suppl. 2, 94.SHENDY, M. A., ASQUITH, J. C., ADAMS, G. E., MICHAEL, B. D., and WATTS, M. E., 1975,
Radiat. Res., 62, 498.TALLENTIRE, A., JONES, A. B., and JACOBS, G. P., 1972, Israel J. Chem., 10, 1185.SILVESTER, J. A., STEERE, E., and BEWLEY, D. K., 1982, Int. J. Radiat. Biol., 41, 449.WATSON, E. R., HALNAN, K. E., DISCHE, S., SAUNDERS, M. I., CADE, I. S., MCEWEN, J. B.,
WIERNIK, G., PERRINS, D. J. D., and SUTHERLAND, I., 1978, Br. J. Radiol., 51, 879.
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