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Systematic evaluation of cellular radiosensitivity parameters

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1976 Phys. Med. Biol. 21 491 (http://iopscience.iop.org/0031-9155/21/4/001) View the table of contents for this issue, or go to the journal homepage for more

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PHYS. MED. BIOL.,

1976,

VOL.

21,

NO.

4,491-503.

@

1976

Systematic Evaluation of Cellular Radiosensitivity Parameters R. A. ROTH, S. C. SHARMAf and ROBERT KATZ$ University of Nebraska, Lincoln, NE 68508, U.S.A.

Received 2 September 1975, in final form 27 November 1975 Cellular radiosensitivity parameters of the track structure theoryof Katz and co-workers are evaluatedfrom a sum of squares minimizing computer program for nonlinear models. Based on these observations, suggestions are presented for efficient experiment design for the determination of these parameters from track-segment bombardments of high LET radiations. ABSTRACT.

1. Introduction

In theextension of the &ray theoryof track structurefrom nuclear emulsions and other l-hit detectors to biological cells, a model was achieved in which a used to set of fourexperimentallyevaluatedradiosensitivityparametersis calculate cellular survivalasafunction of absorbed dose in any radiation environment whose particle-energydistribution (quality)is known (Katz, Ackerson, Homayoonfar and Sharma 1971, Katz and Sharma 1973). Two of the radiosensitivity parameters for a particular cell line in a particular ambience are obtained from the measured survival curve after y-irradiation, and are the extrapolation number of the y-ray survival curve, m (nominally interpreted as being associated with the multiplicity of subcellular sensitive D-37 dose, E, (interpretedasrelatedtothe sites),andtheextrapolated radiosensitivity of the sensitive sites when exposed to secondary and higher U,, and K , generationelectrons).The other tworadiosensitivityparameters, are found from survival measurements after track-segment irradiations with high LET radiations. The first of these is the plateau (or saturation) value of the inactivation cross-section and nominally represents the cross-sectional area of the cell nucleus, while the second is obtained from the array of available high LET bombardments, for the maximal value of the RBE is expected to occur in the neighbourhood of z2/Kf12 = 2, while the plateau value of the cross4, when the cell cultureis section is achieved when thisproductisabout bombarded with beams of particles of effective charge number z at relative speed fl. The quantity K is understood to reflect the linear dimension of the sensitive sites inthe cell, just as z2/f12 reflects the stopping power of the incident ions. Then za/Kf12 reflects the energy deposited in a sensitive site. As in the case of nuclear emulsions we distinguish between the grain-count regime, where inactivations occur along a particle’s path like beads randomly

t Present address: University of Texas System, Cancer Centor, Texas Medical Center, Houston, Texas 77025, U.S.A. $ Address for correspondence. 18

492

R. A . Roth, S. C. Sharma and Robert Katz

distributed on a string, and the track-width regime, where the inactivations are distributed like a ‘hairy rope’ centred on the ion’s path, The transition from the grain-count to the track-width regime takes place in the neighbourhood of z2/Kp2 = 4;a t lower values we are in the grain-countregime, a t higher values in the track-width regime. I n order to accommodate the capacity ofcells to accumulatesub-lethal damage, two modes of inactivation are identified, called ‘ion-kill’ and ‘gammakill’. Cells inactivated by the passage of a single heavy ion aresaid to be inactivated in theion-kill mode, with inactivation cross-section U, whose value is less than U, in the grain-count regime, and greaterthan U, in the track-width regime. The fraction of ‘track-segments’ inactivating cells in the ion-kill mode in the grain-count regime is said to be equal to u/uo, which is taken to be equal to P, the probability for inactivation in the ion-kill mode, and also taken to equal the fraction of the dose deposited inthe ion-kill mode.Cells not inactiva’ted in the ion-kill mode are damaged by the &rays from the passing particle, and can then be inactivated by %rays from other passing ions in the gamma-kill mode, so called because it resembles the mechanism for inactivationby secondary electrons from y-ray photons. In the grain-count regime, the surviving fraction of a cellular population, m, E,, U and , K , aftertrack-segment whose radiosensitivityparametersare irradiation with a dose D of a fluence F of particles of atomic number 2, effective charge number z , relative speed p, and stopping power L (LET,) is found from the expressions

N/N, = ion-kill mode gamma-kill mode and

ni x n,,

(1)

n, = exp ( - uF), II,= l - { l - e x p [ ( l - P ) D / E O ] } ~ P

= u/uo = [l - exp

( -Z ~ / K ~ ? ~ ) ] ~ .

(2)

(3)

(4)

In thetrack-width regime, where P > 0.98, we take

n 7 =1

(5)

and find U from the ‘track width’ (fig. 2 of Katz and Sharma 1973). As in emulsion, the track width increases linearlywith zip, andtheinactivation cross-section with z 2 / P 2 , up to a limit set by the maximal radial penetration of %rays. I n all cases D=FL (6) and z = Z[1 -exp ( - 125pZ-f)l. (7) From survival data obtained with bombardments a t a range of LET values we haveextracted cellular radiosensitivityparameters by visualmethods. 1. These are The values of the ‘best fitting parameters’ are shown in table adjusted from an initial set of parameters found (m and E,) from the survival curve after X-rays (where P = O ) , (U,) from the survival curve after a high LET bombardment (where P E 1, say 2 = 9 a t 10 MeV/amu), and ( K ) from the

of Cellular RadiosensitivityParameters

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survival curve indicating a maximal RBEextrapolated (whereP2:0.5). We are reminded that in the grain-count regime

The facility with which visually fitted parameters may be assigned depends on the quality of the data. The most serious problems arise when there are no X-ray data from which to make independent assignments of m and E,, and no sufficiently high LET data, for then the estimates of U,, K and m interact and cannot be considered independently, as seen in eqn (4). To assign parameters on a quantitative basis, with an evaluation of confidence limits, we haveappliedasum of squares minimizing computer program to the system of equations and the data. The program (J.P. Ochsner and J. M. Eakman, private communication)makes use of a modified Marquardt method for fitting parameters to a nonlinear model, and here incorporates both grain-count and track-widthregimes. 2. Curve fitting

Fitting parameters to nonlinear models by least squares is based on the assumption of regions of local linearity in the nonlinearrelationship. It is assumed that there is a functional relationship between a dependent variable, one or more independentvariables, andtheparameters.The independent variables Xi are assumed to be accuratelymeasurable, andthedependent variable Y is expressed as (Marquardt 1963) y =f(X,,X,,

...,x t ; Bl,B,, ...,Bk)

(9)

where the B, are the parameters to be estimated. We consider the experimental data to be a sample taken from an infinitely large population of possible data. Characteristics of the sample are used to estimate those of the population, so that theparameters determined from the sample (b,) are differentiated from the true values which characterize the population (Bi). Corresponding to each experimental point (K,Xli, XZi, ...,X,%) a theoretical value is calculated, and the sum of squares of their differences is found as (D =

5(y,-p,)2

i=l

where n is the number of experimental points. For the linear case Q, is minimized by taking the partial derivatives of eqn (10) with respect to each parameter, and setting them equal to zero. In thenonlinear case the minimum, or least squares, is obtainedby iteration. The method of Marquardt is a modified steepest descent method, in which the neighbourhood of the minimum is reached by a method of steepest descent, while the best convergence in this region is found by application of a Taylor series expansion to linearize the model in the neighbourhood of a trial set of

SystematicEvaluation of Cellular RadiosensitivityParameters

495

parameters, so as to find corrections tothetrialparameters for thenext iteration (Marquardt 1959, 1963, 1966). In thepresent work we take thefunctional relationship of interest tobe log (NINO) = f(z,

B, D ; m, Eo, K , 0 0 )

(11)

and seek a minimum value in

All data points are assigned equal weights. When supplied with the functional relations of the model, a set of experimental data, anda set of trial values of the parameters, the computer program yields a set of optimal parameters and an estimate of 95% confidence limits. 3. Results Theoptimalparametersobtainedbycomputerleastsquares procedures are shown in table 2, for the cell lines listed in table 1, with the exception 1961) and buddingyeast (Raju, of Artemia eggs (EasterandHutchinson Gnanapurani,Stackler,Madhvanath,Howard,Lyman, Manney and Tobias 1972) for which the computer program did not converge. Since the parameters for mouse bone marrow(Broerse,Engels, LeLieveld, vanPutten,Duncan, Greene, Massey, Gilbert, Hendry and Howard 1971) and anoxically irradiated HeLa cells (Nias, Greene, Fox and Thomas 1967) were determined from X-ray and 14 MeV neutron data, these cases were not considered in this study. I n each case several trials were made for each set of data, using as initial parameters the visually fitted set, or sets departing from the visual parameters by 10 or 20%, in various combinations. The parameters in roman type converged to identical values whatever the starting set of parameters. The parameters in italic converged to somewhat different values, depending on the initial set of parameters. The values listed in the table are derived using visually fitted parametersas the starting point. I n most cases, the other optimal parameters were within 40% of those in table 2. I n most cases which consistently converged to thesame set of parameters the deviation from the visually fitted values is remarkably small. I n table 2 we list the best values of the computer-fitted parameters, the 95% confidence interval expressed as apercentagedeviation, the percentage deviation from the visuallyfittedparameters, the number of experimental ~ , its (RMS) uncertainty. For points, and the value of the product E O o O /and illustrativepurposes,in fig. 1 thedata for anoxicallyirradiatedbacterial spores(Powers,Lyman and Tobias 1968) are compared tothe theoretical curves calculated from the optimal least squares parameters (heavy lines) and from the individual 95% confidence limits on each parameter. The thin lines are computed by fixing three parameters to their optimal values (e.g. K O , and E:) and varying the fourth parameter to the two values determined by the

R. A . Roth, S . C . Sharma and Robert Katz

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confidence interval (e.g. r n o - or mot). For the eight resulting cases (denoted as a, a', b, b', etc.), the value of the varied parameter is listed at the top of the graph. For clarity,only the outermost curve is shown for each bombardment. Where the data permit, attempts are made to determine optimal parameters from subsets of the data. F

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Fig. 1 . Calculated survivals of bacterial spores anoxically irradiat'ed with heavy ions are compared to experimental data (Powers, private communication). The heavy lines are calculated from the optimal parameters (denotedmO, asK O , U: and E:) and having values listed below, from the computer least squares fit. The thinlines are calculated by fixing three parameters at their optimal values (e.g. K O , U: and E:) and assigning the fourth parameter according to the95% confidence limits (e.g. mO-or mo+).These eight sets of parameters are denoted ascase a,a', b, b', etc., and the corresponding value of the varied parameter is listed below for each case. Only the outermost curves are shown for clarity. m0 = 3.98 U; = 2.01 x (cm2) K O = 901 E: = 5.86 x IO8 (erg cm-3) a: mO- = 3.57 b : K O - = 819 a': r n o + = 4.39 b': K O + = 983 C: 0:- = 1.93 X 10-9 d : E:- = 5.52 X 106 C': U:+ = 2.09 X d ' : E$+= 6.19 X lo6

I n table 3 we explore the result of excluding data in different survival decades. Thus we exclude all data in the first decade of survival, or all data in the first and fourth (and lower) decades of survival, or retain only data in the second or in the third decade of survival. A principal result is that data in the first survival decade do not significantly affect the evaluation of parameters. Thismighthave been anticipated,inthatsurvival curvesdisplayasmall initial shoulder.Thecontributions to 0 in the shoulder region arerather insensitive to variationin the parameters. I n many cases the exclusion of experimental points in the first survival decade eliminates half the data, with little effect on the values of the parameters or the values of the confidence limits.Themostvaluable data points seem to bein the second and third decades of survival.

R.A . Roth, S. C . Sharrna and Robert Katz

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Table 4 displays the optimal parameter sets obtained by excluding data by bombardment and/or dose groupings. For these exclusions, clusters of experimental points a t approximately the same dose comprise one dose group. I n section Aall dataare excluded except one dose groupperbombardment, yielding survivalsin the second to third decades. With this elimination of about three quarters of the original data, thecomputer finds parameters within 20% of those found using all the data. Yet another grouping of excluded data, as suggested by the visual fitting procedures, is shown in section B of table 4. For the case of bacterial spores (Powers et al. 1968) we have retained only the data for Z = 1, 5 or 6 and 9, while for Chinese hamster cells (Skarsgard,Kihlman,Parker,Pujaraand for Z = 6 and 10. Richardson 1967) we have retained the X-ray data, and data The results are similar to those in section A. Finally we combine these exclusions. We use only two dose groups (yielding survivals in the second and third decades) per bombardment, and only three bombardments, as above, and with about an 85% reduction in the number 20% of the of experimentalpoints emerge withparametersagainwithin parameters found from all of the data, as shown in table 4, section C. 4. Conclusions

This statistical study has demonstrated a remarkable consistency of theory with experiment. The procedures examined here will serve a dual function, as an alternate method of parameter evaluation, and as tool for the design of efficient and internally consistent experiments. Our procedures require that we make a visual fit of the data as input to thecomputer program. Although the visual fit is somewhat subjective, we would not like to place complete reliance on the computer program. The combination of visual and computer fits to the data avoids the problems associated with sole reliance on either method, and hasthe advantageof a check system on parameter determination. I n most cases visual parameters are very consistent with computer fits. An exception lies in the case of Chinese hamster cells, where a re-evaluation of m from 2.5 to 3 may be appropriate,with an associatedalterationin IC. The results of the exclusion of different decades of data in t'his case imply that the visual fits leaned heavily on agreement in the second decade of survival. In fig. 2 the experimental data for hamster cells (Skarsgard et al. 1967) are shown compared to the theoretical lines for calculations from the visually fitted and computer-determinedparametersets.A similar caseis Chang liver cells (Todd 1975), shown in fig. 3. In each case the least squares parameters yield the better overall fit, although the differences are small. In the dataof Todd (1967) for T-1 human kidney cells anoxically irradiated, the optimal parameters are also inconsistent with the visually fitted parameters. However, the exclusion of first decade data indicates t'hat the visual fit is probably a better choice. The observed effect on parameter evaluation from the exclusion of dat'a by bombardment and by decades of survival suggests the development of a protocol for efficient experimentation to minimize beam time. A survival curve after

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502

R. A . Roth, S . C . Sharma and Robert Katz

X-irradiation must be made, to yield an initial estimate of m and E,. Measurement of the diameter of the cell nucleus should yield an initial estimate of U,. An initial value of K may be estimated from the empirical relationship for E , c ~ ~ / Ktables , 1 and 2. These approximations may be used to estimate the dose required to obtain surviving fractions in the second or third decade of bombardments at RBE maximum ( PN 0.5) and near the plateau value of the inactivation cross-section ( PN 0.95). The resulting data may be used interactively with the minimization program to determine the iterations needed to determine radiosensitivity parameters to the desired precision, and to expose questionable results. It is perhaps worthy of note that the sets of data having the largest number of experimental points (bacterial spores and Chinese hamster cells) offer the narrowest confidence limitsontheparameters. Inboth these cases the extrapolationnumbers seem remarkably close to integers,resurrectinga question long since laid to rest in target theory-Is the extrapolation number the number of cellular targets which must be hit in order to inactivate the cell ? We thank Bruce Fullerton and Christine Landkamer for their help in the course of these investigations. We also thank J. P. Ochsner and J. M. Eakman for their computer program. This work was supported by US ERDA and the NSF(RANN).

R~uME Evaluation systbmatique des parametres de radiosensibilit6 cellulaire Les parametres de radiosensibilit6 cellulaire de la thborie des structures de trajets de Katz et ses collaborateurs sont Bvalubs d’une somme de programmes d’ordinateur de minimisation des carres pour des modeles non-lin6aires. E n se basant sur ces observations, on pr6sente des suggestions pour 1’6tude d’exp6rimentations efficaces pour determiner ces parametres a partir de bombardements segmentaires de trajets de forts rayonnements LET.

ZUSAMMENFASSUNG Systematische Wertung von Zellular-Radiosensitivitats-Parametern

Zellular-Radiosensitivitats-Parameter der Spurstruktur-Theorie von Katz et al. werden aufgrund einer Summe von Quadraten gewertet, die das Computerprogramm fur nichtlineare Modelle auf ein Minimum beschranken. Von diesen Beobachtungenausgehend werden Vorschlage fur wirkungsvolle Versuchskonzeption zur Bestimmung dieser Parameter aus Teilspur-Bombardierungen durch hohe LET-Strahlung unterbreitet.

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503

REFERENCES BARENDSEN, G. W., KOOT,C. J., VAN KERSEN,G. R., BEWLEY,D. K., FIELD, S. B., and PARNELL, C. J., 1966, Int. J . Radiat. Biol., 10, 317. BERRY, R. J.,1970, Radiat. Res., 44, 237. BROERSE, J. J., ENGELS, A. C., LELIEVELD, P., VAK PUTTEN, L. M., DUNCAN, W., GREESE, D., MASSEY, J. B., GILBERT,C. W., HENDRY, J. H., and HOWARD, A., 1971, Int. J . Radiat. Biol., 19, 101. DEERING, R. A., and RICE, R., JR.,1962, Radiat. Res., 17, 774. E a S T E R , S. S., JR.,and HUTCHINSON, F., 1961, Radiat. Res., 15, 333. KATZ, R., ACKERSON, B.,HOMAYOONFAR,M., a ndSHm~A,S. C., 1971, Radiat. Res.,47,402. KATZ,R., and KOBETICH, E. J., 1969, Phys. Rev., 186, 344. KATZ,R.,and SHARMA, S. C., 1973, Nucl. Instrum. Meth., 111, 93. MARQUARDT, D. W., 1959, Chem. Eng. Progr., 55 (6), 65. MARQUARDT,D. W7.,1963, J . Soc. Indust. AppZ. Math., 11, (2), 431. MARQUARDT, D. W., 1966, IBM Share Library, No. 309401. MORTIMER,R.?BRUSTAD, T., and CORMACK, D. V., 1965, Radiat. Res., 26, 465. MUKSOX,R.J ., XEARY,G. J., BRIDGES, B. A., and PRESTON, R. J., 1967, Int. J . Radiat. Biol., 13, 205. XIAS,A. H. W., GREENE,D., Fox, M., and THOMAS, R. L., 1967, Int. J . Radiat. Biol., 13, 449. POWERS,E. L., LYMAN, J. T., and TOBIAS,C. A., 1968, Int. J . Radiat. Biol., 14, 313. RAJC,M. R., GNASAPURANI,M., STACPLER, B., MADHVANATH, U., HOWARD, J., LYMAN, J. T., MANNEY, T. R., and TOBIAS,C. A., 1972, Radiat. Res., 51, 310. M., STACKLER, B., MARTINS, B. T., MADHVANATH, U., RAJC, M. R., GSABAPURAKI, HOWARD, J., LYMAN, J. T., and MORTIMER, R. K., 1971, Radiat. Res., 47, 635. SAYEG, J. A., BIRGE,A. C., BEAM,C. A., and TOBIAS,C. A., 1959, Radiat. Res., 10, 449. SKARSGARD, L. D., KIHLMAN, B. A., PARKER,I., PUJARA, C. M., and RICHARDSON, S., 1967, Radiat. Res. Suppl., 7, 208. TODD, P,, 1964, Ph.D. Thesis, University of California, Berkeley, California 94720. TODD, P.,1967, Radiat. Res. Suppl., 7, 196. TODD, P.,1975, Radiat. Res., 61, 288. YANG,C., CRAISE, L., WELCH, G. P., and TOBIAS, C. A., 1973, Radiat. Res., 55, 523; and in Donner Laboratory Report, LBL-2016, Lawrence Berkeley Laboratory, Berkeley, CA 94720.