flexure dose: the low-dose limit of effective fractionation

$: Flexure dose: The low-dose limit of effective fractionation$
In, J Rodmron Oncology Eiol. Phys Vol 9. pp. 1373-1383 Printed m the U S A. All rights reserved.

036(r3Ol6/83/09l373-1l$03.00/0 Copyright D 1983 Pergamon Press Ltd

0 Oncology Intelligence

FLEXURE DOSE: THE LOW-DOSE LIMIT OF EFFECTIVE FRACTIONATION

SUSAN L. TUCKER, PH.D. AND HOWARD D. THAMES JR., PH.D. Department of Biomathematics, The University ofTexas M.D. Anderson Hospital and Tumor Institute at Houston,

6723 Bertner, Houston, TX 77030

Total radiation dose often can be increased without subsequent increases in the severity of tissue injury by using reduced doses per fraction. The flexure dose, d,, is defined as the largest fractional dose for which further fractionation produces no significant change in the total dose required to reach a specified effect level. Thus, d, is clinically relevant in that it represents the limit of effective dose fractionation. For those tissues in which injury reflects depletion of a critical proportion of target cells, the flexure dose is a measure of the extent of the initial, nearly linear portion of the dose-survival curve. More generally, the flexure dose is a measure of the extent of the initial, nearly linear portion of a dose-response curve in organized tissue, whatever its relationship to clonogenic target cells might be. Several quantitative expressions for d, are derived. The characteristic common to these is that each defines the Rexure dose as a multiple of the ratio a/B of the parameters of the linear-quadratic model of cell survival or dose response, where the multiple is a measure of experimental or statistical resolution. These multiples tend to fall within a limited range, thereby defining the “region of flexure” via the inequality 0.05 ((Y/B) 5 d, 5 0.15 (a//3). Estimates of the region of flexure are presented for a variety of normal and neoplastic tissues.

Dose fractionation, Dose-response curves, Radiation isoeffects, Radiotherapy, Acute and late effects.

INTRODUCTION

In vivo colony assays of survival of proliferating popula- tions of cells have confirmed, for a few normal tissues, that certain radiation endpoints are consistent with depletion of a critical fraction of target cells. This is true, for example, of bone marrow lethality24 and gut lethality” in mice. In keeping with this idea, it is frequently observed, in tissues for which a colony survival assay is lacking and under conditions that minimize proliferation, that the total dose of radiation required for a specified level of injury is increased when the dose is given in multiple small doses with intervals sufficient to allow complete repair of sublethal injury. Further increases in the isoeffective total dose can be achieved by decreasing the size of each fractional dose until the initial, nearly linear region of the survival curve of putative target cells for the tissue injury (or response curve of the tissue) is reached. At this point, the increases i,n total dose required for the isoeffect become negligible and cannot be resolved experimental-

ly. It would be valuable to the radiotherapist to know the

size of the fractional dose at which this occurs, i.e., the limit of effective fractionation, for various normal tissue responses. This idea was first introduced by Dutreix et a1.6 and subsequently elaborated on by Withers,37 who labeled this dose per fraction the “flexure dose,” d,. Its impor-

tance can be appreciated in several clinical situations. For example, the dissociation between acute and late radiation effects after altered dose fractionation can be explained by a smaller flexure dose for late-effects tissues than for acutely responding tissues3’ Furthermore, the use of several treatments per day to improve the therapeutic ratio (hyperfractionation), whereby the size of dose per fraction is significantly reduced, ideally should be guided by estimates of the flexure dose for the dose- limiting normal tissues in the treatment field.”

The purpose of this paper is to introduce a number of possible definitions of the flexure dose and to apply them to published data to calculate a range of flexure doses for a number of therapeutically relevant tissues.

METHODS AND MATERIALS If the severity of tissue response is related to the

fraction of survivors in some target-cell population and conditions are such that proliferation is negligible, then the dependence of total isoeffect dose on the size of dose per fraction is determined by the cell-survival relationship or dose response (Fig. 1). Assuming that each successive dose reduces survival by the same proportion, the total dose increases toward a maximum as the dose per fraction is decreased. If the initial portion of the curve were linear (in semilogarithmic coordinates) for doses up to some dose d, (the “flexure dose”), the maximum total dose

Supported in part by Grants CA-29026 and CA- 11430 from Reprint request to: Dr. S. L. Tucker. the National Cancer Institute, National Institutes of Health. Accepted for publication 19 April 1983.

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1374 Radiation Oncology 0 Biology 0 Physics September 1983, Volume 9, Number 9

Dose

Fig. 1. Influence of dose per fraction on total dose required for an isoeffect (adapted from Withers”). Assuming that equal fractional doses produce equal decrements in tissue function (on a logarithmic scale), the total dose needed to produce the injury increases to a maximum dose D, with diminishing dose per fraction. Since IO doses of size d/and 20 doses of size de produce the same effect, the responses to each of the two doses d, and d, must lie on the initial linear portion of the curve, and D, = IO x d, = 20 x d,. The hypothetical response curve shown above has an initial linear region. If, instead, the curve above were continuously downward-bending, the total dose 20 x d, would be displaced slightly to the right of IO x dy

would be attained whenever the dose per fraction did not exceed d,. However, neither the two-component (TC)’ nor the linear-quadratic (LQ)19 model of cell survival predicts a purely exponential initial region; therefore, according to these models, there is no precisely determined “flexure dose.” Withers3’ used the term “region of flexure” to denote the range of doses in which the survival curve begins to deviate significantly from linearity, and observed that a value can be assigned to d/ by determining the largest dose per fraction that can be further fractionated without requiring a (significant) increase in total dose for an isoeffect. Clearly, any definition of the flexure dose must make precise the sense in which differences in dose or response will be considered “insignificant.”

Although the points made above for cell-survival curves are equally valid for more general dose-response curves, we shall for conciseness use the cell-survival terminology throughout. This does not imply any particular form of the relationship between survival of critical target cells and the proportion of functional cells which contribute to an observed effect on the tissue in situ. A quantitative expression for the flexure dose will be derived from each of several possible definitions. These derivations are based on the following assumptions: (1) that the dose-response relationship has a linear-quadratic form, i.e. In (response) = -((Yx + /3x’); and (2) that equal decrements in log response occur after equal fractional doses.

Estimation of C# Since the ratio (Y/P of the parameters of the LQ model

is essential to each of the derived expressions for the

flexure dose, we include a review of a well-known method for estimating this quantity from isoeffect data. This is the so-called Fe-plot proposed by Douglas and Fowler,4 wherein the reciprocal total dose required for an isoeffect is plotted against dose per fraction.

Let D, = EDSO denote the total radiation dose required in n equal fractions to produce a specified level of injury (an isoeffect) in 50% of the individuals from some population. Let -E be the natural logarithm of the fraction of surviving target cells corresponding to this degree of injury. (E could equally well represent a given level of effect that can be measured as the response of an organized tissue.) It follows easily from the LQ model that there is a linear relationship between the dose per fraction x, = D,/n and the reciprocal of the total isoeffect dose:

I a a X.

D, E+E”

Estimates of the intercept (Y/E and the slope ,6/E can be obtained by linear regression of 1 /D, against x, (Fig. 2). Since, in general, the value of E is not known, the parameters (Y and p cannot be determined, but the ratio

a/P = (aIE)I(PIE) can be calculated independent of the chosen level of effect.

From Equation (I), the limiting value of l/D” as x, tends to zero (i.e. as the fraction number becomes infi- nite) is cx/E (Figure 2). Hence, any fractionated dose greater than D, = E/Q will produce a level of injury exceeding the specified isoeffect. Although the maximal isoeffect dose D, cannot be attained, in principle the total

8 -- r

BL ’ I I I I I I I I 1

0.0 02 0.4 0.6 0.8 1.0 12 1.4 1.0 1.8 20 Dose per Fraction krad)

Fig. 2. Plot of reciprocal total dose for an isoeffect vs. dose per fraction. Points are obtained from data of van der Kogel” and represent the total doses required to produce a specified level of damage following once-daily irradiation of the rat spinal cord. Numerals indicate the number of fractions used in each regimen. The ratio of intercept to slope of the regression line fit to these data (Equation (I)) gives an estimate of a/& where (Y and p are the LQ parameters of the effective dose-response curve of the target cell population. The reciprocal of the intercept is a measure of D,, the maximum dose possible without exceeding the specified level of injury.

Estimation of the flexure dose 0 S. L. TUCKER AND H. D. THAMES JR. 1375

dose could be made arbitrarily close to D, by using sufficiently many dose fractions. In practice, of course, the limiting value of the isoeffective total dose might differ from D, because of factors such as a proliferative response to the injury that would become manifest after sufficient protraction of time.

Expressions for thejexure dose 1. Limited resolution between fractional doses

If the dose-survival curve for some target cell population had an initial linear region given by ln(s.f.) = --ax, for doses x less than some value d, (the “flexure dose”), then the response to a fractional dose x, would lie in this region (i.e. x, I d,) provided that the fraction number n were large enough (Fig. 1). For two sufficiently large fraction numbers n and m, the fractional doses x, and x, (both less than or equal to d,) would satisfy the relationship m(cux,) = n(ax,), or x, = (n/m)x, (e.g., doubling the number of fractions would reduce the dose per fraction by a factor of 2). If, instead, the cell-survival curve is continuously downward-bending (as predicted by the LQ model), the fractional dose x, is always greater than (n/m)x, when m > n. However, the two expressions become more nearly equal as the fraction numbers n and m are increased. This concept is illustrated in Fig. 3, where the doses xs and (1/2)x4 differ less than do the doses x1 and ($)x,. Based on these ideas, the flexure dose could be defined as the largest fractional dose x, for which x, is approximately equal to (n/m)x, for specified values of m > n. The value of d, is determined by the criterion selected for considering doses to be essentially “equal” and by the requirements placed on the fraction numbers m > n for which the approximate equality of x, and

Dose

Observed isoeffect

Fig. 3. Resolution between fractional doses. The same biologic effect is observed when total doses 0. are given in n fractions of size x,, for n = 1, 2, 4, and 8. The dose x, = D, does not correspond to response on the approximately linear initial portion of the curve, since doubling the number of fractions to n = 2 corresponds to a fractional dose x2 which is markedly different from ($)x,. However, the dose (r/r)xq differs only slightly from xg, so xq does not exceed the flexure dose. Analogously, total doses D, and D, differ significantly, but doses D, and D, do not.

(n/m)x, should hold. Two possible definitions for the flexure dose are as follows:

(a) Let d, be the largest fractional dose x, for which (n/m)x, is within a prescribed percentage of x, for every m > n.

Using the LQ model, one can solve for x, from definition (a) to find that d, is a multiple of the ratio CY/@ of the LQ parameters. (Details of the calculations are presented in Appendix A 1.) When the level of resolution is taken to be 596, the multiple has the value 0.053:

d, = (0.053) ; .

(b) Let df be defined as above, with a less stringent condition on the fraction numbers: require that (n/m)x, be within a fixed percentage of x, for the commonly encountered case m = 2n.

Again, one finds (Appendix A 1) that d, is a multiple of (r/p. For a resolution level of 5%. the flexure dose is given by:

d/= (0.118);.

Although the two preceding definitions are given in terms of fractional doses, each is equivalent to requiring that the total dose D, = nx, be within a specified percentage of D, = mx, for certain m > n (Fig. 3).

2. Sacrificed total dose As the dose per fraction decreases, the total dose

required to reach a given biologic effect approaches a limiting value D, (cf. Equation (1) and Fig. 2). The flexure dose could be defined by specifying what percentage the isoeffective total dose should be of the limiting value (Fig. 4). Again, it follows that the flexure dose is a multiple of the ratio CX/P (Appendix A2). When the specified resolution between D, and D, is 596, the flexure dose is given by:

d, = (0.05) ;

Alternatively, one could require that the isoeffective dose D differ from D, by no more than a fixed amount, AD. In this case, the multiple of cz/p defining d, is AD/(D_ - AD) (Appendix A2). This expression is equivalent to the previous one when the sacrificed dose AD is expressed as a percentage of the isoeffective dose D = D, - AD.

3. Statistical formulation In practice, ED50 values are estimated as follows. A

range of fractionated doses is administered to individuals from some population, and the fraction of responders is

1376 Radiation Oncology 0 Biology ??Physics September 1983, Volume 9, Number 9

Dose

Observed isoeffect

AD

Fig. 4. Sacrificed total dose. The total dose D, required to reach a specified biologic effect in n fractions tends monotonically to a maximum D, determined by the initial tangent to the dose-survival curve. The sacrificed total dose, equal to the difference AD = D, ~ D,, approaches zero as the number of dose fractions increases.

scored as an estimate of the probability of response at each dose. For larger values of dose per fraction, the number of fractions is held constant, so that a range of total doses is achieved by varying the size of dose per fraction, while for small fractional doses, the size of dose per fraction can be fixed and the number of fractions

4.0 4.5 5.0 5.5 6.0 6.5 Dose (krad)

7.0 7.5

Fig. 5. Data of van der Kogel showing the proportion of rats with hind leg paresis resulting from once-daily irradiation of the lumbosacral cord (adapted from Figure 6.3 of ref. 33). The slope s at p = 50% was estimated for each of the n-fraction curves above by approximating the increase in dose required to change the proportion of responders from 25 to 75%. The resulting estimates of s were used to obtain a regression estimate (Appen- dix A3 and Figure 8) of the quantity I /c appearing in expression (3) for the flexure dose.

varied. In either case, the graph of response versus total dose has a characteristically sigmoidal shape (c.f. Fig.

5). If the isoeffect assay were repeated, the resulting data

would differ from that obtained previously, and consequently the estimates of ED50 would differ by some amount. All other conditions being equal, the variation among replicate estimates should decrease as the number of doses and the number of animals irradiated at each dose is increased.

If D and D’ are theoretical values of ED50 and D’ corresponds to a smaller dose per fraction than does D, then D’ > D (Fig. 6). Because of statistical variation in the data, however, there is some probability of obtaining

Dose

Fig. 6. Theoretical curves representing the response to fractionated irradiation. Increasing the number of dose fractions and decreasing the size of dose per fraction causes the curves to be displaced to the right toward a limiting (dashed) curve. For small doses per fraction, these curves are less distinct from one another than for large doses per fraction, and it is more difficult to distinguish between ED50 values.

Estimation of the flexure dose ??S. L. TUCKER AND H. D. THAMES JR. 1377

estimates of D and D’ such that 3 5 b. The isoeffect doses D and D’ could be considered essentially equal if the probability of observing a 5 b is sufficiently high. This leads to the following definition of the flexure dose:

Let d, be the largest fractional dose for which the probability of obtaining ED50 estimates band i>l with fi 5 i, exceeds some prescribed valuep, where D is the value of ED50 corresponding to the dose per fraction d,, and where LY is any value of ED50 corresponding to a dose per fraction smaller than d, or to a number of fractons greater than D/d/

In Appendix A3, a quantitative expression is derived for this definition of the flexure dose. The formula is consistent with each of the previously derived expressions for the flexure dose in that d,is given as a multiple of the ratio

aI@:

d, = (constant) i.

The size of the constant is affected by several factors. The first of these is the probability p appearing in the definition of d,. Larger values of p correspond to smaller estimates of the flexure dose, since for smaller doses per fraction, the theoretical values of ED50 are closer to the limiting value D, and are therefore more difficult to resolve from one another.

Second, as the number of animals per dose group and the number of different dose groups is increased, quanta1 response data tend to exhibit less scatter, and correspond- ingly improved estimates of ED50 are obtained. Conse- quently, loss of resolution between values of ED50 occurs at higher fraction numbers and smaller values of d, are predicted.

Third, the size of the flexure dose is influenced by the degree of heterogeneity in the animal population with regard to the response under study. If the proportion of responders increases from 0% to 100% over a narrow dose range (relatively homogeneous population), replicate estimates of ED50 would show less variation than if the quanta1 response curve were shallow (heterogeneous population). Therefore, the point at which distinct ED50 values can no longer be resolved is influenced by the slopes of the underlying response curves, with shallower curves leading to loss of resolution at larger doses per fraction.

4. Deviation from linearity of the dose-surviving fraction curve.

Since changes in total dose resulting from dose fractionation reflect the degree of linearity in the initial portion of the curve, the flexure dose could be defined as that dose at which the dose survival curve begins to bend significantly downward (Fig. 7). For example, the “linear” region could be defined to include those doses for which log cell survival is within a certain percentage of the asymptotic linear response curve (the initial tangent).

Dose x

Fig. 7. Deviation of the survival curve from linearity. As the dose increases, the response in semilogarithmic coordinates (solid curve) diverges continuously from the asymptotic response to fractionated doses (dashed line) determined by the initial slope. The amount of divergence is expressed as a proportion of the linear response.

As shown in Appendix A4, this definition implies that d,is a multiple of cr/p where the multiple is equal to the chosen percentage, e.g., df = .(0.05) a/p or d, = (0.1) a/p.

5. Regression estimate of d, from cell survival data. If low-dose surviving fraction data are available, the

flexure dose could be defined as the largest dose df for which the quadratic coefficient p is insignificant (at some prescribed level) when the LQ model In(s.f.) = -(ax + @x2) is fitted to the data available for doses x 5 dp As expected, experiments with computer simulated data showed that the size of d, depends on the number of observations per dose, the spacing between the doses, and the variance in replicate observations at the same dose. It was found that under fixed experimental conditions (e.g., spacing of doses) and with the theoretical variance estimated from multifraction in vivo colony assays, the estimate of d, from regression was a monotone increasing function of the ratio a/@ of theLQ parameters.

RESULTS To illustrate the procedure by which the flexure dose d,

is computed from definition (3), the statistical formulation, we consider published isoeffect data of van der Kogel,33 for which the endpoint was hind leg paresis following once-daily irradiation of the lumbosacral cord of rats. The quantitative expression for d, derived from the statistical definition, as detailed in Appendix A3, is the following:

(3)

To compute d,, estimates of the parameters a/a, k, and c are needed.


Figure 5 shows the data from van der Kogel’s fractionation experiments, and Figure 2 represents the plot of reciprocal total dose vs. dose per fraction corresponding to these data. The regression estimates of the slope and intercept of the best-fitting straight line are P/E = 0.215 krad-’ and a/E = 0.09 krad-‘, so that CX/~ = 0.42 krad.

The parameter k appearing in Eq. (3) is a measure of the combined influence of two experimental parameters: the number nA of animals irradiated at each dose, and the number n, of distinct total doses delivered in each fractionation experiment, as described in Appendix A3. Esti- mates of k for different values of nA and n, are listed in Table 1 (see Appendix B). Assuming that the average number of doses used to obtain each quanta1 curve was n, = 3 (as indicated in Fig. 5), and that n,,, = 10 animals were irradiated per dose, the value of the parameter k for these data is approximately k = 0.175 (cf. Table 1).

0.0 I.0 10 9.0 4.0 5.0 0.0 7.0 0.0

4l The constant c in Equation (3) is a biologic parameter

related to the degree of heterogeneity in the animal population with respect to the specified response. As described in Appendix A3, linear regression can be used to estimate the quantity (1 /c): it is the slope of the line obtained by plotting the values (a/E) + 2(0/E) (D,/n) against the values s, for various fraction numbers n, where D, is the n-fraction ED50 for the chosen isoeffect, and where s, is the slope at ED50 of the n-fraction quanta1 response curve from which D, is estimated. Figure 8 shows the data used to obtain the least-squares estimate 1 /c = 0.152 for these data. The ordinate for each point was computed using the values a/E and ,6/E from the plot of Figure 2. For the abcissa, the slope s, of each quanta1 curve in Figure 5 was estimated by eye. Substi- tuting the values o/p = 0.42 krad, k = 0.175, and 1 /c = 0.152 in Equation (3) yields the estimate d, = 0.028 krad for the flexure dose of the rat spinal cord.

Fig. 8. Regression estimate of the parameter I/c for a tissue endpoint (c.f. Appendix A3). The quantity c, as described in the text, is a measure of the variation between individuals in reaching some level of injury as the fraction of surviving target cells decreases. An estimate of l/c is required for Equation (3) of the text, which defines the flexure dose in terms of statistical uncertainty in determining the total doses corresponding to an isoeffect. The points were obtained from multifraction isoeffect data of van der Kogel” (Figure 5). The values of a/E and P/E used to compute the ordinate were obtained from Figure 2. For the abcissa, the slope S, of each n-fraction quanta1 response curve of Figure 5 was estimated by eye.

flexure dose from each of the other definitions described in previous sections. This observation suggests that the range of doses d/defined by 0.05 (a/p) 5 d, 5 0.15 (a/P) might be considered the “region of flexure” in the sense of Withers3’ for any dose-response curve.

Other tissues for which data were available’2m’6,22.35 were analyzed in the same way to obtain estimates of the flexure doses of various normal-tissue responses. With one exception,‘4 the parameters k and c were such that the multiples of CX/~ defining the flexure dose in Equation (3) were found to lie in the interval (0.05, 0.15). This range includes the multiples of CY/~ required to calculate the

Estimates of the regions of flexure for a variety of tissue responses are listed in Table 2. In most cases, the a/@ values used to determine df were calculated from Equation (1). For a few,27,29*30,32.38 (Y and p were found directly by fitting the LQ model to cell survival data. In one instance,23 the parameters were determined from the dependence of ED50 on dose rate after exposure to continuous irradiation using a model of Roesch.2s

Table 1. Approximate dependence of the constant k on experimental parameters rrA and n,

nA (number of

animals/dose)

4 6 8

10 12 14 16 18 20

3 4 5

,210 .185 ,165 .I95 .I70 .145 .I85 .I50 .I 15 ,175 .I35 ,105 ,170 ,120 ,090 .I65 .I10 ,085 .160 .I00 .080 ,155 ,090 .075 ,150 ,085 .070

n, (number of doses)

6 7

,145 .135 .I25 ,115 .I05 ,095 .090 .085 .080 .075 .070 .065 ,065 .060 .060 ,055 ,060 ,055

8 9 IO

,125 ,115 .I IO ,100 .090 .085 .085 .075 ,070 ,075 ,065 ,060 ,065 ,060 ,055 ,060 .055 .055 .060 ,055 ,055 ,055 ,050 ,050 ,050 ,050 ,050

Early responses

Late responses

Tumors


Table 2. Estimated regions of flexure

Tissue (endpoint)* References d,(rad)

Bone marrow (LD50) Colon Esophagus (LD50) Jejunum Jejunum Skin (foot deformity) Skin (moist desquamation) Skin (moist desquamation) Spleen Testis Brain (LD50) Kidney (LD50) Kidney (renal function) Kidney (renal function) Lung (LD50) Lung (LD50) Skin (contraction) Spinal cord (hind leg paralysis) Spinal cord (hind leg paralysis) Spinal cord (progressive

radiculopathy) Spinal cord (white matter

necrosis) C3H murine mammary tumor

(ED50 for tumor recurrence) C3H mammary carcinoma

(ED50 for tumor recurrence under clamp hypoxia)

Murine fibrosarcoma Murine fibrosarcoma

Band 2 Band 4

McNeil ef al.” Tucker et al. 32 Hornsey and FieldI Thames et al. I0 Withers et al.38 Denekamp’ Fowler et al.’ Douglas and Fowler4 Wither? Thames and Withers29 Hornsey et a1.14 Caldwell’ Hopewell and Berry” Jordan et al. “.I6 Wara et al.‘4 Hornsey et al. ” Masuda et al.” Masuda et al.2’ White and Hornsey35 van der Kogell’

van der Kogel”

Fowler et al.9

Suit et al.16

Mason and Withers*’ Thames et al.”

45-134 43-l 28 40-l 20 66-198 43-128 54-162 477140 48-143 45-l 34 63-188 1 l-32 2-6 7-20 3-9

I l-32 6618

23-69 25574 2447 I 21-63

13-38

699207

1699507

64- I92

39881,194 148-443

*Where applicable.

DISCUSSION

The flexure dose, d,, of a tissue is a measure of the extent of the initial, approximately linear region of the dose-response curve. Alternatively, it is the largest fractional dose for which further reduction in the size of dose per fraction produces no significant increase in the total dose required to reach a specified level of tissue injury. Any definition of d, must state some criterion by which two isoeffective doses will be considered significantly different, or for determining the point at which the response curve begins to bend significantly downward.

Each of the definitions of the flexure proposed in this paper implies that d, is an increasing function of the ratio (Y/P of the LQ parameters of the relevant dose-response curve. Moreover, the estimates of d, tend to lie in a limited range described approximately by 0.05 ((Y/P) zs df 5 0.15 (a/p). This, then, might be considered the “region of flexure.” For a tissue with an (Y/P ratio of 1 krad, the region of flexure ran,ges from 50 to 150 rad. In this case, increased sparing would be expected for a fixed total dose if doses per fraction greater than 150 rad were further

fractionated, while no benefit would be expected from fractionation of doses less than 50 rad.

The simplest summary of the results presented in Table 2 might be: “the flexure dose is smallest for late-reacting normal tissues, and largest for tumors and acutely- responding normal tissues.” This follows from our finding that the flexure dose is a multiple of the ratio LY/~ of parameters of the linear-quadratic model, and from the fact that its inverse, @ICI, a measure of the sensitivity to change in size of dose per fraction, is largest for late- responding normal tissues3’

The potential significance to clinical radiotherapy of knowing the flexure dose is illustrated by considerations in the optimal design of hyperfractionated regimens.28.39 Hyperfractionation differs from conventional fractionation regimens in the use of small doses per fraction, given two or three times per day, to achieve an increase in total dose given in the same overall time as conventional therapy. It is the late-responding tissues, characterized by higher @/CI ratios,” and consequently smaller flexure doses (c.f. Table 2), that are usually considered dose- limiting. The total dose given in a hyperfractionated


regimen should be selected to produce no increase in severity of late injury (28, Appendix) to normal tissues in the treatment volume. Because of the probable similarity between the response of the tumor and that of the acutely responding normal tissues (characterized by smaller p/a ratios and larger values of d,, c.f. Table 2), however, the degree of cell killing in these would increase. Thus, a therapeutic gain would occur relative to late effects, at the cost of increased severity of acute effects.

estimate “(Y/P” from plots of isoeffect data, though now lacking the earlier precise interpretation, should still provide a reasonable description of the range of doses over which the dose-response curve begins to deviate from being linear.

The above considerations suggest that there are two quantities to be specified in the optimal design of hyperfractionated regimens. The first is the minimum flexure dose for dose-limiting, late-responding normal tissues in the treatment field; this is determined by the maximum P/(Y ratio for these tissues. The second is the total dose that may be given, with no increase in late injury, when the dose per fraction is equal to the minimum flexure dose (28, Appendix).

The LQ model was used to derive each of the expressions for d/ presented herein. It is possible that the dose-response characteristics of a cell population are better described by some other model, such as the TC model. In this case, the LQ model can be viewed as an approximation to the true curve over some dose range; it is not possible to distinguish between these models for cell-survival data in the low-dose region. Consequently, the values of the flexure doses calculated using the

One of the assumptions on which each of the expressions for df is based is that of equal effect per dose fraction during the course of a fractionated irradiation regimen. If this assumption is not valid, then the optimal size of each fractional dose might not equal the flexure dose. For example, if sufficient time is allowed between doses to allow repair of sublethal injury, proliferation is likely to occur in acutely responding tissues during the protracted time of treatment necessary for large numbers of fractions. In this case, larger total doses could be tolerated by the tissue than would be indicated by the response to small numbers of fractions. Conversely, there might be an accumulation of damage, so that the estimates of dYcould be larger than the optimal dose per fraction. Other factors, such as cell synchrony and reoxygenation, might also play a role. The hypothesis of equal effect per fraction is most likely to be valid for the nonproliferative late-responding normal tissues. Since these are usually the dose-limiting tissues in radiotherapy, the concept of the flexure dose is of greatest interest when applied to these tissues.

REFERENCES

I.

2.

3.

4.

5.

6.

7.

8.

9.

IO.

Bender, M.A., Gooch, P.C.: The kinetics of X-ray survival of mammalian cells in vitro. Int. J. Radiat. Biol. 5: l33- 145, 1962. Caldwell, W.L.: Time-dose factors in fatal post-irradiation nephritis. In Cell Survival After Low Doses of Radiation: Theoretical and Clinical Implications. Alper, T. (Ed.). London and New York, John Wiley and Sons, Ltd. 1975, pp. 328-332. Denekamp, J.: Early and late radiation reactions in mouse feet. Brit. J. Cancer 36: 322-329, 1977. Douglas, B.G., Fowler, J.F.: The effect of multiple small doses of X rays on skin reactions in the mouse and a basic interpretation. Radiat. Res. 66: 401-426, 1976. Draper, N.R., Smith, H.: Applied Regression Analysis. New York, John Wiley and Sons. 1966, p. I I. Dutreix, J., Wambersie, A., Bounik, C.: Cellular recovery in human skin reactions: Application to dose fraction number overall time relationship in radiotherapy. Eur. J. Cancer 9: 159-167, 1973. Finney, D.J.: Statistical Method in Biological Assay. Lon- don, Charles Griffin, I97 I. Fowler, J.F., Denekamp, J., Delapeyre, C., Harris, S.R., Sheldon, P.W.: Skin reactions in mice after multifraction X-irradiation. Int. J. Radiat. Biol. 25 (3): 213-223, 1974. Fowler, J.F., Denekamp, J., Sheldon, P.W., Smith, A.M., Begg, A.C., Harris, S.R., Page, A.L.: Optimum fractionation in X-ray treatment of C3H mouse mammary tumours. Brit. J. Radioi. 41: 78 l-789, 1974. Hendry, J.H., Polten, C.S., Roberts, N.D.: The gastrointes- tinal syndrome and mucosal clonogenic cells: Relationships between target cell sensitivities, LD50 and cell survival, and their modification by antibiotics. Radiat. Res. (In press) 1983.

I I.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Hopewell, J.W., Berry, R.J.: Radiation tolerance of the pig kidney: a model for determining overall time and fraction factors for preserving renal function. Int. J. Radiat. Oncof. Biol. Phys. 1: 61-68, 1976. Hornsey, S., Field, S.B.: The effects of single and fractionated doses of X-rays and neutrons on the oesophagus. Eur. J. Cancer 15: 491498, 1979. Hornsey, S., Kutsutani, Y., Field, S.B.: Damage to mouse lung with fractionated neutrons and X rays. Radiology 116: l7l-174,1975. Hornsey, S., Morris, C.C., Myers, R.: The relationship between fractionation and total dose for X ray induced brain damage. Int. J. Radiat. Oncol. Biol. Phys. 7: 393- 396, 1981. Jordan, SW., Yuhas, J.M., Butler, J.L.B., Kligerman, M.M.: Dependence of RBE on fraction size for negative pi-meson induced renal injury. Int. J. Radiat. Oncol. Biol. Phys. 7: 223-227, I98 I. Jordan, S.W., Yuhas, J.M., Key, C.R., Hogstrom, K.R., Butler, J.L.B., Kligerman, M.M.: Comparative late effects of X-rays and negative pi-mesons on the mouse kidney. Am. J. Pathol. 97: 3 15-326, 1979. Kendall, M.G., Stuart, A.: The Advanced Theory of Statis- tics Vol. I, 2nd edition. London, Charles Griffin. 1968, pp. 231-232. Larson, H.J.: Introduction to Probability Theory and Sta- tistical Inference, NY, John Wiley and Sons, Inc. 1969, p. 177. Lea, D.E.: Actions of Radiations on Living Cells, 2nd edition. Cambridge, Cambridge University Press. 1947, p. 256. Mason, K.A., Withers, H.R.: RBE of neutrons generated by 50 MeV deuterons on beryllium for control of artificial

21.

22.

23.

24.

25.

26.

27.

28.

29.

pulmonary metastases of a mouse fibrosarcoma. Brit. J. Radiol. 50: 652-651, 1977. Masuda, K., Hunter, N., Withers, H.R.: Late effect in mouse skin following single and multifractionated irradiation. Int. J. Radiat. Oncol. Biol. Phys. 6: 1539-l 544, 1980. Masuda, K., Reid, B.O., Withers, H.R.: Dose effect relationship for epilation and late effects on spinal cord in rats exposed to gamma rays. Radiology 122: 239-242, 1917. McNeil, J., Thames, H.D., Stewart, J.R., Peters, L.J.: Estimates of repair halftime and dose-survival parameters of bone-marrow stem cells from the dependence of LD50/ 30 on dose rate. (In preparation.) Puro, E.A.. Clark, G.M.: The effect of exposure rate on animal lethality and spleen colony cell survival. Radiat. Res. 52: I 15-I 29, F972. Roesch, W.C.: Models of the radiation sensitivity of mammalian cells. In Proceedings Third Symposium on Neutron Dosimetql in Biolagy and Medicine, G. Burger and H.G. Ebert (Eds.). Luxembourg, Commission of the European Communities, 1978, pp. l-27. Suit, H.D., Howes, A.E., Hunter, N.: Dependence of response of a C3H mammary carcinoma to fractionated irradiation on fractionation number and intertreatment interval. Radiat. Res. 12: 440-454, 1971. Thames, H.D., Grdina, D.J., Milas, L.: Response of fibrosarcoma cell subpopulations to small doses of radiation delivered in situ. Int. J. Radiat. Oncol. Biol. Phys. 9: 217-220, 1983. Thames, H.D., Peters, L.J., Withers, H.R., Fletcher, G.H.: Accelerated fractionation vs. hyperfractionation: rationales for several treatments per day. Int. J. Radiat. Oncol. Biol. Phys. 9: 127-l 38, 1983. Thames, H.D., Withers, H.R.: Test of equal effect per fraction and estimation of initial clonogen number in micro-


colony assays of survival after fractionated irradiation. Brit. J. Radiol. 53: 1071-1077, 1980.

30. Thames, H.D., Withers, H.R., Mason, K.A., Reid, B.O.: Dose-survival characteristics of mouse jejunal crypt cells. Int. J. Rddiat. Oncol. Biol. Phys. 7: 1591-1591, 1981.

31. Thames, H.D.. Withers, H.R., Peters, L.J., Fletcher, G.H.: Changes in early and late radiation responses with altered dose fractionation: Implications for dose-survival relationships. Int. J. Radiat. Oncol. Biol. Phys. 8: 219-226, 1982.

32. Tucker, S.L., Withers, H.R., Mason, K.A., Thames, H.D.: A dose-surviving fraction curve for mouse colonic mucosa. Eur. J. Cancer Clin. Oncol. 19: 433-437, 1983.

33. van der Kogel, A.J.: Late effects of radiation on the spinal cord: dose-effect relationships and pathogenesis. Thesis, University of Amsterdam, Radiobiological Institute TNO, Rijswijk, the Netherlands, 1979.

34. Wara, W.M., Phillips, T.L., Margolis, L.W., Smith, V.: Radiation pneumonitis: a new approach to the derivation of time-dose factors. Cancer 32: 547-552, 1973.

35. White, A., Hornsey, S.: Radiation damage to the rat spinal cord: the effects of single and fractionated doses of x ray. Brit. J. Radiol. 51: 515-523, 1978.

36. Withers, H.R.: lsoeffect curves for various proliferative tissues in experimental animals. In Proceedings ofconfer- ence on Time-Dose Relationships in Clinical Radiothera- py. Madison, Wisconsin, Madison Printing and Publishing Co. 1975, pp. 30-38.

31. Withers, H.R.: Responses of tissues to multiple small dose fractions. Radiat. Res. 71: 24-33, 1977.

38. Withers, H.R., Chu, A.M., Reid, B.O., Hussey, D.H.: Response of mouse jejunum to multifraction radiation. Inl. J. Radiat. Oncol. Biol. Phys. 1: 41-52, 1975.

39. Withers, H.R., Peters, L.J., Thames, H.D., Fletcher, G.H.: Hyperfractionation (an editorial). Int. J. Radiat. Oncol. Biol. Phys. 8: 1807-I 809, 1982.

APPENDIX A

Quantitative expressions for thejexure dose Al. Limited resolution between fractional doses.

The flexure dose df is defined to be the largest fractional dose x, for which (n/m)x, is within p x 100% of x, for certain fraction numbers m > n, i.e.,

xm (1 - P) 5 f (x,1.

Since n doses of size x, produce the same response as m doses of size x, for any choice of m and n, it follows that n(crx, + oxi) = m(a x, + fl xi). Solving this expression for x, and substituting into inequality (4) one obtains:

x, i- (mln)~ (1 -P) 1 1 ‘ye

(m/n) (1 - p)” - 1 P (5)

For a fixed value of n, Equation (5) holds for every m > n provided that:

I

X<PcY n I 1 1 -P P’

When p = 0.05, this is equivalent to x 5 (0.053) (a/p).

If Equation (5) is required to hold only for m 5 2n, then

x*S 1 2P(l -P> f.x 2(1 _p)‘_ 1 1 8’

For p = 0.05, this implies that x, 5 (0.118) ((Y/P).

A2. Sacrificed total dose.

The limiting dose D, is within p x 100% of the isoeffective dose D when the dose per fraction x satisfies:

E E a((1 +p) ___

1 1 (Y + px

(c.f. Equation (1) of the text) or, equivalently,

X<(P)? P

Alternatively, D, is within AD rad of the isoeffective dose provided


A3. Statistical formulation.

Suppose that D and D’ are theoretical values of ED50 such that the dose D’ is given in more fractions, with smaller doses per fraction, than total dose D. Then theoretically, D < D’. The doses D and D’ are considered “significantly” different if the probability of obtaining estimates b and 0’ with b’ 5 b is less than some prescribed level p. To evaluate Prob {b,’ 5 b} for doses D and D’, is first necessary to derive an equation for the theoretical variance of 0. For simplicity, we assume throughout that the estimates 6 are obtained using logit analysis. In practice, there is usually very little difference between the logit and probit estimates of ED50.

The logistic regression model of the sigmoidal relationship between dose D and the proportion P of responders at dose D is given by

In ep = A + B In(D) (6)

where A and B are constants7 The quantity Y = In(P/l - P) is called the logit of P. Once estimates of A and B are obtained, the dose D = ED50 at which the theoretical proportion of responders is 50% (P = .5) is approximated by solving the equation 0 = 2 + B In h,, i.e., b = exp( -i/B). Therefore,

var(b) = exp( -2A/B) var(i/B)

= (ED50)*var(h/B). (7)

Let P,, . . . Pq be the observed proportions of responders at doses D, through D,, respectively. Let Y, be the logit of Pi fori= l,... , q. The regression estimates 2 and B satisfy A = Y - Bm,’ where r and InD are the arithmetic means of the Y:s and the D;s, respectively. Hence

var(A/B) = var(lnD - r/B) = var(?/B,.

The expression on the right can be approximated” by

var(r/B) = $ varY+

If the experimental doses Di

(A + BlnD)* var(k)

B2 I. have been selected so that

1nD is close to ln(ED50) (e.g., if the responses range from 0% to lOO%), then A + 1nD = 0, so var(y/B) = (l/B*) var(r). Substituting this expression into Equation (7) yields the approximation var(b) = (ED50/B)* - var(?).

Next, we substitute for the logistic parameter B in the expression for var(b) as follows. The slope s at ED50 of a quanta1 response curve modeled by Equation by the derivative of P = eY/(l + eY), where In(D):

dP B

’ = dD D=EDSO = 4(ED50) ’

(6) is given Y=A+B

(8)

Therefore, var(b) = (k/s)*, where k = (var(Y)/4), i.e., the variance of the estimate fi of ED50 is inversely proportional to the square of the slope (at ED50) of the underlying quanta1 response curve. The variance of the mean r depends on two factors: the number 4 = n, of summands Yi, and the variance of each quantity Pi from which Y, is computed. Since the variance of Pi depends on the number n, of animals per dose group, it follows that the proportionality constant k is determined by the two experimental parameters n, and n,. (See Appendix B for a description of the simulation method used to estimate the dependence of k on n, and n, (presented in Table 1.)

The probability P that an individual in the population will exhibit a particular radiation endpoint is a function of the amount of tissue injury at a given dose. Equivalently, P = J/( -ln(response)). Using the LQ model, we obtain an expression for the slope s of the curve describing quantai response to fractionated irradiation. If the response curve is obtained by fixing the number n of dose fractions and varying the size of dose per fraction, then:

dP d $(D(a + Ox)) s =d6 D_EDSO = dD E

where x = EDSO/n, -E is the theoretical effect level corresponding to n fractions of size x, and the constant c is equal to $‘(E) - E. If, instead, the dose per fraction is fixed but the number of fractions varies, then the slope of the quanta1 response curve is:

dP

s = z D_ED50 =+‘(E)(a+px)=c ;+;x . (10) 1 1

Therefore, the variance in replicate estimates of ED50 from quanta1 data is given either by

(k/c) * (a/E) + 2(B/E)x 1

or by

(fixed number of fractions) ( 11)

- 1 (k/c) * var(D) = (a/E) + @/E)x 1

(fixed dose per fraction). ( 12)

From Equations (11) and (12), we see that the variance of b is approximately constant for small doses per fraction. We will use the approximation var(b) = (kE)/(ca)’ to simplify the remaining calculations. In addition, we assume that the logit estimate b of ED50 is a normal random variable with mean ED50. Computer simulations of quanta1 response data suggest that this assumption is valid. It follows from statistical theory” that Prob (b,’ - b 5 O] 2

p when (D - D’)/dm 1 z, where z is the standard- ized normal to p. Doses D and D’ satisfy D - D’ z z

where x and x’ are the isoeffective doses per fraction corresponding to D and D’, respectively. .The largest fractional dose x for which the above condition is satisfied for all x’ 5 x is

( - JZz(k/c) ff df= 1 + JZz(k/c) j’ )

For p = 0.05, the corresponding value of z is z = - 1.645,

so the flexure dose is given by

d/ =


(13)

Values of k are obtained from Table 1 for the appropri- ate n, and n,. The constant I/e is obtained as follows. The quantities (Y/E and P/E are known from the F,-plot (Equation (I)), and the slope s at ED50 of a quanta1 response curve can be estimated using the output from the logit analyses (Equation (6)), or by eye (by estimating the increment in total dose required to increase response near 50% by a certain amount, e.g., from 40% to 60%). Then l/c is the slope of the regression line obtained by plotting s against (a/E) + 2 (P/E) x (Equation (9)) or against

(a/E) + (PIE) x (Equation (lo)), where x is the isoeffective dose per fraction for 50% response determined by logit analysis of the quanta1 curve.

APPENDIX B

Derivation of Table I Calculations in Appendix A3 show that the standard

deviation of replicate estimates of ED50 from logit analysis is inversely proportional to the slope s at ED50 of the quanta1 response curve, i.e. s.d.(b) = k/s. Approximate values of the proportionality constant k are given in Table 1 for various choices of the experimental parameters n, and n,.

Initial estimates of k (for fixed n, and n,) were obtained by doing a least-squares fit of estimates of s.d.(b) against l/s for 10 values of s ranging from 0.5 to 5.0. The sample variances var(b) were obtained as described below.

The slope s at ED50 of a quanta1 response curve modeled by Equation (6) can be computed easily from the parameters A and B (see Appendix A3). Given values for A and B determining a particular value of s, quanta1 data were simulated using a binomial random deviate genera-

tor, with probabilities determined by Equation (6) for each dose. For each data set, it was necessary to select the doses and to stipulate the number of animals per dose from which response is estimated. An estimate b of ED50 was obtained for each data set using logit analysis. Six such estimates of ED50 were used to compute the sample variance var(b)). Although the number n, of doses was fixed for the approximation of var(b), the doses were selected differently for each of the six data sets. Two estimates of the standard deviation of b were obtained for each choice of logistic parameters A and B, and two (A,B) pairs were used for each value of the slope s.

After initial estimates of k were obtained, the results were “smoothed” as follows. For each value of n,., = 4, 6, . . . ( 20, the estimates of k were plotted against n,. Smooth curves were fit by eye to the data, and the values of k appearing in Table 1 were read from these curves as the nearest multiple of 0.005.

flexure dose: the low-dose limit of effective fractionation

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