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Effect of incomplete repair on normal tissue complication probability in the spinal cord
Int. J. Radiation Oncology Biol. Phys., Vol. 46, No. 3, pp. 631– 638, 2000 Copyright © 2000 Elsevier Science Inc. Printed in the USA. All rights reserved 0360-3016/00/$–see front matter
PII S0360-3016(99)00372-7
BIOLOGY CONTRIBUTION
EFFECT OF INCOMPLETE REPAIR ON NORMAL TISSUE COMPLICATION PROBABILITY IN THE SPINAL CORD DAPHNE LEVIN-PLOTNIK, M.SC.,*† ANDRZEJ NIEMIERKO, PH.D.,‡
AND
SOLANGE AKSELROD, PH.D.*
*Abramson Institute of Medical Physics, Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Israel; †Department of Radiation and Cellular Oncology, University of Chicago, Chicago, IL; and ‡Department of Oncology, Massachusetts General Hospital, Harvard Medical School, Boston, MA Purpose: To incorporate the effects of repair into a model for normal tissue complication probability (NTCP) in the spinal cord. Methods and Materials: We used an existing model of NTCP for the spinal cord, based on a critical volume concept, into which we incorporated an incomplete repair (IR) scheme. Values for the repair half time were taken from existing experimental data. Repair corrections were expanded to account for the possibility of biphasic repair, namely the existence of long and short components of repair. Results: We found that the model predicts complete repair to occur at approximately 15 hours, consistent with experimental data. The dependence of the model on the value of the dose per fraction was also studied. It was found that there is a sparing effect as the dose per fraction is decreased below 2 Gy. Surface plots of the NTCP as a function of both the interfraction interval (IFI) and the dose per fraction were generated. We investigated “iso-NTCP” curves, which may allow freedom in choice of treatment plans in terms of the optimal IFI and dose per fraction. As for biphasic repair, as the relative weights of the long and short components of repair were varied, the NTCP changed as well. The model showed little difference between mono- and bi-exponential repair in the time to complete repair, due to a dominance of the long component at long IFIs. Conclusions: Incorporating IR into NTCP modeling of the spinal cord is consistent with current experimental data. The concept of iso-NTCP curves is an approach which may be clinically useful. © 2000 Elsevier Science Inc. Normal tissue complication probability, Radiation myelopathy, Incomplete repair, Mathematical models.
INTRODUCTION
(1– 8). Various theoretical models have been proposed (9, 10) to explain the clinical and experimental results, but these are mostly phenomenological, using the probit or logistic statistical fits. A single model which does attempt to incorporate both the physical structure of the spinal cord and the biological mechanism of cell killing has been proposed by Yaes and Kalend (11), and will be described in more detail in the Methods. However, a theoretical understanding of normal tissue complication probability (NTCP) in the spinal cord is still lacking. It is not certain, for example, which cells are the target cells whose depletion results in myelopathy, and indeed it is even possible that damage to different types of cells leads to different types of lesions with different latencies (12). This issue becomes even more acute when one recalls the incidence of radiation myelopathy reported with continuous, hyperfractionated, accelerated radiation therapy (CHART), which was higher than expected with the existing models of NTCP (13–15). The aim of the current work was to modify an existing one-dimensional model of the spinal cord (11), to incorpo-
Radiotherapy entails delivering high doses of ionizing radiation to a localized area of the body, where the tumor is located. In this process, irradiation of normal tissue is unavoidable, since the ionizing beams deposit energy along their path, not only in the tumor, and because the treatment volume encompasses normal as well as tumor tissue. This radiation can cause normal tissue complications of varying severity. Complications in the spinal cord are very severe, and are manifested after radiation treatment is complete, affecting patients who have effectively been cured of the tumor. For this reason, radiation to the spinal cord is limited to doses that are known from clinical experience not to exceed cord tolerance, perhaps at the expense of a satisfactory dose distribution to the tumor, and thus at the expense of tumor control. Much experimental data on radiation complications in the spinal cord have been accumulated in laboratory animals Reprint requests to: Daphne Levin-Plotnik, Department of Radiation and Cellular Oncology, 5758 South Maryland Ave., MC 9006, Chicago, IL 60637. E-mail: [email protected] Acknowledgments—The author wishes to thank Prof. A. Gotsman,
Dept. of High Energy Physics, Tel Aviv University, Israel, for helpful discussions over the course of this work. Accepted for publication 30 August 1999. 631
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rate effects of repair. Both mono- and bi-exponential repair models were explored. This yields a tool with which the NTCP in the spinal cord can be studied as a function of various, controllable variables, such as dose, dose per fraction, and the interfraction interval (IFI). We found results in agreement with previously published experimental data. We also introduce the possibility of choosing different fractionation regimes, in terms of values of doses per fraction and IFIs, leading to the same complication probability. METHODS AND MATERIALS The present work makes use of two models that deal with NTCP in the spinal cord and repair (11, 16, 17). These models are presented briefly in the following paragraphs. Local stem cell depletion model This model, suggested by Yaes and Kalend (11), assumes only one type of target stem cell population, distributed uniformly along a one-dimensional spinal cord. Because the cord appears to be a serial structure, it is assumed that there is some “critical length” of cord that must be completely depleted of cells in order for a complication to be expressed. If at least one stem cell within such a critical length survives, it will be repopulated, and no complication will occur, and if less than a critical length is depleted, then it will be repopulated due to migration from neighboring areas. This critical length is referred to as a “slice” of thickness t. A finite length of cord L contains m such slices, each containing N stem cells. Assuming the linear-quadratic (LQ) relationship applies for cellular survival probability, the surviving fraction of cells after a course of radiation of n fractions and dose per fraction d is given by S ⫽ exp 共⫺E兲
(1)
E ⫽ n共 ␣ d ⫹  d 2兲,
(2)
where
and ␣ and  are the LQ model parameters. Using Poisson statistics the probability of all cells within a slice dying is given by P ⫽ exp 共⫺NS兲,
(3)
and the probability of at least one cell surviving is simply (1 ⫺ P). According to the model assumptions, complication will not occur if in each slice there is at least one surviving cell, therefore the probability Q of no complication occurring is Q ⫽ 共1 ⫺ P兲
m
which can be approximated for practical limits to be
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Q ⫽ exp 共⫺mP兲
(5)
(For a discussion of the validity of this derivation see Ref. 11.) Since the model uses the LQ relation for cellular damage after n equal fractions, complete repair between fractions is implicitly assumed (18). In the following, we attempted to relax this constraint by explicitly incorporating repair effects into the model. Originally the above model (11) was fitted to rodent data published by White and Hornsey (19) for a single dose and for two fractions with an IFI of 24 hours, and the agreement with the data was good. Incomplete Repair (IR) model We used the Incomplete Repair model first proposed by Oliver (20) and then generalized by Thames to include n fractions (16). The original model introduced the concept of an “effective dose” that decays exponentially over time, or a “dose equivalent of incomplete repair.” The idea is that after a dose of size d, the injury induced by some fraction of the dose is still unrepaired by the time an additional dose is given. The fraction is assumed to decay exponentially in time, according to
⫽ exp 共⫺ ⌬t兲,
(6)
where ⌬t is the IFI, and is the repair constant, related to the half-life of repair (T1/2) by
⫽ ln 2/T 1/ 2.
(7)
Thames (16) generalized this idea to include n equal fractions each of dose d. Assuming that proliferation is negligible between doses, and that the LQ model is an accurate survival function, one obtains the following: ln S n共d, 兲 ⫽ ⫺n共 ␣ d ⫹  d 2兲 ⫺ n  d 2h n共 兲, h n共 兲 ⫽
冉 冊冋 册冋 2 n
1⫺
n⫺
册
1 ⫺ n , 1⫺
(8) (9)
where d is the dose per fraction, and Sn is the survival function as given by the LQ model (Eq. 1), with ␣ and  the LQ model parameters. It can be seen that for large ⌬t, where complete repair is expected to occur,
⫽ 0,
h n共 兲 ⫽ 0,
(10)
and ln S n共d, 0兲 ⫽ ⫺n共 ␣ d ⫹  d 2兲,
(11)
(4) indeed corresponding to complete repair. For ⌬t ⫽ 0 we obtain
IR effect on NTCP in the spinal cord
⫽ 1,
h n共1兲 ⫽ n ⫺ 1,
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(12)
and ln S n共d, 1兲 ⫽ ⫺␣ nd ⫺  n 2d 2
(13)
corresponding to a single dose of nd, i.e., no repair. Bi-exponential repair. Recently new data have raised the possibility that there are in fact two components of repair in the spinal cord, one short and one long (4, 7). A modified version of the above repair model was introduced to address this possibility (17). This is expressed by modifying eq. 8 as follows: ln S n共d, 1, 2兲 ⫽ ⫺n共 ␣ d ⫹  d 2兲 ⫺ n  1d 2h n共 1兲 ⫺ n  2d 2h n共 2兲;
1 ⫽ exp 共⫺ 1⌬t兲;
2 ⫽ exp 共⫺ 2⌬t兲;
(14)
 ⫽ 1 ⫹  2.
(15)
Here 1 and 2 are the repair constants corresponding to the long and short repair times, T 1/ 2 i ⫽ ln 2/ i . It is also assumed that there are two independent types of lesions characterized by 1 and 2. Introducing repair into the stem cell model The basic assumptions of the original stem cell depletion model are unchanged by introduction of repair, that is, we still assume the existence of critical lengths that must be totally depleted of cells in order for a complication to be expressed. Based on eq. 5 the NTCP is given by NTCP ⫽ 1 ⫺ Q ⫽ 1 ⫺ exp 共⫺mP兲.
(16)
Incorporating repair relaxes the constraint that complete repair occurs between consecutive fractions, and introduces a dependence of the NTCP on parameters such as the IFI (⌬T), dose per fraction (d) and the repair constant (). The repair terms are incorporated at the cellular level, modifying the cell survival probability in the following manner: E ⫽ n共 ␣ d ⫹  d 2兲 ⫹ n  d 2h n共 兲
(17)
for mono-exponential repair (eqs. 2 and 8), and E ⫽ n共 ␣ d ⫹  d 2兲 ⫹ n  1d 2h n共 1兲 ⫹ n  2d 2h n共 2兲 (18) for bi-exponential repair (eqs. 2 and 14). The subsequent equations, leading to the calculation of the NTCP, remain unchanged in form, and these are Eqs. 1, 3, and 5. The important difference is in the fact that there is now an explicit dependence of the NTCP on the IFI and repair constants, which also affects the dependence on the dose per
Fig. 1. NTCP as a function of total dose for 2 (left group of curves) and 30 (right group of curves) fractions at different IFIs. The solid line represents the Yaes and Kalend model (assuming complete repair), crosses depict our model with IR at ⌬T ⫽ 15 hr, dotted line ⌬T ⫽ 8 hr, and dashed-dotted line ⌬T ⫽ 2 hr. Note that the effect of incomplete repair is more pronounced for 30 fractions. Repair constant was taken to be ⫽ 0.2888[hr]⫺1 from Ref. 4.
fraction. We investigated the effects of the two types of repair on the NTCP. Specifically, the effect of varying the IFI and repair constants () was studied, as well as the behavior of the NTCP as a function of the IFI and dose per fraction together. Parameters were taken following the original model (11); however, some changes were made. The most important of these is that the ratio ␣/ was taken to be 2, in accordance with most of the available experimental rodent data (4, 7, 21). For the repair constants we used the values given by Ang et al. (4) for both mono and biexponential repair. RESULTS Mono-exponential repair Figure 1 shows the NTCP for different IFIs for 2 and 30 equal fractions, as compared to the curve generated from the Yaes and Kalend model (henceforth to be referred to as the “full repair” curve). The curves were generated using eqs. 16 and 17. It can be seen that as the IFI is decreased, the curves generated by our model (nonsolid curves) are shifted to the left relative to the “full repair” curve (solid line). This is an expected effect of IR. The effect is more pronounced for the case of 30 fractions, since the total dose is larger for a larger
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Fig. 2. NTCP as a function of total dose for 2 (left group of curves) and 30 (right group of curves) fractions at different ’s. Curves were generated using eqs. 16 and 17. Solid lines are “full repair” curves; dashed line indicates model at ⫽ 0.1[hr]⫺1 (T 1/ 2 ⫽ 6.93 hr); and the crosses, which coincide with the “full repair” curve, are for ⫽ 0.28 [hr]⫺1 (T 1/ 2 ⫽ 2.4 hr), the value reported by Ang et al. (4). The IFI was taken to be ⌬T ⫽ 24 hr.
number of fractions. Varying the value of the IFI to the point where our curve coincides with the “full repair” curve yields the time at which complete repair has occurred. This time was found to be ⬃15 hours for the case of two fractions, and somewhat longer for 30 fractions. To check for model consistency we varied the repair constant at a fixed IFI. It is important to note that is a biological parameter, whose value cannot be varied externally by the physicist, so that this is more in the nature of a “gedanken” experiment. We chose the IFI to be ⌬T ⫽ 24 hr, to ensure that indeed we were in the region of complete repair (this value is in agreement with the results published by Landuyt et al. [7]). The results are shown in Fig. 2. It can be seen that as the value of decreases, corresponding to a longer repair time (cf. eq. 7), the dashed curves are shifted to the left relative to the “full repair” curve (solid), so that the NTCP is increased at the same dose values. The value of for which our curve coincides with the “full repair” curve is ⬃ 0.28 (plus curve). This value corresponds to T 1/ 2 ⫽ 2.4 hr, in agreement with the value given by Ang et al. (4). In Fig. 3 the NTCP is plotted as a function of the IFI, for different fixed doses. It can be seen that as the IFI is increased, the NTCP decreases, possibly indicating that more repair occurs between fractions. Once the IFI reaches a value of ⬃15 hr, for all dose values, a further increase of
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Fig. 3. NTCP as a function of the IFI for different fixed doses, for 30 fractions. It can be seen that as the IFI increases the NTCP decreases, until a minimal value of NTCP is reached. The value of was taken at 0.2888 [hr]⫺1, from Ref. (4).
the IFI no longer yields any benefit to the NTCP. This is consistent with complete repair having been achieved. Dependence on dose per fraction Figure 4 shows a surface plot illustrating the combined effect of varying the dose per fraction and the IFI on the NTCP, for a total dose of 100 Gy. On this surface “isoNTCP” curves are overlaid. All points on each such curve represent different possible treatment schedules, all with the same NTCP. From these iso-NTCP curves it can be learned that for a particular choice of total dose and NTCP value, there is a maximal value of the dose per fraction, attained when the IFI reaches the complete repair region. Increasing the dose per fraction beyond this value will necessarily lead to an increase in the complication probability. Mathematically, this is expressed by the fact that the equation which relates the NTCP to the dose per fraction and the IFI has no solutions beyond that value of dose per fraction (see Discussion, eq. 19). Figure 5 plots such iso-NTCP curves for total doses of 100 and 150 Gy. The solid curves are for an NTCP of 0.5 and the dashed curves represent an NTCP value of 0.05. These curves express clearly the asymptotic behavior of the dose per fraction as a function of the IFI. Additional information, which can be gleaned from the figure, is that as the total dose increases the maximal dose per fraction leading to a specific value of NTCP decreases rapidly. Thus, for example, for 100 Gy and NTCP ⫽ 0.5 the maximum dose per
IR effect on NTCP in the spinal cord
Fig. 4. Surface plot of NTCP as a function of dose per fraction and IFI at a total dose of 100 Gy. The solid lines overlaid on the plot represent iso-NTCP curves, calculated for NTCP values of 0.05, 0.1, 0.25, 0.5, 0.75, 0.9, in ascending order.
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Fig. 6. Bi-exponential compared to mono-exponential repair. The solid curve is the “full-repair” curve, shown for reference. The dotted curve is the mono-exponential repair curve, at ␣/ ⫽ 2 Gy, ⌬T ⫽ 4 hr, and ⫽ 0.2888 hr⫺1. The dashed– dotted curve is for bi-exponential repair, with 1 ⫽ 0.006 Gy⫺2 and 2 ⫽ 0.003 Gy⫺2, and the dashed curve is for 1 ⫽ 0.003 Gy⫺2 and 2 ⫽ 0.006 Gy⫺2. The repair constants were taken from Ang et al. (4) as 1 ⫽ 0.99 hr⫺1 and 2 ⫽ 0.182 hr⫺1. The curves were generated using eqs. 16 and 18.
fraction is 4.3 Gy, whereas the corresponding number for a total dose of 150 Gy is only 2.1 Gy.
Fig. 5. Iso-NTCP curves for NTCP values of 5% and 50%, for total doses of 100 and 150 Gy. Every point on each of the curves represents a different treatment schedule possibility in terms of choice of IFI and dose per fraction, with all points on a given curve having the same complication probability. Note the saturation of the dose per fraction, indicating complete repair has occurred.
Bi-exponential repair The values for the repair constants were taken from Ang et al. (4) for these calculations as well. We kept the value of  equal to the one assumed for mono-exponential repair kinetics and varied 1 and 2 such that the relation 1 ⫹ 2 ⫽  held. Figure 6 demonstrates the results for different values of 1 and 2. The solid curve is the “full-repair” curve, shown for reference, the dotted line is the monoexponential repair curve, and the dash– dot and dash curves are for bi-exponential repair. When the values of 1 and 2 were varied, so that 1, associated with the shorter repair time, was greater than 2, associated with the longer repair time, the curve was shifted to the right (dash– dot curve), indicating a lower NTCP at the same dose, i.e., more repair occurring over the given period of time. When 1 was less than 2 the NTCP curve was shifted to the left, indicating that more time was needed for repair. For 1 equal to 2 the bi-exponential and mono-exponential repair curves coincided, as expected. The difference between the bi- and mono-exponential
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repair curves decreased as the IFI was increased, so that the complete repair time, as given by the bi-exponential repair was only slightly shorter than that predicted for monoexponential repair. This was the case even when the values of 1 and 2 were varied so that 1 was much larger than 2, increasing the role of the short component of repair. DISCUSSION Incorporating repair into the stem cell model In the present model we incorporated the possibility of incomplete repair of clonogenic cells between consecutive doses of radiation, in order to predict NTCP in the spinal cord. The results of our model are in agreement with those of a previous model, which does not account for the phenomenon of incomplete repair, but rather implicitly assumes that complete repair occurs between dose fractions. This can be seen from the fact that when the IFI was varied from very short values to longer ones, it was found that the models coincided. This occurred at a value of approximately 15 hr, varying slightly, depending on the number of fractions used in the calculation. This value is consistent with values obtained in different experiments (4, 7), where rats were irradiated at different IFIs ranging between 0.5 and 8 hr, as well as at an interval of 24 hr. Both studies concluded that repair was incomplete at 8 hr, the longest IFI they used, but was already complete at 24 hr. However, neither experiment used IFI values of more than 8 hr. When we incorporated a biphasic repair model, the end results were not markedly different than those obtained from mono-exponential repair kinetics. The different “weights” assigned to 1 and 2 (cf. eq. 14) emphasized the short or long repair components respectively, but at values of IFI in the complete repair region both repair mechanisms were saturated, so that no difference between mono- or bi-exponential repair kinetics could be detected. This should be expected, since while even a small component of long repair remains, as we approach the time to complete repair this is the component of repair that will dominate. The short repair component saturates, in our model, with the repair constant taken from the experimental data (4), at a value of 5 hr. After that time the time to complete repair is affected only by the long repair component. Because the long repair constant is similar to that of the mono-exponential repair constant, we would expect the time to complete repair to be similar, regardless of the weighting of the two ’s. The difference between the different scenarios will be in the rapidity of the approach to the complete repair time, not in the actual time complete repair is achieved. The possible clinical implication of this observation is that if the treatment goal is to allow for complete repair between fractions, so as to minimize the risk of complication, then it is more important to determine the length of the long repair component, rather than that of the shorter one. This issue becomes even more acute if the proportion of damage repaired by the long component is larger than the one repaired by the short component, as suggested by Ang
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et al. (4). The clinical experience with myelopathy cases in CHART trials seems to bear out these conclusions. In these trials, the IFI was shortened to 6 hr (13) and 3 hr (15), and a higher than expected (from the LQ model) incidence of complications was observed. It may be that these IFI values allowed for short-term repair, while not allowing for the long repair, which occurs only after ⬃15 hr, according to our model. In order to distinguish between mono- and bi-exponential repair kinetics, our model needs reliable estimates of the two repair constants, as well as two ␣/ ratios. Many experiments have been performed lately to establish whether bi-exponential repair kinetics do indeed fit the data better, and what the half life values of repair are (4, 7, 8, 22). However, there is still much controversy about the results, and it is not clear which model more reliably describes the available experimental data. In addition, while some experiments do fit two different repair half times, as of yet there are no reliable values for two different ␣/ ratios. Indeed, several authors (4, 7) supply only one value of this ratio which they fit from the data along with the two repair times. However, it is reasonable to assume that if there are two repair constants, they are due to two different biological mechanisms, hence have two different  values, and thus two different ratios (17). Dynamic range In agreement with the experimental and clinical experience, as the number of fractions increases, so does the dynamic range of the NTCP, i.e., the slope of the NTCP curves becomes less steep. It is interesting to note that the dynamic range is relatively independent of the IFI, and is correlated mostly with the total dose. Varying the IFI changes the dose boundaries of the NTCP dynamic range—as the IFI decreases, both the dose at which the NTCP deviates from zero, and the dose at which the NTCP reaches a value of one, decrease, but the range between them remains almost constant. It is not intuitively obvious that this should be so. One might expect that as the amount of repair taking place between fractions decreases (corresponding to a decrease in IFI) the lower bound, at which the NTCP first deviates from zero, will decrease more than the upper bound, at which NTCP ⫽ 1, resulting in a change in the slope, or dynamic range. According to our model, this is not the case. Dose per fraction effect The dose per fraction is a significant factor affecting the NTCP. The Yaes and Kalend model used in this work (11) dealt with total dose and number of fractions as the independent variables, determining completely the dose per fraction (d ⫽ D/n). In our approach we attempted to study the interrelation of IFI and dose per fraction, and how it affects the NTCP. In the surface plot in Fig. 4 the region of interest, where both IFI and dose per fraction are important, is the area of curvature. Once the IFIs are in the region where complete
IR effect on NTCP in the spinal cord
repair has occurred, the problem is effectively reduced to a one variable equation, and varying the dose per fraction will have the same effect, regardless of the IFI. Thus, 2-D cuts at constant NTCP are parallel to each other. However, the concept of taking a cut of the surface at an arbitrary value of NTCP may have useful clinical aspects. Each curve in Fig. 5 is simply a solution of the equation NTCP 共⌬T, d兲 ⫽ const,
(19)
where the constant is any value of interest. These iso-NTCP curves allow us the latitude of choosing between different (⌬T,d) pairs, which may lead to differing tumor control probabilities (TCP), but all have the same NTCP. In other words, this adds an additional degree of freedom for the physician when prescribing a treatment schedule. Ideally, the next step would be to generate TCP surfaces for the tumor being treated, as a function of the same two variables—IFI and dose per fraction. One would then solve the equation TCP 共⌬T, d兲 ⫽ const,
(20)
where the right-hand side of the equation is generally set at 100%. When overlaying the resultant curve on the relevant curve of the type depicted in Fig. 5, the point at which the two curves cross can be identified as the optimal IFI and dose per fraction at which treatment should be given so as to yield both the TCP and NTCP desired. The concept presented here can, however, be experimentally verified or disproved. One would have to choose a complication endpoint, and a value of NTCP, and then perform different experiments using schedules corresponding to two or more different points on the relevant isoNTCP curve. Another feature which can be seen from Fig. 4 is that the model predicts a sparing effect even at doses below 2 Gy per fraction. This is demonstrated by choosing a particular value of IFI, and following the NTCP as a function of dose per fraction. The NTCP is a monotonically decreasing function of the dose per fraction all the way down to a value of zero (the value of zero itself is a singular point). Currently there is no consensus on whether this is a true picture or not. Withers (23) first introduced the concept of “flexure dose,” which was later expanded by Tucker et al. (24). This is the value beyond which decreasing the dose per fraction will have no further sparing effect. In the framework of the LQ model, this is simply a measure of the extent of the linear part of the dose–response curve, which is governed by single hit cell killing, a nonrepairable mechanism. For the spinal cord, they estimated the value of said flexure dose as ⬃1.7 Gy. Experimental work on sparing effects at low doses per fraction has not yet yielded clear results. Although early work appeared to show no sparing as the dose per fraction was decreased (25, 26), subsequent reanalysis in-
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corporating IR showed that the LQ model can indeed account for the data at doses per fraction lower than 2 Gy (27). More recent work indicates a sparing effect which is either consistent with the LQ model (4) or more than the LQ model predicts (21). Thus, with currently available experimental data, it is not possible to determine the validity of this prediction. Certainly, we might expect a monotonically decreasing function of NTCP with dose per fraction, since even when the linear component of cell killing dominates, there is still a part that is two hit cell kill, and therefore capable of repair. The question to be determined is: Is the slope we obtain valid, or does it overestimate or underestimate the true sparing effect as the dose per fraction is decreased? The model proposed here is based on the assumption that the spinal cord is a serial structure. Recently data have been published that indicate the existence of a volume effect in canine spinal cords (28), indicating that the cord may behave like a parallel organ. Other work, in primates, pigs, or rats, has shown either a marginal, or no volume effect, supporting the serial organ hypothesis (29, 5, 30). Our unpublished analysis of the data in canine spinal cords (28), using both serial and parallel models, demonstrates that both models are consistent with the data, and thus the data cannot discriminate between the two. The current model assumes a critical volume that must be completely depleted of stem cells in order for it to “die.” The distinction between parallel and serial structures is in the number of such critical volumes that must die before a complication is expressed. For radiation myelopathy this number was chosen here as 1, corresponding to a serial structure. This choice leads to the triple exponential dependence of the NTCP on the surviving fraction of cells. In principle, a different choice can be made, and the cord can be treated as a parallel organ. The NTCP model will then have a different mathematical dependence on the surviving fraction of cells, and is likely to have a much different form.
CONCLUSIONS A model for NTCP in the spinal cord has been presented. The model has a biological basis and relies on the LQ model of cell survival. The model has been shown to be consistent with available experimental data in terms of complete repair time. However, currently there are insufficient reliable data to test the model in depth. The model can incorporate either mono- or bi-exponential repair, yet cannot distinguish between them using the currently available values of repair constants. We emphasize the concept of iso-NTCP curves, which potentially introduce an additional degree of freedom for physicians in their choice of treatment schedules. In the future, it may prove advantageous to calculate iso-TCP curves in a similar manner, and to find the optimal treatment schedule, in terms of the IFI and the dose per fraction.
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REFERENCES 1. van der Kogel AJ. Radiation tolerance of the rat spinal cord: Time– dose relationships. Radiology 1977;122:505–509. 2. Ang KK, Thames HD, van der Kogel AJ, et al. Is the rate of repair of radiation-induced sublethal damage in rat spinal cord dependent on the size of dose per fraction? Int J Radiat Biol Oncol Phys 1987;13:557–562. 3. Schultheiss TE, Stephens LC, Jiang G, et al. Radiation myelopathy in primates treated with conventional fractionation. Int J Radiat Oncol Biol Phys 1990;19:935–940. 4. Ang KK, Jiang GL, Guttenberger R, et al. Impact of spinal cord repair kinetics on the practice of altered fractionation schedules. Radiother Oncol 1992;25:287–294. 5. Schultheiss TE, Stephens LC, Ang KK, et al. Volume effects in the rhesus monkey spinal cord. Int J Radiat Oncol Biol Phys 1994;29:67–72. 6. Ruifrok ACC, Stephens LC, van der Kogel AJ. Radiation response of the rat cervical cord after irradiation at different ages: tolerance, latency and pathology. Int J Radiat Oncol Biol Phys 1994;29:73–79. 7. Landuyt W, Fowler J, Ruifrok A, et al. Kinetics of repair in the spinal cord of the rat. Radiother Oncol 1997;45:55– 62. 8. Pop LAM, van der Plas M, Ruifrok ACC, et al. Tolerance of rat spinal cord to continuous interstitial irradiation. Int J Radiat Oncol Biol Phys 1998;40:681– 689. 9. Schultheiss TE, Orton CG, Peck RA. Models in radiotherapy: Volume effects. Med Phys 1983;10:410 – 415. 10. Niemierko A, Goitein M. Calculation of normal tissue complication probability and dose volume histogram reduction schemes for tissues with a critical element structure. Radiother Oncol 1991;20:166 –176. 11. Yaes RJ, Kalend A. Local stem cell depletion model for radiation myelitis. Int J Radiat Oncol Biol Phys 1988;14: 1247–1259. 12. Schultheiss TE, Stephens LC. Invited Review: Permanent radiation myelopathy. Br J Radiol 1992;65:737–753. 13. Dische S, Saunders M. Continuous, hyperfractionated, accelerated radiotherapy (CHART): An interim report upon late morbidity. Radiother Oncol 1989;16:67–74. 14. Dische S. Accelerated treatment and radiation myelitis. Radiother Oncol 1991;20:1–2. 15. Wong CS, Van Dyk J, Simpson WJ. Myelopathy following hyperfractionated accelerated radiotherapy for anaplastic thyroid carcinoma. Radiother Oncol 1991;20:3–9. 16. Thames HD. An ‘incomplete-repair’ model for survival after fractionated and continuous irradiations. Int J Radiat Biol 1985;47:319 –339.
17. Thames HD. Repair kinetics in tissues: Alternative models. Radiother Oncol 1989;14:321–327. 18. Barendsen GW. Dose fractionation, dose rate and iso-effect relationships for normal tissues responses. Int J Radiat Oncol Biol Phys 1982;8:1981–1997. 19. White A, Hornsey S. Radiation damage to the rat spinal cord: The effect of single and fractionated doses of X rays. Br J Radiol 1978;51:515–523. 20. Oliver R. A comparison of the effects of acute and protracted gamma-radiation on the growth of seedlings of Vicia faba. Part II. Theoretical calculations. Int J Rad Biol 1964;8:475– 488. 21. Wong CS, Minkin S, Hill RP. Linear-quadratic model underestimates sparing effect of small doses per fraction in rat spinal cord. Radiother Oncol 1992;23:176 –184. 22. Kim JJ, Hao Y, Jang D, et al. Lack of influence of sequence of top-up doses on repair kinetics in rat spinal cord. Radiother Oncol 1997;43:211–217. 23. Withers HR. Response of tissue to multiple small dose fractions. Rad Res 1977;71:24 –33. 24. Tucker SL, Thames HD. Flexure dose: The low-dose limit of effective fractionation. Int J Radiat Oncol Biol Phys 1983;9: 1373–1383. 25. Ang KK, van der Kogel AJ, van der Schueren E. Lack of evidence for increased tolerance of rat spinal cord with decreasing fraction doses below 2 Gy. Int J Radiat Oncol Biol Phys 1985;11:105–110. 26. Lavey RS, Johnstone AK, Taylor JMG, et al. The effect of hyperfractionation on spinal cord response to radiation. Int J Radiat Oncol Biol Phys 1992;24:681– 686. 27. Thames HD, Ang KK, Stewart FA, et al. Does incomplete repair explain the apparent failure of the basic LQ model to predict spinal cord and kidney responses to low doses per fraction? Int J Radiat Biol 1988;54:13–19. 28. Powers BE, Thames HD, Sharon MG, et al. Volume effects in the irradiated canine spinal cord: Do they exist when the probability of injury is low? Radiother Oncol 1998;46:297– 306. 29. van der Aardweg GJ, Hopewell JW, Whitehouse EM. The radiation response of the cervical spinal cord of the pig: Effects of changing the irradiated volume. Int J Radiat Oncol Biol Phys 1995;31:51–55. 30. van der Kogel AJ. Dose volume effects in the spinal cord. Radiother Oncol 1993;29:105–109.
PII S0360-3016(99)00372-7
BIOLOGY CONTRIBUTION
EFFECT OF INCOMPLETE REPAIR ON NORMAL TISSUE COMPLICATION PROBABILITY IN THE SPINAL CORD DAPHNE LEVIN-PLOTNIK, M.SC.,*† ANDRZEJ NIEMIERKO, PH.D.,‡
AND
SOLANGE AKSELROD, PH.D.*
*Abramson Institute of Medical Physics, Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Israel; †Department of Radiation and Cellular Oncology, University of Chicago, Chicago, IL; and ‡Department of Oncology, Massachusetts General Hospital, Harvard Medical School, Boston, MA Purpose: To incorporate the effects of repair into a model for normal tissue complication probability (NTCP) in the spinal cord. Methods and Materials: We used an existing model of NTCP for the spinal cord, based on a critical volume concept, into which we incorporated an incomplete repair (IR) scheme. Values for the repair half time were taken from existing experimental data. Repair corrections were expanded to account for the possibility of biphasic repair, namely the existence of long and short components of repair. Results: We found that the model predicts complete repair to occur at approximately 15 hours, consistent with experimental data. The dependence of the model on the value of the dose per fraction was also studied. It was found that there is a sparing effect as the dose per fraction is decreased below 2 Gy. Surface plots of the NTCP as a function of both the interfraction interval (IFI) and the dose per fraction were generated. We investigated “iso-NTCP” curves, which may allow freedom in choice of treatment plans in terms of the optimal IFI and dose per fraction. As for biphasic repair, as the relative weights of the long and short components of repair were varied, the NTCP changed as well. The model showed little difference between mono- and bi-exponential repair in the time to complete repair, due to a dominance of the long component at long IFIs. Conclusions: Incorporating IR into NTCP modeling of the spinal cord is consistent with current experimental data. The concept of iso-NTCP curves is an approach which may be clinically useful. © 2000 Elsevier Science Inc. Normal tissue complication probability, Radiation myelopathy, Incomplete repair, Mathematical models.
INTRODUCTION
(1– 8). Various theoretical models have been proposed (9, 10) to explain the clinical and experimental results, but these are mostly phenomenological, using the probit or logistic statistical fits. A single model which does attempt to incorporate both the physical structure of the spinal cord and the biological mechanism of cell killing has been proposed by Yaes and Kalend (11), and will be described in more detail in the Methods. However, a theoretical understanding of normal tissue complication probability (NTCP) in the spinal cord is still lacking. It is not certain, for example, which cells are the target cells whose depletion results in myelopathy, and indeed it is even possible that damage to different types of cells leads to different types of lesions with different latencies (12). This issue becomes even more acute when one recalls the incidence of radiation myelopathy reported with continuous, hyperfractionated, accelerated radiation therapy (CHART), which was higher than expected with the existing models of NTCP (13–15). The aim of the current work was to modify an existing one-dimensional model of the spinal cord (11), to incorpo-
Radiotherapy entails delivering high doses of ionizing radiation to a localized area of the body, where the tumor is located. In this process, irradiation of normal tissue is unavoidable, since the ionizing beams deposit energy along their path, not only in the tumor, and because the treatment volume encompasses normal as well as tumor tissue. This radiation can cause normal tissue complications of varying severity. Complications in the spinal cord are very severe, and are manifested after radiation treatment is complete, affecting patients who have effectively been cured of the tumor. For this reason, radiation to the spinal cord is limited to doses that are known from clinical experience not to exceed cord tolerance, perhaps at the expense of a satisfactory dose distribution to the tumor, and thus at the expense of tumor control. Much experimental data on radiation complications in the spinal cord have been accumulated in laboratory animals Reprint requests to: Daphne Levin-Plotnik, Department of Radiation and Cellular Oncology, 5758 South Maryland Ave., MC 9006, Chicago, IL 60637. E-mail: [email protected] Acknowledgments—The author wishes to thank Prof. A. Gotsman,
Dept. of High Energy Physics, Tel Aviv University, Israel, for helpful discussions over the course of this work. Accepted for publication 30 August 1999. 631
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rate effects of repair. Both mono- and bi-exponential repair models were explored. This yields a tool with which the NTCP in the spinal cord can be studied as a function of various, controllable variables, such as dose, dose per fraction, and the interfraction interval (IFI). We found results in agreement with previously published experimental data. We also introduce the possibility of choosing different fractionation regimes, in terms of values of doses per fraction and IFIs, leading to the same complication probability. METHODS AND MATERIALS The present work makes use of two models that deal with NTCP in the spinal cord and repair (11, 16, 17). These models are presented briefly in the following paragraphs. Local stem cell depletion model This model, suggested by Yaes and Kalend (11), assumes only one type of target stem cell population, distributed uniformly along a one-dimensional spinal cord. Because the cord appears to be a serial structure, it is assumed that there is some “critical length” of cord that must be completely depleted of cells in order for a complication to be expressed. If at least one stem cell within such a critical length survives, it will be repopulated, and no complication will occur, and if less than a critical length is depleted, then it will be repopulated due to migration from neighboring areas. This critical length is referred to as a “slice” of thickness t. A finite length of cord L contains m such slices, each containing N stem cells. Assuming the linear-quadratic (LQ) relationship applies for cellular survival probability, the surviving fraction of cells after a course of radiation of n fractions and dose per fraction d is given by S ⫽ exp 共⫺E兲
(1)
E ⫽ n共 ␣ d ⫹  d 2兲,
(2)
where
and ␣ and  are the LQ model parameters. Using Poisson statistics the probability of all cells within a slice dying is given by P ⫽ exp 共⫺NS兲,
(3)
and the probability of at least one cell surviving is simply (1 ⫺ P). According to the model assumptions, complication will not occur if in each slice there is at least one surviving cell, therefore the probability Q of no complication occurring is Q ⫽ 共1 ⫺ P兲
m
which can be approximated for practical limits to be
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Q ⫽ exp 共⫺mP兲
(5)
(For a discussion of the validity of this derivation see Ref. 11.) Since the model uses the LQ relation for cellular damage after n equal fractions, complete repair between fractions is implicitly assumed (18). In the following, we attempted to relax this constraint by explicitly incorporating repair effects into the model. Originally the above model (11) was fitted to rodent data published by White and Hornsey (19) for a single dose and for two fractions with an IFI of 24 hours, and the agreement with the data was good. Incomplete Repair (IR) model We used the Incomplete Repair model first proposed by Oliver (20) and then generalized by Thames to include n fractions (16). The original model introduced the concept of an “effective dose” that decays exponentially over time, or a “dose equivalent of incomplete repair.” The idea is that after a dose of size d, the injury induced by some fraction of the dose is still unrepaired by the time an additional dose is given. The fraction is assumed to decay exponentially in time, according to
⫽ exp 共⫺ ⌬t兲,
(6)
where ⌬t is the IFI, and is the repair constant, related to the half-life of repair (T1/2) by
⫽ ln 2/T 1/ 2.
(7)
Thames (16) generalized this idea to include n equal fractions each of dose d. Assuming that proliferation is negligible between doses, and that the LQ model is an accurate survival function, one obtains the following: ln S n共d, 兲 ⫽ ⫺n共 ␣ d ⫹  d 2兲 ⫺ n  d 2h n共 兲, h n共 兲 ⫽
冉 冊冋 册冋 2 n
1⫺
n⫺
册
1 ⫺ n , 1⫺
(8) (9)
where d is the dose per fraction, and Sn is the survival function as given by the LQ model (Eq. 1), with ␣ and  the LQ model parameters. It can be seen that for large ⌬t, where complete repair is expected to occur,
⫽ 0,
h n共 兲 ⫽ 0,
(10)
and ln S n共d, 0兲 ⫽ ⫺n共 ␣ d ⫹  d 2兲,
(11)
(4) indeed corresponding to complete repair. For ⌬t ⫽ 0 we obtain
IR effect on NTCP in the spinal cord
⫽ 1,
h n共1兲 ⫽ n ⫺ 1,
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(12)
and ln S n共d, 1兲 ⫽ ⫺␣ nd ⫺  n 2d 2
(13)
corresponding to a single dose of nd, i.e., no repair. Bi-exponential repair. Recently new data have raised the possibility that there are in fact two components of repair in the spinal cord, one short and one long (4, 7). A modified version of the above repair model was introduced to address this possibility (17). This is expressed by modifying eq. 8 as follows: ln S n共d, 1, 2兲 ⫽ ⫺n共 ␣ d ⫹  d 2兲 ⫺ n  1d 2h n共 1兲 ⫺ n  2d 2h n共 2兲;
1 ⫽ exp 共⫺ 1⌬t兲;
2 ⫽ exp 共⫺ 2⌬t兲;
(14)
 ⫽ 1 ⫹  2.
(15)
Here 1 and 2 are the repair constants corresponding to the long and short repair times, T 1/ 2 i ⫽ ln 2/ i . It is also assumed that there are two independent types of lesions characterized by 1 and 2. Introducing repair into the stem cell model The basic assumptions of the original stem cell depletion model are unchanged by introduction of repair, that is, we still assume the existence of critical lengths that must be totally depleted of cells in order for a complication to be expressed. Based on eq. 5 the NTCP is given by NTCP ⫽ 1 ⫺ Q ⫽ 1 ⫺ exp 共⫺mP兲.
(16)
Incorporating repair relaxes the constraint that complete repair occurs between consecutive fractions, and introduces a dependence of the NTCP on parameters such as the IFI (⌬T), dose per fraction (d) and the repair constant (). The repair terms are incorporated at the cellular level, modifying the cell survival probability in the following manner: E ⫽ n共 ␣ d ⫹  d 2兲 ⫹ n  d 2h n共 兲
(17)
for mono-exponential repair (eqs. 2 and 8), and E ⫽ n共 ␣ d ⫹  d 2兲 ⫹ n  1d 2h n共 1兲 ⫹ n  2d 2h n共 2兲 (18) for bi-exponential repair (eqs. 2 and 14). The subsequent equations, leading to the calculation of the NTCP, remain unchanged in form, and these are Eqs. 1, 3, and 5. The important difference is in the fact that there is now an explicit dependence of the NTCP on the IFI and repair constants, which also affects the dependence on the dose per
Fig. 1. NTCP as a function of total dose for 2 (left group of curves) and 30 (right group of curves) fractions at different IFIs. The solid line represents the Yaes and Kalend model (assuming complete repair), crosses depict our model with IR at ⌬T ⫽ 15 hr, dotted line ⌬T ⫽ 8 hr, and dashed-dotted line ⌬T ⫽ 2 hr. Note that the effect of incomplete repair is more pronounced for 30 fractions. Repair constant was taken to be ⫽ 0.2888[hr]⫺1 from Ref. 4.
fraction. We investigated the effects of the two types of repair on the NTCP. Specifically, the effect of varying the IFI and repair constants () was studied, as well as the behavior of the NTCP as a function of the IFI and dose per fraction together. Parameters were taken following the original model (11); however, some changes were made. The most important of these is that the ratio ␣/ was taken to be 2, in accordance with most of the available experimental rodent data (4, 7, 21). For the repair constants we used the values given by Ang et al. (4) for both mono and biexponential repair. RESULTS Mono-exponential repair Figure 1 shows the NTCP for different IFIs for 2 and 30 equal fractions, as compared to the curve generated from the Yaes and Kalend model (henceforth to be referred to as the “full repair” curve). The curves were generated using eqs. 16 and 17. It can be seen that as the IFI is decreased, the curves generated by our model (nonsolid curves) are shifted to the left relative to the “full repair” curve (solid line). This is an expected effect of IR. The effect is more pronounced for the case of 30 fractions, since the total dose is larger for a larger
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Fig. 2. NTCP as a function of total dose for 2 (left group of curves) and 30 (right group of curves) fractions at different ’s. Curves were generated using eqs. 16 and 17. Solid lines are “full repair” curves; dashed line indicates model at ⫽ 0.1[hr]⫺1 (T 1/ 2 ⫽ 6.93 hr); and the crosses, which coincide with the “full repair” curve, are for ⫽ 0.28 [hr]⫺1 (T 1/ 2 ⫽ 2.4 hr), the value reported by Ang et al. (4). The IFI was taken to be ⌬T ⫽ 24 hr.
number of fractions. Varying the value of the IFI to the point where our curve coincides with the “full repair” curve yields the time at which complete repair has occurred. This time was found to be ⬃15 hours for the case of two fractions, and somewhat longer for 30 fractions. To check for model consistency we varied the repair constant at a fixed IFI. It is important to note that is a biological parameter, whose value cannot be varied externally by the physicist, so that this is more in the nature of a “gedanken” experiment. We chose the IFI to be ⌬T ⫽ 24 hr, to ensure that indeed we were in the region of complete repair (this value is in agreement with the results published by Landuyt et al. [7]). The results are shown in Fig. 2. It can be seen that as the value of decreases, corresponding to a longer repair time (cf. eq. 7), the dashed curves are shifted to the left relative to the “full repair” curve (solid), so that the NTCP is increased at the same dose values. The value of for which our curve coincides with the “full repair” curve is ⬃ 0.28 (plus curve). This value corresponds to T 1/ 2 ⫽ 2.4 hr, in agreement with the value given by Ang et al. (4). In Fig. 3 the NTCP is plotted as a function of the IFI, for different fixed doses. It can be seen that as the IFI is increased, the NTCP decreases, possibly indicating that more repair occurs between fractions. Once the IFI reaches a value of ⬃15 hr, for all dose values, a further increase of
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Fig. 3. NTCP as a function of the IFI for different fixed doses, for 30 fractions. It can be seen that as the IFI increases the NTCP decreases, until a minimal value of NTCP is reached. The value of was taken at 0.2888 [hr]⫺1, from Ref. (4).
the IFI no longer yields any benefit to the NTCP. This is consistent with complete repair having been achieved. Dependence on dose per fraction Figure 4 shows a surface plot illustrating the combined effect of varying the dose per fraction and the IFI on the NTCP, for a total dose of 100 Gy. On this surface “isoNTCP” curves are overlaid. All points on each such curve represent different possible treatment schedules, all with the same NTCP. From these iso-NTCP curves it can be learned that for a particular choice of total dose and NTCP value, there is a maximal value of the dose per fraction, attained when the IFI reaches the complete repair region. Increasing the dose per fraction beyond this value will necessarily lead to an increase in the complication probability. Mathematically, this is expressed by the fact that the equation which relates the NTCP to the dose per fraction and the IFI has no solutions beyond that value of dose per fraction (see Discussion, eq. 19). Figure 5 plots such iso-NTCP curves for total doses of 100 and 150 Gy. The solid curves are for an NTCP of 0.5 and the dashed curves represent an NTCP value of 0.05. These curves express clearly the asymptotic behavior of the dose per fraction as a function of the IFI. Additional information, which can be gleaned from the figure, is that as the total dose increases the maximal dose per fraction leading to a specific value of NTCP decreases rapidly. Thus, for example, for 100 Gy and NTCP ⫽ 0.5 the maximum dose per
IR effect on NTCP in the spinal cord
Fig. 4. Surface plot of NTCP as a function of dose per fraction and IFI at a total dose of 100 Gy. The solid lines overlaid on the plot represent iso-NTCP curves, calculated for NTCP values of 0.05, 0.1, 0.25, 0.5, 0.75, 0.9, in ascending order.
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Fig. 6. Bi-exponential compared to mono-exponential repair. The solid curve is the “full-repair” curve, shown for reference. The dotted curve is the mono-exponential repair curve, at ␣/ ⫽ 2 Gy, ⌬T ⫽ 4 hr, and ⫽ 0.2888 hr⫺1. The dashed– dotted curve is for bi-exponential repair, with 1 ⫽ 0.006 Gy⫺2 and 2 ⫽ 0.003 Gy⫺2, and the dashed curve is for 1 ⫽ 0.003 Gy⫺2 and 2 ⫽ 0.006 Gy⫺2. The repair constants were taken from Ang et al. (4) as 1 ⫽ 0.99 hr⫺1 and 2 ⫽ 0.182 hr⫺1. The curves were generated using eqs. 16 and 18.
fraction is 4.3 Gy, whereas the corresponding number for a total dose of 150 Gy is only 2.1 Gy.
Fig. 5. Iso-NTCP curves for NTCP values of 5% and 50%, for total doses of 100 and 150 Gy. Every point on each of the curves represents a different treatment schedule possibility in terms of choice of IFI and dose per fraction, with all points on a given curve having the same complication probability. Note the saturation of the dose per fraction, indicating complete repair has occurred.
Bi-exponential repair The values for the repair constants were taken from Ang et al. (4) for these calculations as well. We kept the value of  equal to the one assumed for mono-exponential repair kinetics and varied 1 and 2 such that the relation 1 ⫹ 2 ⫽  held. Figure 6 demonstrates the results for different values of 1 and 2. The solid curve is the “full-repair” curve, shown for reference, the dotted line is the monoexponential repair curve, and the dash– dot and dash curves are for bi-exponential repair. When the values of 1 and 2 were varied, so that 1, associated with the shorter repair time, was greater than 2, associated with the longer repair time, the curve was shifted to the right (dash– dot curve), indicating a lower NTCP at the same dose, i.e., more repair occurring over the given period of time. When 1 was less than 2 the NTCP curve was shifted to the left, indicating that more time was needed for repair. For 1 equal to 2 the bi-exponential and mono-exponential repair curves coincided, as expected. The difference between the bi- and mono-exponential
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repair curves decreased as the IFI was increased, so that the complete repair time, as given by the bi-exponential repair was only slightly shorter than that predicted for monoexponential repair. This was the case even when the values of 1 and 2 were varied so that 1 was much larger than 2, increasing the role of the short component of repair. DISCUSSION Incorporating repair into the stem cell model In the present model we incorporated the possibility of incomplete repair of clonogenic cells between consecutive doses of radiation, in order to predict NTCP in the spinal cord. The results of our model are in agreement with those of a previous model, which does not account for the phenomenon of incomplete repair, but rather implicitly assumes that complete repair occurs between dose fractions. This can be seen from the fact that when the IFI was varied from very short values to longer ones, it was found that the models coincided. This occurred at a value of approximately 15 hr, varying slightly, depending on the number of fractions used in the calculation. This value is consistent with values obtained in different experiments (4, 7), where rats were irradiated at different IFIs ranging between 0.5 and 8 hr, as well as at an interval of 24 hr. Both studies concluded that repair was incomplete at 8 hr, the longest IFI they used, but was already complete at 24 hr. However, neither experiment used IFI values of more than 8 hr. When we incorporated a biphasic repair model, the end results were not markedly different than those obtained from mono-exponential repair kinetics. The different “weights” assigned to 1 and 2 (cf. eq. 14) emphasized the short or long repair components respectively, but at values of IFI in the complete repair region both repair mechanisms were saturated, so that no difference between mono- or bi-exponential repair kinetics could be detected. This should be expected, since while even a small component of long repair remains, as we approach the time to complete repair this is the component of repair that will dominate. The short repair component saturates, in our model, with the repair constant taken from the experimental data (4), at a value of 5 hr. After that time the time to complete repair is affected only by the long repair component. Because the long repair constant is similar to that of the mono-exponential repair constant, we would expect the time to complete repair to be similar, regardless of the weighting of the two ’s. The difference between the different scenarios will be in the rapidity of the approach to the complete repair time, not in the actual time complete repair is achieved. The possible clinical implication of this observation is that if the treatment goal is to allow for complete repair between fractions, so as to minimize the risk of complication, then it is more important to determine the length of the long repair component, rather than that of the shorter one. This issue becomes even more acute if the proportion of damage repaired by the long component is larger than the one repaired by the short component, as suggested by Ang
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et al. (4). The clinical experience with myelopathy cases in CHART trials seems to bear out these conclusions. In these trials, the IFI was shortened to 6 hr (13) and 3 hr (15), and a higher than expected (from the LQ model) incidence of complications was observed. It may be that these IFI values allowed for short-term repair, while not allowing for the long repair, which occurs only after ⬃15 hr, according to our model. In order to distinguish between mono- and bi-exponential repair kinetics, our model needs reliable estimates of the two repair constants, as well as two ␣/ ratios. Many experiments have been performed lately to establish whether bi-exponential repair kinetics do indeed fit the data better, and what the half life values of repair are (4, 7, 8, 22). However, there is still much controversy about the results, and it is not clear which model more reliably describes the available experimental data. In addition, while some experiments do fit two different repair half times, as of yet there are no reliable values for two different ␣/ ratios. Indeed, several authors (4, 7) supply only one value of this ratio which they fit from the data along with the two repair times. However, it is reasonable to assume that if there are two repair constants, they are due to two different biological mechanisms, hence have two different  values, and thus two different ratios (17). Dynamic range In agreement with the experimental and clinical experience, as the number of fractions increases, so does the dynamic range of the NTCP, i.e., the slope of the NTCP curves becomes less steep. It is interesting to note that the dynamic range is relatively independent of the IFI, and is correlated mostly with the total dose. Varying the IFI changes the dose boundaries of the NTCP dynamic range—as the IFI decreases, both the dose at which the NTCP deviates from zero, and the dose at which the NTCP reaches a value of one, decrease, but the range between them remains almost constant. It is not intuitively obvious that this should be so. One might expect that as the amount of repair taking place between fractions decreases (corresponding to a decrease in IFI) the lower bound, at which the NTCP first deviates from zero, will decrease more than the upper bound, at which NTCP ⫽ 1, resulting in a change in the slope, or dynamic range. According to our model, this is not the case. Dose per fraction effect The dose per fraction is a significant factor affecting the NTCP. The Yaes and Kalend model used in this work (11) dealt with total dose and number of fractions as the independent variables, determining completely the dose per fraction (d ⫽ D/n). In our approach we attempted to study the interrelation of IFI and dose per fraction, and how it affects the NTCP. In the surface plot in Fig. 4 the region of interest, where both IFI and dose per fraction are important, is the area of curvature. Once the IFIs are in the region where complete
IR effect on NTCP in the spinal cord
repair has occurred, the problem is effectively reduced to a one variable equation, and varying the dose per fraction will have the same effect, regardless of the IFI. Thus, 2-D cuts at constant NTCP are parallel to each other. However, the concept of taking a cut of the surface at an arbitrary value of NTCP may have useful clinical aspects. Each curve in Fig. 5 is simply a solution of the equation NTCP 共⌬T, d兲 ⫽ const,
(19)
where the constant is any value of interest. These iso-NTCP curves allow us the latitude of choosing between different (⌬T,d) pairs, which may lead to differing tumor control probabilities (TCP), but all have the same NTCP. In other words, this adds an additional degree of freedom for the physician when prescribing a treatment schedule. Ideally, the next step would be to generate TCP surfaces for the tumor being treated, as a function of the same two variables—IFI and dose per fraction. One would then solve the equation TCP 共⌬T, d兲 ⫽ const,
(20)
where the right-hand side of the equation is generally set at 100%. When overlaying the resultant curve on the relevant curve of the type depicted in Fig. 5, the point at which the two curves cross can be identified as the optimal IFI and dose per fraction at which treatment should be given so as to yield both the TCP and NTCP desired. The concept presented here can, however, be experimentally verified or disproved. One would have to choose a complication endpoint, and a value of NTCP, and then perform different experiments using schedules corresponding to two or more different points on the relevant isoNTCP curve. Another feature which can be seen from Fig. 4 is that the model predicts a sparing effect even at doses below 2 Gy per fraction. This is demonstrated by choosing a particular value of IFI, and following the NTCP as a function of dose per fraction. The NTCP is a monotonically decreasing function of the dose per fraction all the way down to a value of zero (the value of zero itself is a singular point). Currently there is no consensus on whether this is a true picture or not. Withers (23) first introduced the concept of “flexure dose,” which was later expanded by Tucker et al. (24). This is the value beyond which decreasing the dose per fraction will have no further sparing effect. In the framework of the LQ model, this is simply a measure of the extent of the linear part of the dose–response curve, which is governed by single hit cell killing, a nonrepairable mechanism. For the spinal cord, they estimated the value of said flexure dose as ⬃1.7 Gy. Experimental work on sparing effects at low doses per fraction has not yet yielded clear results. Although early work appeared to show no sparing as the dose per fraction was decreased (25, 26), subsequent reanalysis in-
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corporating IR showed that the LQ model can indeed account for the data at doses per fraction lower than 2 Gy (27). More recent work indicates a sparing effect which is either consistent with the LQ model (4) or more than the LQ model predicts (21). Thus, with currently available experimental data, it is not possible to determine the validity of this prediction. Certainly, we might expect a monotonically decreasing function of NTCP with dose per fraction, since even when the linear component of cell killing dominates, there is still a part that is two hit cell kill, and therefore capable of repair. The question to be determined is: Is the slope we obtain valid, or does it overestimate or underestimate the true sparing effect as the dose per fraction is decreased? The model proposed here is based on the assumption that the spinal cord is a serial structure. Recently data have been published that indicate the existence of a volume effect in canine spinal cords (28), indicating that the cord may behave like a parallel organ. Other work, in primates, pigs, or rats, has shown either a marginal, or no volume effect, supporting the serial organ hypothesis (29, 5, 30). Our unpublished analysis of the data in canine spinal cords (28), using both serial and parallel models, demonstrates that both models are consistent with the data, and thus the data cannot discriminate between the two. The current model assumes a critical volume that must be completely depleted of stem cells in order for it to “die.” The distinction between parallel and serial structures is in the number of such critical volumes that must die before a complication is expressed. For radiation myelopathy this number was chosen here as 1, corresponding to a serial structure. This choice leads to the triple exponential dependence of the NTCP on the surviving fraction of cells. In principle, a different choice can be made, and the cord can be treated as a parallel organ. The NTCP model will then have a different mathematical dependence on the surviving fraction of cells, and is likely to have a much different form.
CONCLUSIONS A model for NTCP in the spinal cord has been presented. The model has a biological basis and relies on the LQ model of cell survival. The model has been shown to be consistent with available experimental data in terms of complete repair time. However, currently there are insufficient reliable data to test the model in depth. The model can incorporate either mono- or bi-exponential repair, yet cannot distinguish between them using the currently available values of repair constants. We emphasize the concept of iso-NTCP curves, which potentially introduce an additional degree of freedom for physicians in their choice of treatment schedules. In the future, it may prove advantageous to calculate iso-TCP curves in a similar manner, and to find the optimal treatment schedule, in terms of the IFI and the dose per fraction.
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