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Repopulation of mouse jejunal crypt cells
Radiotherapy and Oncology, 20 (1991) 181-190 Elsevier
181
RADION00818
Repopulation of mouse jejunal crypt cells J e r e m y M . G . T a y l o r , H . R o d n e y W i t h e r s , K a t h y A. M a s o n a n d Cally A. D a v i s Department of Radiation Oncology and Jonsson Comprehensive Cancer Center, UCLA Medical Center, Los Angeles, CA, U.S.A.
(Received 26 March 1990, revision received 6 November 1990, accepted 5 December 1990)
Key words: Acute effects; Mathematical model; Logistic growth; Regeneration; Mitotic delay
Summary The regenerative response to radiation of mouse jejunal crypt cells was investigated using a three fraction experiment. The time between the second and third fraction was varied between 5 and 60 h, and the dose in each of the three fractions was different. The data show that the onset of regeneration is within 14 h of the first dose and possibly earlier. The doubling time of the clonogenic cells during repopulation is estimated to be between 5 and 10 h, with the most likely value approximately 6 to 7 h. The data also show that the time course of repopulation in an acutely responding tissue depended in a complex way on the fractionation scheme. The implications of this for radiotherapy are that simple formulas are unlikely to be accurate predictors of acute effects in altered fractionation schemes. Detailed mathematical modelling of the data is undertaken using a model which consists of a single dose survival curve, a part to incorporate the regenerative response and a part to accommodate the delayed onset of regeneration. The model is shown to give a good, although not perfect, fit to the data. A mathematical derivation is given of the expected number of crypts following the 3 dose radiation schedule. This derivation takes into account the fact that any crypt could be denuded of cells prior to the final dose and hence not repopulate, and thus the formula given is necessarily more complex than previous formulas which were based on simpler models.
Introduction With the increasing use of accelerated treatment in radiotherapy, the acute effects are becoming more important and may be dose limiting for some regimens. This suggests the need for a complete understanding of acute effects and in particular the kinetics of the regenerative response to radiation insult during treatment. Previous data [18,19] using a jejunal colony assay system had suggested that there was a burst of repopulation one to two days after the initial dose in experiments which consisted of 2 doses separated by varying time intervals. However, other data for colonic crypt cells [20] and mathematical modelling of fractionated and more recent data [ 11] suggest that the onset of regeneration is earlier than one to two days after the initial dose and could be as early as 3 h [9]. These data [ 11 ] also suggest that the delay before the onset of regeneration was dependent on the dose and
could be explained by a cell cycle delay proportional to the size of the dose. A further experiment was designed to explicitly test these findings.
Materials and methods Animals
Female C 3 H f / S e d / / K a m specific pathogen-free mice aged 9 to 15weeks were used (n = 761). A 12h light/dark cycle was used. There were 20 animals at most of the dose points. Irradiation
Groups of 10 mice were simultaneously whole-body irradiated using a G a m m a Cell 40 (AECL) cesium source at dose rates of 111 to 113 cGy per minute.
Addressfor correspondence:J. M. G. Taylor, Department of Radiation Oncology,UCLA Medical Center, Los Angeles, CA 90024-1714, U.S.A.
0167-8140/91/$03.50 © 1991 Elsevier Science Publishers B.V. (Biomedical Division)
182 Microcolony assay
Statistical methods and mathematical models
The microcolony as say of Withers and Elkind [ 18,19 ] was used. Briefly, animals were sacrificed 3 days and 14 h after the midpoint of each experimental regimen. A 2-3 cm length of jejunum was excised and fixed in neutral buffered formalin. Following routine histological processing, sections were cut at 5 #m and stained with hemotoxylin and eosin. Surviving crypts per circumference of jejunum were scored microscopically using 100 x magnification. Three to five transverse sections of jejunum were scored for each animal. All histological sections were scored by one observer (K. A. M.). The average score for each animal was used in the analysis.
Means and 95 ~ confidence intervals for the number of crypts per circumference are shown in Figs. 1 and 2. The rate of repopulation between the second and third doses is calculated independently for each of the 4 experiments by assuming an exponential growth over the range of At between 5 and 20 h. The model
Experimental design Following a pilot study, the experiment consisted of three fractions, an initial dose of 6 Gy followed 4 h later by dose d2 Gy and then a further time At hours later a dose of size d3 Gy. Atis either 5, 10, 15, 20, 25, 30, 35, 40, 50, or 60 h, and there were 4 sets of the (d2,d3) pair, (d2,d3) = (2.0,10.7), (10.7,2.0), (5.1,8.4) or (8.4,5.1). In addition, data from a range of single doses was collected. Also, data was collected from a pair of two fraction dose points, in which the initial dose of 6 Gy was followed 4 h later by either 10.7 Gy or 8.4 Gy. The minimum separation between two doses is 4 h which is sufficient time for complete repair of sublethal injury. The rationale for this 3-dose design, with a fixed priming dose followed by two test doses, was to insure that the priming dose rather than either of the test doses was responsible for activating the mechanisms which turn on repopulation. The initial dose of 6 Gy was considered sufficiently large to reduce cell survival to a low enough level ( < 0.2) so that the mechanisms controlling repopulation would be activated. The choice of the pairs (2.0,10.7) and (5.1,8.4) was so that, in the absence ofrepopulation, all 4 sets should give approximately the same cell killing. The relatively short times for At were chosen to emphasize the onset of regeneration and the initial rate of repopulation. Both orders of the pair 2.0 Gy and 10.7 Gy and the pair 5.1 Gy and 8.4 Gy were used to ascertain how the fractionation scheme affected the onset of regeneration. Previous analysis [ 11] had suggested that there was a cell cycle delay following each irradiation proportional to the dose per fraction, in which case the onset of regeneration would be later in the scheme with the second dose larger than the third.
log( - log(1 - C/V) = A + At. log2/DT was fit to the data separately for each of the four designs using non-linear least squares. In this model, the parameters DT ( = doubling time) and A are estimated, C is the observed crypt score and V is the number of crypts containing at least one clonogen after the second dose. The assumed values of V were 160, 150, 82, and 32 for d2 = 2.0, 5.1, 8.4, and 10.7, respectively. In addition, a variety of mathematical models were fit to the complete data in an attempt to jointly estimate the effects of cell killing and regeneration. These models and the associated statistical methods are slight adaptations of those described in Taylor etal. [11], and readers are referred to that article for complete details. The models are similar in spirit, but different in detail, to the models described by Cohen [4] and van de Geijn [ 16]. Briefly, the basic model consists of a single dose survival curve, either the linear-quadratic model or the two-component model. Time to the onset of regeneration is determined by either a dose-dependent mitotic delay following each irradiation dose or a fixed time delay after the initial dose, or both, or neither; and the repopulation kinetics is determined by either logistic or Gompertz growth. The possibility of a fixed delay was included in the model to represent the time interval following the initial dose before which damage in the stem cell compartments is recognized. In these models the final cell survival level is calculated by tracking the cell survival level throughout the fractionation scheme. This requires allowing each separate dose to cause a reduction in the cell survival level and for a possible increase in the cell survival level due to repopulation during the periods between fractions. With this model, the final cell survival level can be explicitly specified in a possibly complex equation. Three changes were made to the mathematical models previously described [ 11 ] in order that it will more correctly represent the underlying biology. The first change is that periods of mitotic delay are now assumed to be cumulative. For example, if a dose of 6 Gy causes 6 h mitotic delay and if2 doses of 6 Gy are separated by 4 h, then the total delay is assumed 12 h rather than the 10 h as was used in the previous analysis.
183 The second change is that the length of the mitotic delay after each dose, which is proportional to dose per fraction and the duration of the cell cycle, is reduced during repopulation due to the shortened cell cycle time, whereas in the previous analysis the length of the mitotic delay was not reduced during periods of rapid repopulation. In particular, mitotic delay = r × dose × cell cycle time, where the cell cycle time is the average cycle time between mitoses during repopulation and is given by cell cycle time = minimum (18 h, log(2P) × doubling time/log(2)), where P is the fraction of cells which remain in the stem cell compartment after division, and the cell doubling time is obtained from the kinetic growth model of repopulation. The parameter r is estimated from the data. The quantity P is in the range 0.5 (steady state) to 1.0 (maximum possible) and 18 h is the assumed steady state cell cycle time and also the maximum possible cell cycle time during repopulation. P is 0.5 before repopulation begins, and we assumed that P = 0.9 during repopulation. We chose a high value of P because previous data had suggested that the doubling time would be about 6 to 8 h [1,11]. As the cell cycle time is unlikely to decrease below about 5 h, a high value of P is necessary to ensure a low doubling time. Sensitivity analyses were performed to assess the effect of different values of P and the steady state cell cycle time. For the logistic growth model dlog S / d t = 2(1 - S), where S is the cell surviving fraction; the doubling time = log 2/(2(1 - S)). The calculation of the doubling times in the tables is based on the situation where S is small, i.e. the doubling time is calculated as log 2/2. The third change is that the functional subunit structure of the crypts within the jejunum is taken into account. Previous modelling assumed that the whole section of jejunum could repopulate given sufficient time however low the cell surviving fraction became during the course of the irradiations. However, in reality, there are a finite number of cells in each crypt so that once a crypt is completely denuded of cells it cannot repopulate itself and hence will never be scored in the assay. We also assumed that repopulation from adjacent crypts cannot occur within the duration of the experiment. The following equation for the expected crypt score incorporates this structure of the crypt: 1 - E(Crypt score)/160 = Q1K + (1 - Q392)Q2g' {(Q1 + e~o2h') 1¢ - Q K} + 03 g2 (Q2 + P203*2)g' {(Q, + P, (Qz + PzQ3h2)*') r - Qi/':} •
(1)
where 160 is the control number of crypts per circumference, Pi is the surviving fraction of cells from a single dose of size di, Qi = 1 - Pi, K is the initial number of clonogenic cells per crypt, gi and h i determine the approximate increase in the number of cells due to repopulation between the ith and (i + 1)th dose. We assumed the value of K to be 100 for the majority of the analyses, but also considered both higher and lower values in a sensitivity analysis. The Appendix to this paper contains the derivations of Eqn. 1 and the expressions for gi and h i. Throughout this paper, unless otherwise stated, we used the linear-quadratic model for the single dose survival curve and logistic growth to model the kinetics of regeneration. We also always included both a fixed time delay and a radiation-induced mitotic delay within the model. The parameters of the models were fitted by nonlinear unweighted least-squares regression on the square root scale of crypt score using the NLIN procedure from the SAS statistical software. That is, we minimized with respect to the parameters Z(Cj 1/2 2, where Cj is the observed crypt score for animal j and E(Cj) is the expected crypt score for animal j which depends on the unknown parameters of the mathematical model and is given by Eqn. 1. The rationale for this least squares approach has been described previously [ 11 ]. There are five parameters in the model: ~ and ~//3 which determine the linearquadratic model, r which determines the mitotic delay, Tf which determines the fixed time delay and 2 which determines the rate of repopulation and hence the doubling time. The mean squared error is used to assess each model, the smaller value indicating a better fit. All the data, including the pilot study (n = 12) and the single dose control data (n = 45), is used in fitting the models.
--(E(Cj))'/2)
Results
Figures 1 and 2 show the results for the main part of the 3-fraction experiment. The figures do not show the data for the single dose or pilot experiment, but they do show the results of the two dose experiments consisting of 6 Gy plus d2 (denoted by A on the horizontal axis). The features of the graphs which are of particular interest are (i) the onset time of the increased response, assumed to be due to repopulation, is within 14 h of the initial dose of 6 Gy for all 4 curves and could be much earlier with the possible exception of the design in which d 2 = 10.7 Gy. (ii) There is a very noticeable difference between curves depending upon the order of the different sized second and third doses. When the larger
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dose is the second of the three, the response reaches a plateau level for large At, which is similar to the score that results when only the first two doses are given (At -- A in Figs. 1 and 2). The reason for this plateau is that the second dose empties a large number of the crypts of all stem cells, so these crypts can never be scored however large At. A notable exception to this is for d2 = 10.7 Gy and At -- 50 h. We have no established explanation for this point, which was duplicated in two separate experiments. It could result if in 50 h approximately 60 % of the crypts surviving the first two doses (6 Gy + 10.7 Gy) were able to repopulate adjacent crypts. When the larger dose is the third of the three, there is no observed plateau because the first two doses
would not depopulate completely any of the 160 crypts per circumference. (iii) When At = 5 h, presumably before any significant regeneration occurs, the response differs by a small but statistically significant amount depending upon the order in which the doses are given. In particular, the crypt survival is slightly less when dE is the smaller of the two possible doses. A lack of difference would support the hypothesis of the independent cell killing effect of each dose or the lack of interaction between successive doses. So the data in Figs. 1 and 2 suggest possibly a small inter-dependence of the cell killing effect of separate doses. (iv)The rate of increase of the curves suggests a considerable amount of repopulation. (v) The horizontal separation between the initial ascending segments of the two lines on each graph (at At < 25 h) could be attributed to mitotic delay but it is also confounded by the plateau effect discussed above. Also, the separation of the curves is not constant, making it impossible to determine a single value for the size of the mitotic delay just from the graph. The approximate doubling times in the period At ~< 20 h calculated separately from the four designs assuming an exponential growth are given in Table I. The values in the table suggest that the doubling time is longer when d2 is larger. This is a somewhat surprising result if more severe injury were expected to lead to more rapid recovery. It is possible, however, that cellular injury from the higher dose impairs the ability to repopulate in a manner similar to that which leads to the dose dependent formation of small (petite) colonies after irradiation of cells in vitro [ 10]. Table II shows the parameter estimates for the best fitting regeneration model for the whole data set (MSE = 0.84). The values of a and ct/fl are in reasonable agreement with previous estimates. The estimated doubling time of 10 h is slightly higher than the results in Table I for 3 of the 4 independent experiments. The estimates for fixed delay and mitotic delay are less satisfactory. The mitotic delay estimate of 3 to 4% of the cell cycle time per Gy is equivalent to roughly 0.4 h per Gy. This is about half the value of previous estimates of 1 h per Gy [5,6,11] and appreciably less than TABLE I Estimated doubling times for times ~<20 h between second and third dose. Design 6 6 6 6
Gy Gy Gy Gy
+ + + +
2 Gy + 10.7 Gy 5.1 Gy + 8.4 Gy 8.4 Gy + 5.1 Gy 10.7 Gy + 2 Gy
Doubling time (h)
S.E.
6.5 5.9 8.5 15.8
(6.0, 7.1) (5.5, 6.4)
(7.0, lO.O) (9.8, 21.7)
185 TABLE II
TABLE III
Parameter estimates from best fitting model to the complete data set (n = 761).
Parameter estimates from best fitting model to data with 5 ~< At ~< 20 h (n = 394).
c~( G y - ~) ~/fl (Gy) Doubling time (h) Fixed time delay (h) Mitotic delay per Gy per cell cycle (Gy-~)
Estimate
S.E.
0.228 14.4 10.0 4.4
0.002 0.4 0.3 1.4
0.032
0.006
the in vivo estimates of 1 to 2.8 h per Gy [3]. The estimates of the fixed delay and the mitotic delay are, however, very sensitive to the functional form of the model. For example, if a Gompertzian growth is used instead of a logistic growth, then the estimates of the fixed time delay and the mitotic delay are 0 h and 6~o of the cell cycle time per Gy, respectively. In contrast, the total amount of delay for a particular fractionation scheme, defined as the fixed delay plus the mitotic delay, is less sensitive to the functional form of the model. For example, the total delay after two doses when dl = 6 Gy and d2 = 5.1 Gy, is estimated to be 10.8 h when the logistic model is used and 12.6 h when the Gompertz model is used. Figure 3 a - f shows the observed crypt count and the model predicted counts (solid line) using the parameters in Table II. It can be seen that overall the model is able to match the important features of the data with reasonable accuracy. There are some discrepancies, in particular in Fig. 3b at At = 50 h and to a lesser extent at At = 40 h the model underestimates the crypt score. Also, in all of the figures, the model slightly overestimates the crypt score at At = 5 h, and the curves are slightly too shallow, reflecting the overestimate (10 h) in the doubling time in the early period. In an attempt to obtain an estimate of doubling time which is more consistent with the data in the range of At between 5 and 20 h, the model was refitted to the subset of the data obtained by omitting all animals with At greater than 20 h. This gave a better fit to the data (dashed lines, M S E = 0.42). The parameter estimates are given in Table III. The estimated doubling times are at the lower end of those in Table I, but now we estimate there to be no dose dependent mitotic delay although there is a fixed time delay. Discussion
The in vivo data and mathematical modelling presented in this paper were used to investigate the relationship
(Gy - ! )
o~/fl (Gy) Doubling time (h) Fixed time delay (h) Mitotic delay per Gy per cell cycle (Gy ~)
Estimate
S.E.
0.239 14.8 5.8 11.5
0.005 0.8 0.3 2.8
0.000
between regeneration kinetics (fixed delay, radiationinduced mitotic delay, and clonogen doubling time) and the pattern of dose delivery (dose per fraction, fractionation interval, total dose). The results illustrate that there is a complex relationship between the specific design of the dose regimen and the time course of repopulation. Thus, simple formulas such as the N S D or the linear quadratic model with a time component [14] will not accurately predict acute effects from altered fractionation schemes.
Delay (lag-time) to repopulation The data do not allow firm or consistent conclusions regarding mitotic delay. Repopulation began within 14 h of the initial dose of 6 Gy, but it is not possible to separate the relative roles of mitotic delay and fixed time delay. Our concept of a fixed delay and time dependent mitotic delay determining the time of onset of repopulation may be too simplistic, and other factors such as cell cycle redistribution between dose fractions may play an important, but unrecognized, role. The estimates of lag time in this report are similar to our recent estimates [11] but differ from some other published data. Results from a 2-fraction experiment using a macrocolony cell survival assay [ 18] had suggested a delay of nearly two days after a dose of 6.6 Gy X-rays, although there was an earlier, unexplained wave of recovery. Other experiments [8], using a multifraction scheme, suggested an onset of regeneration at about 18 h after the initial dose. Recent data [9] from labelling index studies after an initial dose of 8 Gy suggested that regeneration may begin after as short a lag time as 3 h. Low values (e.g. < 14 h) for the duration of the lag time before onset of repopulation favor a feedback mechanism from within the crypt [9], rather than as a result of depletion of villi, a process requiring longer times, e.g. 1-3 days.
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Fig. 3. Comparison of observed crypt counts and model predicted counts using the parameters in Table II and III. O = observed data, the solid line is the model prediction, the dashed line is the model prediction when all data with At > 20 h is omitted. (a) (d~ ,d2,d3) = (6 Gy, 2 Gy, 10.7 Gy); (b) (all ,d2,d3) = (6 Gy, 10.7 Gy, 2 Gy); (c) (d~ ,d2,d3) = (6 Gy, 5.1 Gy, 8.4 Gy); (d) (d~ ,d2,d3) = (6 Gy, 8.4 Gy, 5.1 Gy); (e) single dose; (f) pilot experiment, (dl,d2,d3) = (6 Gy, 2 Gy, 10.3 Gy).
187
Clonogenic cell doubling time The present experiments suggest a clonogenic cell doubling time as short as 6 or 7 h, similar to our earlier results [ 11 ], those from a macrocolony assay [ 18], and recent results from studies [ 1] using the same microcolony assay as that used in the present series of experiments. Cairnie [2] estimated a slightly longer doubling time from labeling studies. Potten et al. [ 8] estimated a clonogenic cell doubling time of 21 h, but this seems incompatible with the rate at which microcolonies reappear and grow during the 3 1/2 days needed for the microcolony assay [19] or the 13 days needed for regrowth of large (e.g. > 1 mm) macrocolonies [ 18]. A possible cause for the divergent results obtained by Potten et al. [8] is that they use the crypt clonogenic assay at the extremes ofmeasureable crypt counts (e.g. near zero and greater than 80-100 per circumference), where the usual Poisson assumption may be less "~alid. The notion that lower cell survival levels, that is, greater radiation injury leads to a more active regenerative response, was supported by our previous finding [ 11] that Gompertzian growth gave a better fit than logistic growth, but is not supported by our present results (Table I) and the fact that the Gompertzian model gave a slightly worse fit to the current data than the logistic model.
The mathematical model The mathematical model which we fit to the data gave a good overall description of the data and was able to match the main features of the data, although it was inadequate in some of the finer details. This suggests that refinements or slight modifications might be appropriate, and further experiments may be necessary to develop and test any modifications. The model enabled us to estimate the parameters of the linear-quadratic model and to quantify the rate of repopulation. However, it could not distinguish between the role of mitotic delay and fixed delay in determining the onset of regeneration. This should not be viewed necessarily as a problem with the model but rather as an indication that the data did not provide enough information concerning this issue. The standard errors given in the tables are unrealistically small, and should not be interpreted as a precise measure of our uncertainty in the parameter estimates. They are based on the assumption that the proposed mathematical model gives a good fit to the data, whereas it is clear that the model is inadequate in explaining every detail of the whole data set. Many assumptions were made in developing the mathematical model. The sensitivity of the conclusions
to some of these assumptions was tested by changing the assumption and refitting the model. K value. We assumed there were 100 stem cells per crypt (K) whereas others have suggested higher values of approximately 150 [ 13] or lower values of approximately 40 [7]. Unpublished data from many multifraction neutron experiments at various facilities around the world suggest that the value of K is approximately 100 to 120 [21]. Refitting the model assuming 150 cells per crypt gave a similar but slightly improved fit (MSE = 0.83), whereas assuming 50 cells per crypt gave a worse fit (MSE = 0.86). The estimates of the five parameters changed when different values of K were assumed, but for no parameter was the change larger than 14~o, so for neither value of K would the overall conclusion from the experiment be significantly changed. Survival curve model. We assumed the linear-quadratic model to describe the cell killing, however it is believed that this model is not accurate at high doses. To investigate this, the two-component model (log S = - 0~d + log(1 - (1 - exp( - ~d))M)) was used instead. This resulted in a better fit (MSE = 0.74). However, the estimated parameters of the model were such that the initial exponential shoulder of the survival curve extended to the unreasonably large value of 8 Gy. In addition, the rate of repopulation was very similar to that obtained when the linear-quadratic model was assumed. For these reasons, the two-component model was not pursued further. Probability of post-division clonogenicity. We assumed that P, the fraction of the cells which remain in the stem cell compartment after division during repopulation, was 0.9; changing P to 0.6 or 0.99 made an insignificant small difference to the parameter estimates given in Table II. Cell cycle time. We assumed that the steady state cell cycle time was 18 h; changing this to 12 or 24 h made no difference to the fit of the model, although it did change the mitotic delay per Gy as a fraction of the cell cycle time, but it did not change the estimated amount of mitotic delay following each dose in the experiment. At first sight it may be hard to believe that changing P or the steady state cycle time does not affect the doubling time. However, in the model, these quantities directly affect only the mitotic delay and not the doubling time. During regeneration, there could be both an increase in P and a decrease in the cell cycle time. However, the estimate of the doubling time is measured directly from the data, thus any change in P will have negligible effect on the doubling time. Its only effect within the model will be to alter the estimated cell cycle time to give the same amount of repopulation. Varying the origin of the repopulation curve. In deriving
188 the doubling rates in Table I we assumed a certain number (V) of surviving crypts after the second dose. Varying V by up to 15 ~o made a small difference to the values in Table I but did not affect the overall findings.
Functional subunits and the model One finding from our experiment is the importance of the subunit structure of crypts within the jejunum in determining the response. The structure is the reason the response reached a plateau level for some of the curves in Fig. 1. It is important that this structure be taken into consideration when analyzing data. For the jejunal crypt assay and other colony forming assays this can be achieved using methods of the type described in the Appendix. For other tissues where the structure is known in less detail or the assay is not a direct clonal measure of cell survival, such as skin reactions, the structure may still have an important influence on the measured response. However, it is not obvious how it should be properly taken into consideration because the type of detailed computations given in the Appendix are not possible.
Alternative mathematical models The mathematical model discussed in the present paper has five parameters. As with any model, it is only an empirical approximation to the underlying mechanism. Many other factors could be taken into consideration which would make the model more complex and hopefully more realistic. For example, the known differences in cell cycle sensitivities could be incorporated, or a pipeline [16] or compartment of differentiated cells which influence the repopulation kinetics of the clonogenic cells could be included. Such a compartment may be necessary for any modelling of skin reactions because for this assay the measured response is thought to be directly related to the depletion of the functional cell compartment and only indirectly related to the stem cell comparment. Mathematical models with many more than five parameters have the advantage of being flexible and capable of fitting almost any data set. However they have the overwhelming disadvantage of requiring extremely large and high quality data sets to obtain reasonable estimates of the parameters. For clinical data, it is extremely difficult to obtain accurate
parameter estimates for models which contain as few as three parameters. In addition, predictions from mathematical models with many parameters tend to be highly sensitive to the choice of parameter values in addition to being sensitive to the specific choice of the equations defining the mathematical model. Similar reservations about the problems of multiparameter mathematical models have been expressed previously [12]. Thus, complex nonlinear models with more than about seven parameters are unlikely to be of much use for predicting acute effects in the clinical setting. The role of such models should be confined to that of qualitatively suggesting further experiments or new fractionation schemes to investigate.
Clinical relevance of present experiments The particular fraction sizes and fraction numbers used in the experiment described in this article differ greatly from those used in conventional radiotherapy. As such, direct extrapolation to the clinic is not possible and was not the aim of the experiment. The purpose of the experiment was to gain some understanding of the regenerative response of an acutely responding tissue in the time period immediately following the initial radiation insult. The conclusion important to radiotherapists is that even in a simple fractionation experiment the acute response depends in a complex way on the fractionation scheme. Therefore, in a clinical setting, a simple formula which claims to predict the acute response is unlikely to be accurate. Before any fractionation scheme in radiotherapy is based on a mathematical model of acute effects, the model and possible competitors should be thoroughly and properly tested on several large real data sets. Since such data sets do not exist, it will be necessary for radiotherapists to collect detailed clinical data on acute effects, with levels of injury documented, throughout entire fractionation schemes.
Acknowledgements This work was supported by PHS Grant numbers CA-45216 and CA-29644 awarded by the National Cancer Institute, DHHS. The authors thank Jan Haas, Yeh-Chi Lo and Vijaya Vegesna for assisting with this work.
189
References 1 Blott, P. and Trott, K.R. The effect of actinomycin D on splitdose recovery and repopulation in jejunal crypt cells in vivo. Radiother. Oncol. 15: 73-78, 1989. 2 Cairnie, A. B., Lamerton, L. F. and Steel, G.G. Cell proliferation studies in the intestinal epithelium of the rat. I. Determination of the kinetic parameters. Exp. Cell Res. 39: 528-538, 1965. 3 Chwalinski, S. and Potten, C. S. Radiation-induced mitotic delay: duration, dose and cell position dependence in the crypts of the small intestine in the mouse. Int. J. Radiat. Biol. 49:809-819, 1986. 4 Cohen, L. A cell population kinetic model for fractionated radiation therapy. I. Normal tissues. Radiology 101: 419-427, 1971. 5 Elkind, M. M., Han, A. and Volz, K.W. Radiation response of mammalian cells grown in culture. IV. Dose dependence of division delay and post irradiation growth of surviving and nonsurviving Chinese hamster cells. J. Natl. Cancer Inst. 30: 705-721, 1963. 6 Hegazy, M. A. H. and Fowler, J.F. Cell population kinetics of plucked and unplucked mouse skin. II. Irradiated skin. Cell Tissue Kinet. 6: 587-602, 1973. 7 Hendry, J. H., Potten, C. S., Ghafoor, A., Moore, J. V., Roberts, S.A. and Williams, P.C. The response of murine intestinal crypts to short-range Promethium-147 irradiation: deductions concerning clonogenic cell numbers and positions. Radiat. Res. 118: 364-374, 1989. 8 Potten, C. S., Taylor, Y. and Hendry, J.H. The doubling time of regenerating clonogenic cells in the crypts of the irradiated mouse small intestine. Int. J. Radiat. Biol. 54: 1041-1051, 1988. 9 Potten, C. S., Owen, G. and Roberts, S.A. The temporal and spatial changes in cell proliferation within the irradiated crypts of the murine small intestine. Int. J. Radiat. Biol. 57: 185-199, 1990.
10 Sinclair, W.K. X induced heritable damage (small colony formation) in cultured mammalian cells. Radiat. Res. 21:584-611, 1964. 11 Taylor, J. M. G., Withers, H. R. and Hu, Z. A comparison of mathematical models for regeneration in acutely responding tissues. Int. J. Radiat. Oncol. Biol. Phys. 15: 1389-1400, 1988. 12 Thames, H.D. A new model of proliferative response to fractionated irradiation. Radiother. Oncol. 13: 311-313, 1988. 13 Thames, H.D., Withers, H.R., Mason, K.A. and Reid, B.O. Dose-survival characteristics of mouse jejunal crypt cells. Int. J. Radiat. Oncol. Biol. Phys. 7: 1591-1597, 1981. 14 Travis, E. and Tucker, S. Isoeffect models and fractionated radiotherapy. Int. J. Radiat. Oncol. Biol. Phys. 13: 283-287, 1987. 15 Tucker, S. L., Thames, H. D. and Taylor, J. M.G. How well is the probability of tumor cure after fractionated irradiation described by Poisson statistics? Radiat. Res. 124: 273-282, 1990. 16 van de Geijn, J. Time-dose response of human tumors and normal tissues during and after fractionated radiation treatment. A new model. Radiother. Oncol. 12: 57-78, 1988. 17 Withers, H. R., Chu, A. M., Reid, B. O. and Hussey, D.H. Response of mouse jejunum to multifraction radiation. Int. J. Radiat. Oncol. Biol, Phys. 1: 41-52, 1975. 18 Withers, H. R. and Elkind, M.M. Radiosensitivity and fractionation response of crypt cells of mouse jejunum. Radiat. Res. 38: 598-613, 1969. 19 Withers, H. R. and Elkind, M.M. Microcolony survival assay for cells of mouse intestinal mucosa exposed to radiation. Int. J. Radiat. Biol. 17: 261-267, 1970. 20 Withers, H. R. and Mason, K.A. The kinetics of recovery in irradiated colonic mucosa of the mouse. Cancer 34: 896-903, 1974. 21 Withers, H. R., Mason, K. A. and Taylor, J. M.G. The number of clonogenic cells in a mouse jejunum crypt. Radiother. Oncol., submitted, 1990.
Appendix Derivation o f the Eqn. 1, the expected crypt score in a 3-dose experiment involving repopulation. The measured score in the assay is the n u m b e r o f surviving crypts, treating each of the 160 crypts as a separate unit. The assumption is that the crypt is scored if it has one or more surviving clonogenic cells following the final dose o f irradiation. A corollary to this is that for a crypt to be scored it must maintain at least one clonogenic cell throughout the course o f the regimen. So, for example, if half o f the crypts are completely depleted o f cells prior to the final dose, then however long a time period is allowed for regeneration, these crypts can never repopulate themselves so they will not contribute to the assayed crypt count. In summary, the functional subunit structure o f the jejunum is influencing the measured response. The following mathematical derivation is an a p p r o a c h which allows for the possible depletion of crypts during the course o f the regimen. We will specialize to the 3-dose experiment described in this
paper, although the same a p p r o a c h can be applied for other fractionation schemes. Let d~, d2 and d3 be the 3 doses, and let Pi be the surviving fraction o f cells from a single dose o f size di and Qi = 1 - P i . T h r o u g h o u t we will be considering only clonogenic cells. Consider a single crypt, let K be the initial cell count, let Z~ be the cell c o u n t immediately following dl, let Z2b be the cell c o u n t just preceding d2, let Z2a be the cell count just after d2, let Z3b be the cell count just prior to d 3 and let Z3a be the cell count just after d3. So if Z3a >/1 the crypt is scored, but if Z3a = 0 then the crypt is not scored. The five Z s and K are related in the following way: the distribution o f Z~ is binomial (K, P~ ). The values of Zzb and Z 1 are related by the equation Z2b = alZ~, where al >f 1, with al > 1 if repopulation has occurred between first and second dose. Because al will typically depend on the value of Z~, it was more convenient to use the approximate equation Zzb = gl + hlZ~, where gl and h~ do not
190 depend on Z1. Conditional on the value of Zzb, the distribution of Z=, is binomial (Z2b, /2). The values of Z3b and Z2~ are related by the equation Z3b = a2Z2a, where a2 t> 1, with a2 > 1 if repopulation has occurred between the second and third dose. Again, it was more convenient to use the approximate equation Z3b = g2 + hzZza" Conditional on the value of Z3b, the distribution of Z3~ is binomial (Zsb , P3)" The moment generating function G(U) of the random variable Z3a is defined as G(U) = E (UZ3~), where the expectation is taken with respect to the distribution of Z3a- After lengthy algebra it can be shown that G(U) = Q I~ + Q gl {(QI + elQzh~)K _ Q r } + (Q3 + e 3 u ) g~ [ - Q2gl {(Q1 + e , Q = h , : _ Q r } + (Q2 + e2 (Q3 + e 3 u ) h 2 ) ~' {(Q1 + el (02 + e2 (03 + e 3 u ) h 2 ) h l ) g -- Q,K}I
A property of moment generating functions is that G(0) equals the probability that Z3~ = 0. So the crypt will be counted with probability 1 - G(0) and will not be counted with probability G(0). Thus, the expected number of crypts which will be counted is: E ( C ) = 160(1 - G(0)), where G(O) = Q1x + (I - Q3g2)Qg ' {(Q1 + ? l Q 2 h l ) K -- Q1K} + Q3g2 (Q2 + P2Q3h~)gl {(Q, + ? , (Q2 + ?2Q3h:)hl) K - Qar} •
To validate Eqn. 1 consider some special cases. (i) No repopulation at all during the experiment, then a l = a 2 = 1, g i = g 2 = 0 , h i = h 2 = 1 then E ( C ) becomes 160(1 - G(0)) = 160{1 - [1 - e l e 2 e 3 ] r } , which is the correct form for the expected number of crypts from 3 doses acting independently and cumulatively. (ii) There are very long times between all three fractions so that al and a 2 are both large, gl = g2 = K and hi - - h 2 = 0, and that P2 and P3 a r e both large enough so that either the second or third dose cannot completely empty any full crypts of cells. Then E ( C ) = 1 6 0 { 1 - ( 1 - P 1 ) r } , which is the expected crypt count from a single dose of size d~. (iii) If there is a short time between the first and second doses but a long time between the second and third dose, so that a 1 ---- 1, gl = 0, h I = 1, but a 2 is large and g2 = K, h 2 = 0 and the third dose is not too large, then E ( C ) = 1 6 0 ( 1 - ( 1 - P 1 P 2 ) r ) , which is the expected crypt count from 2 doses of size d 1 and d2. The values orgy, g2, hi and h 2 in Eqn. 1 depend on the time interval between the doses and the kinetic
model of repopulation. If the logistic repopulation is used and let S(t) be the fraction of clonogenic cells in each crypt, then d S / d t = 2S(1 - S), with solution S(t) = Sol(So + (1 - So)e-at).
If So ( = Z ~ / K ) is the fraction of cells just after d~ and S(t) ( = Z2b/K ) is the fraction of cells just before d2 then Z2b = Z 1 ( g / ( z I + ( g - Z i ) e - a t ) ) , where t is the time available for repopulation between dl and dE. Thus, the factor a~ is equal to K/(Z1 + ( K - Z 1 ) e - ~ t ) . Notice that al depends on Z~, but we have assumed that a~ is a constant, but if we replace Z1 by its expected value (= K e l ) then al = 1/(P1 + (1 - el)e-~t). In a similar way we can show that a2 -- 1/(aiPiP2 + (1 - alP1P2)e - at) where t is the time available for repopulation between the second and third doses. We can also show that Z2b = g~ + h~Z~ is a good approximation where gl = Kp12al 2 (1 - e -at) and hl = (a~ - P~ai 2 (1 - e-at));
similarly Z3b = g2 + h2Z2a
where g2 = Ka12PlZp22az2 (1 - e -at) and h2 = (az - alPiP2a22 (1 - e - a t ) )
The validity of these approximations was checked by comparing the expected crypt score using the approximation to the average of some Monte Carlo simulations of Z3a using the exact binomial distributions for Z1, Z2a and Z3a described above. The approximations were found to be very good. Slightly different expressions for gi, gz, hl and h 2 are needed if Gompertz growth or growth based on the equation dS/dt = 2 J . S 0 - S ~/2) is used instead of logistic growth. Another approximation in the derivation of Eqn. 1 is that all the Z's must be integers, but the relationships ZZb = gl + hlZ~ and Z3b = gz + haZza do not necessarily guarantee this because gl, g2, hi, h2 are continuous functions. We will assume that the problems associated with this lack of discreteness are of minor importance. The approach described in this appendix can be adapted to obtain estimates of the probability of cure in growing tumors [15].
181
RADION00818
Repopulation of mouse jejunal crypt cells J e r e m y M . G . T a y l o r , H . R o d n e y W i t h e r s , K a t h y A. M a s o n a n d Cally A. D a v i s Department of Radiation Oncology and Jonsson Comprehensive Cancer Center, UCLA Medical Center, Los Angeles, CA, U.S.A.
(Received 26 March 1990, revision received 6 November 1990, accepted 5 December 1990)
Key words: Acute effects; Mathematical model; Logistic growth; Regeneration; Mitotic delay
Summary The regenerative response to radiation of mouse jejunal crypt cells was investigated using a three fraction experiment. The time between the second and third fraction was varied between 5 and 60 h, and the dose in each of the three fractions was different. The data show that the onset of regeneration is within 14 h of the first dose and possibly earlier. The doubling time of the clonogenic cells during repopulation is estimated to be between 5 and 10 h, with the most likely value approximately 6 to 7 h. The data also show that the time course of repopulation in an acutely responding tissue depended in a complex way on the fractionation scheme. The implications of this for radiotherapy are that simple formulas are unlikely to be accurate predictors of acute effects in altered fractionation schemes. Detailed mathematical modelling of the data is undertaken using a model which consists of a single dose survival curve, a part to incorporate the regenerative response and a part to accommodate the delayed onset of regeneration. The model is shown to give a good, although not perfect, fit to the data. A mathematical derivation is given of the expected number of crypts following the 3 dose radiation schedule. This derivation takes into account the fact that any crypt could be denuded of cells prior to the final dose and hence not repopulate, and thus the formula given is necessarily more complex than previous formulas which were based on simpler models.
Introduction With the increasing use of accelerated treatment in radiotherapy, the acute effects are becoming more important and may be dose limiting for some regimens. This suggests the need for a complete understanding of acute effects and in particular the kinetics of the regenerative response to radiation insult during treatment. Previous data [18,19] using a jejunal colony assay system had suggested that there was a burst of repopulation one to two days after the initial dose in experiments which consisted of 2 doses separated by varying time intervals. However, other data for colonic crypt cells [20] and mathematical modelling of fractionated and more recent data [ 11] suggest that the onset of regeneration is earlier than one to two days after the initial dose and could be as early as 3 h [9]. These data [ 11 ] also suggest that the delay before the onset of regeneration was dependent on the dose and
could be explained by a cell cycle delay proportional to the size of the dose. A further experiment was designed to explicitly test these findings.
Materials and methods Animals
Female C 3 H f / S e d / / K a m specific pathogen-free mice aged 9 to 15weeks were used (n = 761). A 12h light/dark cycle was used. There were 20 animals at most of the dose points. Irradiation
Groups of 10 mice were simultaneously whole-body irradiated using a G a m m a Cell 40 (AECL) cesium source at dose rates of 111 to 113 cGy per minute.
Addressfor correspondence:J. M. G. Taylor, Department of Radiation Oncology,UCLA Medical Center, Los Angeles, CA 90024-1714, U.S.A.
0167-8140/91/$03.50 © 1991 Elsevier Science Publishers B.V. (Biomedical Division)
182 Microcolony assay
Statistical methods and mathematical models
The microcolony as say of Withers and Elkind [ 18,19 ] was used. Briefly, animals were sacrificed 3 days and 14 h after the midpoint of each experimental regimen. A 2-3 cm length of jejunum was excised and fixed in neutral buffered formalin. Following routine histological processing, sections were cut at 5 #m and stained with hemotoxylin and eosin. Surviving crypts per circumference of jejunum were scored microscopically using 100 x magnification. Three to five transverse sections of jejunum were scored for each animal. All histological sections were scored by one observer (K. A. M.). The average score for each animal was used in the analysis.
Means and 95 ~ confidence intervals for the number of crypts per circumference are shown in Figs. 1 and 2. The rate of repopulation between the second and third doses is calculated independently for each of the 4 experiments by assuming an exponential growth over the range of At between 5 and 20 h. The model
Experimental design Following a pilot study, the experiment consisted of three fractions, an initial dose of 6 Gy followed 4 h later by dose d2 Gy and then a further time At hours later a dose of size d3 Gy. Atis either 5, 10, 15, 20, 25, 30, 35, 40, 50, or 60 h, and there were 4 sets of the (d2,d3) pair, (d2,d3) = (2.0,10.7), (10.7,2.0), (5.1,8.4) or (8.4,5.1). In addition, data from a range of single doses was collected. Also, data was collected from a pair of two fraction dose points, in which the initial dose of 6 Gy was followed 4 h later by either 10.7 Gy or 8.4 Gy. The minimum separation between two doses is 4 h which is sufficient time for complete repair of sublethal injury. The rationale for this 3-dose design, with a fixed priming dose followed by two test doses, was to insure that the priming dose rather than either of the test doses was responsible for activating the mechanisms which turn on repopulation. The initial dose of 6 Gy was considered sufficiently large to reduce cell survival to a low enough level ( < 0.2) so that the mechanisms controlling repopulation would be activated. The choice of the pairs (2.0,10.7) and (5.1,8.4) was so that, in the absence ofrepopulation, all 4 sets should give approximately the same cell killing. The relatively short times for At were chosen to emphasize the onset of regeneration and the initial rate of repopulation. Both orders of the pair 2.0 Gy and 10.7 Gy and the pair 5.1 Gy and 8.4 Gy were used to ascertain how the fractionation scheme affected the onset of regeneration. Previous analysis [ 11] had suggested that there was a cell cycle delay following each irradiation proportional to the dose per fraction, in which case the onset of regeneration would be later in the scheme with the second dose larger than the third.
log( - log(1 - C/V) = A + At. log2/DT was fit to the data separately for each of the four designs using non-linear least squares. In this model, the parameters DT ( = doubling time) and A are estimated, C is the observed crypt score and V is the number of crypts containing at least one clonogen after the second dose. The assumed values of V were 160, 150, 82, and 32 for d2 = 2.0, 5.1, 8.4, and 10.7, respectively. In addition, a variety of mathematical models were fit to the complete data in an attempt to jointly estimate the effects of cell killing and regeneration. These models and the associated statistical methods are slight adaptations of those described in Taylor etal. [11], and readers are referred to that article for complete details. The models are similar in spirit, but different in detail, to the models described by Cohen [4] and van de Geijn [ 16]. Briefly, the basic model consists of a single dose survival curve, either the linear-quadratic model or the two-component model. Time to the onset of regeneration is determined by either a dose-dependent mitotic delay following each irradiation dose or a fixed time delay after the initial dose, or both, or neither; and the repopulation kinetics is determined by either logistic or Gompertz growth. The possibility of a fixed delay was included in the model to represent the time interval following the initial dose before which damage in the stem cell compartments is recognized. In these models the final cell survival level is calculated by tracking the cell survival level throughout the fractionation scheme. This requires allowing each separate dose to cause a reduction in the cell survival level and for a possible increase in the cell survival level due to repopulation during the periods between fractions. With this model, the final cell survival level can be explicitly specified in a possibly complex equation. Three changes were made to the mathematical models previously described [ 11 ] in order that it will more correctly represent the underlying biology. The first change is that periods of mitotic delay are now assumed to be cumulative. For example, if a dose of 6 Gy causes 6 h mitotic delay and if2 doses of 6 Gy are separated by 4 h, then the total delay is assumed 12 h rather than the 10 h as was used in the previous analysis.
183 The second change is that the length of the mitotic delay after each dose, which is proportional to dose per fraction and the duration of the cell cycle, is reduced during repopulation due to the shortened cell cycle time, whereas in the previous analysis the length of the mitotic delay was not reduced during periods of rapid repopulation. In particular, mitotic delay = r × dose × cell cycle time, where the cell cycle time is the average cycle time between mitoses during repopulation and is given by cell cycle time = minimum (18 h, log(2P) × doubling time/log(2)), where P is the fraction of cells which remain in the stem cell compartment after division, and the cell doubling time is obtained from the kinetic growth model of repopulation. The parameter r is estimated from the data. The quantity P is in the range 0.5 (steady state) to 1.0 (maximum possible) and 18 h is the assumed steady state cell cycle time and also the maximum possible cell cycle time during repopulation. P is 0.5 before repopulation begins, and we assumed that P = 0.9 during repopulation. We chose a high value of P because previous data had suggested that the doubling time would be about 6 to 8 h [1,11]. As the cell cycle time is unlikely to decrease below about 5 h, a high value of P is necessary to ensure a low doubling time. Sensitivity analyses were performed to assess the effect of different values of P and the steady state cell cycle time. For the logistic growth model dlog S / d t = 2(1 - S), where S is the cell surviving fraction; the doubling time = log 2/(2(1 - S)). The calculation of the doubling times in the tables is based on the situation where S is small, i.e. the doubling time is calculated as log 2/2. The third change is that the functional subunit structure of the crypts within the jejunum is taken into account. Previous modelling assumed that the whole section of jejunum could repopulate given sufficient time however low the cell surviving fraction became during the course of the irradiations. However, in reality, there are a finite number of cells in each crypt so that once a crypt is completely denuded of cells it cannot repopulate itself and hence will never be scored in the assay. We also assumed that repopulation from adjacent crypts cannot occur within the duration of the experiment. The following equation for the expected crypt score incorporates this structure of the crypt: 1 - E(Crypt score)/160 = Q1K + (1 - Q392)Q2g' {(Q1 + e~o2h') 1¢ - Q K} + 03 g2 (Q2 + P203*2)g' {(Q, + P, (Qz + PzQ3h2)*') r - Qi/':} •
(1)
where 160 is the control number of crypts per circumference, Pi is the surviving fraction of cells from a single dose of size di, Qi = 1 - Pi, K is the initial number of clonogenic cells per crypt, gi and h i determine the approximate increase in the number of cells due to repopulation between the ith and (i + 1)th dose. We assumed the value of K to be 100 for the majority of the analyses, but also considered both higher and lower values in a sensitivity analysis. The Appendix to this paper contains the derivations of Eqn. 1 and the expressions for gi and h i. Throughout this paper, unless otherwise stated, we used the linear-quadratic model for the single dose survival curve and logistic growth to model the kinetics of regeneration. We also always included both a fixed time delay and a radiation-induced mitotic delay within the model. The parameters of the models were fitted by nonlinear unweighted least-squares regression on the square root scale of crypt score using the NLIN procedure from the SAS statistical software. That is, we minimized with respect to the parameters Z(Cj 1/2 2, where Cj is the observed crypt score for animal j and E(Cj) is the expected crypt score for animal j which depends on the unknown parameters of the mathematical model and is given by Eqn. 1. The rationale for this least squares approach has been described previously [ 11 ]. There are five parameters in the model: ~ and ~//3 which determine the linearquadratic model, r which determines the mitotic delay, Tf which determines the fixed time delay and 2 which determines the rate of repopulation and hence the doubling time. The mean squared error is used to assess each model, the smaller value indicating a better fit. All the data, including the pilot study (n = 12) and the single dose control data (n = 45), is used in fitting the models.
--(E(Cj))'/2)
Results
Figures 1 and 2 show the results for the main part of the 3-fraction experiment. The figures do not show the data for the single dose or pilot experiment, but they do show the results of the two dose experiments consisting of 6 Gy plus d2 (denoted by A on the horizontal axis). The features of the graphs which are of particular interest are (i) the onset time of the increased response, assumed to be due to repopulation, is within 14 h of the initial dose of 6 Gy for all 4 curves and could be much earlier with the possible exception of the design in which d 2 = 10.7 Gy. (ii) There is a very noticeable difference between curves depending upon the order of the different sized second and third doses. When the larger
184 ~ IO0-
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E Z
5 0
IO
,
,
,
,
20
50
40
50
H
60
,
A
At = Time between 2nd ond 5rd fraction (hours)
Fig. 1. Average crypt counts plotted against At for a 3-fraction experiment (dl, d2, d3). d l = 6 Gy, the time interval between d~ and d2 is 4 h, the time interval between d2 and d3 is At h. © is d2 = 2 Gy, d3 = 10.7 Gy. A is d2 = 10.7 Gy, d3 = 2 Gy. The error bars are 95% confidence intervals. The point at At = A is the score from the 2-fraction experiment with (dr = 6 Gy and d2 = 10.7 Gy).
o lO0u
E 50
~
IO-
E 5
L
o
,o At = Time
,
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II
3o 2nd
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io
i
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5rd fraction (hours)
Fig. 2. Average crypt counts plotted against At for a 3-fraction experiment (dl, d2, d3). d~ = 6 Gy, the time interval between dl and d2 is 4 h , the time interval between d2 and d3 is At h. O is d2 = 5.1 Gy, d3 = 8.4 Gy, A is d2 = 8.4 Gy, d3 = 5.1 Gy. The error bars are 95% confidence intervals. The point at At = A is the score from the 2-fraction experiment with (d~ = 6 Gy and d2 = 8.4 Gy).
dose is the second of the three, the response reaches a plateau level for large At, which is similar to the score that results when only the first two doses are given (At -- A in Figs. 1 and 2). The reason for this plateau is that the second dose empties a large number of the crypts of all stem cells, so these crypts can never be scored however large At. A notable exception to this is for d2 = 10.7 Gy and At -- 50 h. We have no established explanation for this point, which was duplicated in two separate experiments. It could result if in 50 h approximately 60 % of the crypts surviving the first two doses (6 Gy + 10.7 Gy) were able to repopulate adjacent crypts. When the larger dose is the third of the three, there is no observed plateau because the first two doses
would not depopulate completely any of the 160 crypts per circumference. (iii) When At = 5 h, presumably before any significant regeneration occurs, the response differs by a small but statistically significant amount depending upon the order in which the doses are given. In particular, the crypt survival is slightly less when dE is the smaller of the two possible doses. A lack of difference would support the hypothesis of the independent cell killing effect of each dose or the lack of interaction between successive doses. So the data in Figs. 1 and 2 suggest possibly a small inter-dependence of the cell killing effect of separate doses. (iv)The rate of increase of the curves suggests a considerable amount of repopulation. (v) The horizontal separation between the initial ascending segments of the two lines on each graph (at At < 25 h) could be attributed to mitotic delay but it is also confounded by the plateau effect discussed above. Also, the separation of the curves is not constant, making it impossible to determine a single value for the size of the mitotic delay just from the graph. The approximate doubling times in the period At ~< 20 h calculated separately from the four designs assuming an exponential growth are given in Table I. The values in the table suggest that the doubling time is longer when d2 is larger. This is a somewhat surprising result if more severe injury were expected to lead to more rapid recovery. It is possible, however, that cellular injury from the higher dose impairs the ability to repopulate in a manner similar to that which leads to the dose dependent formation of small (petite) colonies after irradiation of cells in vitro [ 10]. Table II shows the parameter estimates for the best fitting regeneration model for the whole data set (MSE = 0.84). The values of a and ct/fl are in reasonable agreement with previous estimates. The estimated doubling time of 10 h is slightly higher than the results in Table I for 3 of the 4 independent experiments. The estimates for fixed delay and mitotic delay are less satisfactory. The mitotic delay estimate of 3 to 4% of the cell cycle time per Gy is equivalent to roughly 0.4 h per Gy. This is about half the value of previous estimates of 1 h per Gy [5,6,11] and appreciably less than TABLE I Estimated doubling times for times ~<20 h between second and third dose. Design 6 6 6 6
Gy Gy Gy Gy
+ + + +
2 Gy + 10.7 Gy 5.1 Gy + 8.4 Gy 8.4 Gy + 5.1 Gy 10.7 Gy + 2 Gy
Doubling time (h)
S.E.
6.5 5.9 8.5 15.8
(6.0, 7.1) (5.5, 6.4)
(7.0, lO.O) (9.8, 21.7)
185 TABLE II
TABLE III
Parameter estimates from best fitting model to the complete data set (n = 761).
Parameter estimates from best fitting model to data with 5 ~< At ~< 20 h (n = 394).
c~( G y - ~) ~/fl (Gy) Doubling time (h) Fixed time delay (h) Mitotic delay per Gy per cell cycle (Gy-~)
Estimate
S.E.
0.228 14.4 10.0 4.4
0.002 0.4 0.3 1.4
0.032
0.006
the in vivo estimates of 1 to 2.8 h per Gy [3]. The estimates of the fixed delay and the mitotic delay are, however, very sensitive to the functional form of the model. For example, if a Gompertzian growth is used instead of a logistic growth, then the estimates of the fixed time delay and the mitotic delay are 0 h and 6~o of the cell cycle time per Gy, respectively. In contrast, the total amount of delay for a particular fractionation scheme, defined as the fixed delay plus the mitotic delay, is less sensitive to the functional form of the model. For example, the total delay after two doses when dl = 6 Gy and d2 = 5.1 Gy, is estimated to be 10.8 h when the logistic model is used and 12.6 h when the Gompertz model is used. Figure 3 a - f shows the observed crypt count and the model predicted counts (solid line) using the parameters in Table II. It can be seen that overall the model is able to match the important features of the data with reasonable accuracy. There are some discrepancies, in particular in Fig. 3b at At = 50 h and to a lesser extent at At = 40 h the model underestimates the crypt score. Also, in all of the figures, the model slightly overestimates the crypt score at At = 5 h, and the curves are slightly too shallow, reflecting the overestimate (10 h) in the doubling time in the early period. In an attempt to obtain an estimate of doubling time which is more consistent with the data in the range of At between 5 and 20 h, the model was refitted to the subset of the data obtained by omitting all animals with At greater than 20 h. This gave a better fit to the data (dashed lines, M S E = 0.42). The parameter estimates are given in Table III. The estimated doubling times are at the lower end of those in Table I, but now we estimate there to be no dose dependent mitotic delay although there is a fixed time delay. Discussion
The in vivo data and mathematical modelling presented in this paper were used to investigate the relationship
(Gy - ! )
o~/fl (Gy) Doubling time (h) Fixed time delay (h) Mitotic delay per Gy per cell cycle (Gy ~)
Estimate
S.E.
0.239 14.8 5.8 11.5
0.005 0.8 0.3 2.8
0.000
between regeneration kinetics (fixed delay, radiationinduced mitotic delay, and clonogen doubling time) and the pattern of dose delivery (dose per fraction, fractionation interval, total dose). The results illustrate that there is a complex relationship between the specific design of the dose regimen and the time course of repopulation. Thus, simple formulas such as the N S D or the linear quadratic model with a time component [14] will not accurately predict acute effects from altered fractionation schemes.
Delay (lag-time) to repopulation The data do not allow firm or consistent conclusions regarding mitotic delay. Repopulation began within 14 h of the initial dose of 6 Gy, but it is not possible to separate the relative roles of mitotic delay and fixed time delay. Our concept of a fixed delay and time dependent mitotic delay determining the time of onset of repopulation may be too simplistic, and other factors such as cell cycle redistribution between dose fractions may play an important, but unrecognized, role. The estimates of lag time in this report are similar to our recent estimates [11] but differ from some other published data. Results from a 2-fraction experiment using a macrocolony cell survival assay [ 18] had suggested a delay of nearly two days after a dose of 6.6 Gy X-rays, although there was an earlier, unexplained wave of recovery. Other experiments [8], using a multifraction scheme, suggested an onset of regeneration at about 18 h after the initial dose. Recent data [9] from labelling index studies after an initial dose of 8 Gy suggested that regeneration may begin after as short a lag time as 3 h. Low values (e.g. < 14 h) for the duration of the lag time before onset of repopulation favor a feedback mechanism from within the crypt [9], rather than as a result of depletion of villi, a process requiring longer times, e.g. 1-3 days.
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Fig. 3. Comparison of observed crypt counts and model predicted counts using the parameters in Table II and III. O = observed data, the solid line is the model prediction, the dashed line is the model prediction when all data with At > 20 h is omitted. (a) (d~ ,d2,d3) = (6 Gy, 2 Gy, 10.7 Gy); (b) (all ,d2,d3) = (6 Gy, 10.7 Gy, 2 Gy); (c) (d~ ,d2,d3) = (6 Gy, 5.1 Gy, 8.4 Gy); (d) (d~ ,d2,d3) = (6 Gy, 8.4 Gy, 5.1 Gy); (e) single dose; (f) pilot experiment, (dl,d2,d3) = (6 Gy, 2 Gy, 10.3 Gy).
187
Clonogenic cell doubling time The present experiments suggest a clonogenic cell doubling time as short as 6 or 7 h, similar to our earlier results [ 11 ], those from a macrocolony assay [ 18], and recent results from studies [ 1] using the same microcolony assay as that used in the present series of experiments. Cairnie [2] estimated a slightly longer doubling time from labeling studies. Potten et al. [ 8] estimated a clonogenic cell doubling time of 21 h, but this seems incompatible with the rate at which microcolonies reappear and grow during the 3 1/2 days needed for the microcolony assay [19] or the 13 days needed for regrowth of large (e.g. > 1 mm) macrocolonies [ 18]. A possible cause for the divergent results obtained by Potten et al. [8] is that they use the crypt clonogenic assay at the extremes ofmeasureable crypt counts (e.g. near zero and greater than 80-100 per circumference), where the usual Poisson assumption may be less "~alid. The notion that lower cell survival levels, that is, greater radiation injury leads to a more active regenerative response, was supported by our previous finding [ 11] that Gompertzian growth gave a better fit than logistic growth, but is not supported by our present results (Table I) and the fact that the Gompertzian model gave a slightly worse fit to the current data than the logistic model.
The mathematical model The mathematical model which we fit to the data gave a good overall description of the data and was able to match the main features of the data, although it was inadequate in some of the finer details. This suggests that refinements or slight modifications might be appropriate, and further experiments may be necessary to develop and test any modifications. The model enabled us to estimate the parameters of the linear-quadratic model and to quantify the rate of repopulation. However, it could not distinguish between the role of mitotic delay and fixed delay in determining the onset of regeneration. This should not be viewed necessarily as a problem with the model but rather as an indication that the data did not provide enough information concerning this issue. The standard errors given in the tables are unrealistically small, and should not be interpreted as a precise measure of our uncertainty in the parameter estimates. They are based on the assumption that the proposed mathematical model gives a good fit to the data, whereas it is clear that the model is inadequate in explaining every detail of the whole data set. Many assumptions were made in developing the mathematical model. The sensitivity of the conclusions
to some of these assumptions was tested by changing the assumption and refitting the model. K value. We assumed there were 100 stem cells per crypt (K) whereas others have suggested higher values of approximately 150 [ 13] or lower values of approximately 40 [7]. Unpublished data from many multifraction neutron experiments at various facilities around the world suggest that the value of K is approximately 100 to 120 [21]. Refitting the model assuming 150 cells per crypt gave a similar but slightly improved fit (MSE = 0.83), whereas assuming 50 cells per crypt gave a worse fit (MSE = 0.86). The estimates of the five parameters changed when different values of K were assumed, but for no parameter was the change larger than 14~o, so for neither value of K would the overall conclusion from the experiment be significantly changed. Survival curve model. We assumed the linear-quadratic model to describe the cell killing, however it is believed that this model is not accurate at high doses. To investigate this, the two-component model (log S = - 0~d + log(1 - (1 - exp( - ~d))M)) was used instead. This resulted in a better fit (MSE = 0.74). However, the estimated parameters of the model were such that the initial exponential shoulder of the survival curve extended to the unreasonably large value of 8 Gy. In addition, the rate of repopulation was very similar to that obtained when the linear-quadratic model was assumed. For these reasons, the two-component model was not pursued further. Probability of post-division clonogenicity. We assumed that P, the fraction of the cells which remain in the stem cell compartment after division during repopulation, was 0.9; changing P to 0.6 or 0.99 made an insignificant small difference to the parameter estimates given in Table II. Cell cycle time. We assumed that the steady state cell cycle time was 18 h; changing this to 12 or 24 h made no difference to the fit of the model, although it did change the mitotic delay per Gy as a fraction of the cell cycle time, but it did not change the estimated amount of mitotic delay following each dose in the experiment. At first sight it may be hard to believe that changing P or the steady state cycle time does not affect the doubling time. However, in the model, these quantities directly affect only the mitotic delay and not the doubling time. During regeneration, there could be both an increase in P and a decrease in the cell cycle time. However, the estimate of the doubling time is measured directly from the data, thus any change in P will have negligible effect on the doubling time. Its only effect within the model will be to alter the estimated cell cycle time to give the same amount of repopulation. Varying the origin of the repopulation curve. In deriving
188 the doubling rates in Table I we assumed a certain number (V) of surviving crypts after the second dose. Varying V by up to 15 ~o made a small difference to the values in Table I but did not affect the overall findings.
Functional subunits and the model One finding from our experiment is the importance of the subunit structure of crypts within the jejunum in determining the response. The structure is the reason the response reached a plateau level for some of the curves in Fig. 1. It is important that this structure be taken into consideration when analyzing data. For the jejunal crypt assay and other colony forming assays this can be achieved using methods of the type described in the Appendix. For other tissues where the structure is known in less detail or the assay is not a direct clonal measure of cell survival, such as skin reactions, the structure may still have an important influence on the measured response. However, it is not obvious how it should be properly taken into consideration because the type of detailed computations given in the Appendix are not possible.
Alternative mathematical models The mathematical model discussed in the present paper has five parameters. As with any model, it is only an empirical approximation to the underlying mechanism. Many other factors could be taken into consideration which would make the model more complex and hopefully more realistic. For example, the known differences in cell cycle sensitivities could be incorporated, or a pipeline [16] or compartment of differentiated cells which influence the repopulation kinetics of the clonogenic cells could be included. Such a compartment may be necessary for any modelling of skin reactions because for this assay the measured response is thought to be directly related to the depletion of the functional cell compartment and only indirectly related to the stem cell comparment. Mathematical models with many more than five parameters have the advantage of being flexible and capable of fitting almost any data set. However they have the overwhelming disadvantage of requiring extremely large and high quality data sets to obtain reasonable estimates of the parameters. For clinical data, it is extremely difficult to obtain accurate
parameter estimates for models which contain as few as three parameters. In addition, predictions from mathematical models with many parameters tend to be highly sensitive to the choice of parameter values in addition to being sensitive to the specific choice of the equations defining the mathematical model. Similar reservations about the problems of multiparameter mathematical models have been expressed previously [12]. Thus, complex nonlinear models with more than about seven parameters are unlikely to be of much use for predicting acute effects in the clinical setting. The role of such models should be confined to that of qualitatively suggesting further experiments or new fractionation schemes to investigate.
Clinical relevance of present experiments The particular fraction sizes and fraction numbers used in the experiment described in this article differ greatly from those used in conventional radiotherapy. As such, direct extrapolation to the clinic is not possible and was not the aim of the experiment. The purpose of the experiment was to gain some understanding of the regenerative response of an acutely responding tissue in the time period immediately following the initial radiation insult. The conclusion important to radiotherapists is that even in a simple fractionation experiment the acute response depends in a complex way on the fractionation scheme. Therefore, in a clinical setting, a simple formula which claims to predict the acute response is unlikely to be accurate. Before any fractionation scheme in radiotherapy is based on a mathematical model of acute effects, the model and possible competitors should be thoroughly and properly tested on several large real data sets. Since such data sets do not exist, it will be necessary for radiotherapists to collect detailed clinical data on acute effects, with levels of injury documented, throughout entire fractionation schemes.
Acknowledgements This work was supported by PHS Grant numbers CA-45216 and CA-29644 awarded by the National Cancer Institute, DHHS. The authors thank Jan Haas, Yeh-Chi Lo and Vijaya Vegesna for assisting with this work.
189
References 1 Blott, P. and Trott, K.R. The effect of actinomycin D on splitdose recovery and repopulation in jejunal crypt cells in vivo. Radiother. Oncol. 15: 73-78, 1989. 2 Cairnie, A. B., Lamerton, L. F. and Steel, G.G. Cell proliferation studies in the intestinal epithelium of the rat. I. Determination of the kinetic parameters. Exp. Cell Res. 39: 528-538, 1965. 3 Chwalinski, S. and Potten, C. S. Radiation-induced mitotic delay: duration, dose and cell position dependence in the crypts of the small intestine in the mouse. Int. J. Radiat. Biol. 49:809-819, 1986. 4 Cohen, L. A cell population kinetic model for fractionated radiation therapy. I. Normal tissues. Radiology 101: 419-427, 1971. 5 Elkind, M. M., Han, A. and Volz, K.W. Radiation response of mammalian cells grown in culture. IV. Dose dependence of division delay and post irradiation growth of surviving and nonsurviving Chinese hamster cells. J. Natl. Cancer Inst. 30: 705-721, 1963. 6 Hegazy, M. A. H. and Fowler, J.F. Cell population kinetics of plucked and unplucked mouse skin. II. Irradiated skin. Cell Tissue Kinet. 6: 587-602, 1973. 7 Hendry, J. H., Potten, C. S., Ghafoor, A., Moore, J. V., Roberts, S.A. and Williams, P.C. The response of murine intestinal crypts to short-range Promethium-147 irradiation: deductions concerning clonogenic cell numbers and positions. Radiat. Res. 118: 364-374, 1989. 8 Potten, C. S., Taylor, Y. and Hendry, J.H. The doubling time of regenerating clonogenic cells in the crypts of the irradiated mouse small intestine. Int. J. Radiat. Biol. 54: 1041-1051, 1988. 9 Potten, C. S., Owen, G. and Roberts, S.A. The temporal and spatial changes in cell proliferation within the irradiated crypts of the murine small intestine. Int. J. Radiat. Biol. 57: 185-199, 1990.
10 Sinclair, W.K. X induced heritable damage (small colony formation) in cultured mammalian cells. Radiat. Res. 21:584-611, 1964. 11 Taylor, J. M. G., Withers, H. R. and Hu, Z. A comparison of mathematical models for regeneration in acutely responding tissues. Int. J. Radiat. Oncol. Biol. Phys. 15: 1389-1400, 1988. 12 Thames, H.D. A new model of proliferative response to fractionated irradiation. Radiother. Oncol. 13: 311-313, 1988. 13 Thames, H.D., Withers, H.R., Mason, K.A. and Reid, B.O. Dose-survival characteristics of mouse jejunal crypt cells. Int. J. Radiat. Oncol. Biol. Phys. 7: 1591-1597, 1981. 14 Travis, E. and Tucker, S. Isoeffect models and fractionated radiotherapy. Int. J. Radiat. Oncol. Biol. Phys. 13: 283-287, 1987. 15 Tucker, S. L., Thames, H. D. and Taylor, J. M.G. How well is the probability of tumor cure after fractionated irradiation described by Poisson statistics? Radiat. Res. 124: 273-282, 1990. 16 van de Geijn, J. Time-dose response of human tumors and normal tissues during and after fractionated radiation treatment. A new model. Radiother. Oncol. 12: 57-78, 1988. 17 Withers, H. R., Chu, A. M., Reid, B. O. and Hussey, D.H. Response of mouse jejunum to multifraction radiation. Int. J. Radiat. Oncol. Biol, Phys. 1: 41-52, 1975. 18 Withers, H. R. and Elkind, M.M. Radiosensitivity and fractionation response of crypt cells of mouse jejunum. Radiat. Res. 38: 598-613, 1969. 19 Withers, H. R. and Elkind, M.M. Microcolony survival assay for cells of mouse intestinal mucosa exposed to radiation. Int. J. Radiat. Biol. 17: 261-267, 1970. 20 Withers, H. R. and Mason, K.A. The kinetics of recovery in irradiated colonic mucosa of the mouse. Cancer 34: 896-903, 1974. 21 Withers, H. R., Mason, K. A. and Taylor, J. M.G. The number of clonogenic cells in a mouse jejunum crypt. Radiother. Oncol., submitted, 1990.
Appendix Derivation o f the Eqn. 1, the expected crypt score in a 3-dose experiment involving repopulation. The measured score in the assay is the n u m b e r o f surviving crypts, treating each of the 160 crypts as a separate unit. The assumption is that the crypt is scored if it has one or more surviving clonogenic cells following the final dose o f irradiation. A corollary to this is that for a crypt to be scored it must maintain at least one clonogenic cell throughout the course o f the regimen. So, for example, if half o f the crypts are completely depleted o f cells prior to the final dose, then however long a time period is allowed for regeneration, these crypts can never repopulate themselves so they will not contribute to the assayed crypt count. In summary, the functional subunit structure o f the jejunum is influencing the measured response. The following mathematical derivation is an a p p r o a c h which allows for the possible depletion of crypts during the course o f the regimen. We will specialize to the 3-dose experiment described in this
paper, although the same a p p r o a c h can be applied for other fractionation schemes. Let d~, d2 and d3 be the 3 doses, and let Pi be the surviving fraction o f cells from a single dose o f size di and Qi = 1 - P i . T h r o u g h o u t we will be considering only clonogenic cells. Consider a single crypt, let K be the initial cell count, let Z~ be the cell c o u n t immediately following dl, let Z2b be the cell c o u n t just preceding d2, let Z2a be the cell count just after d2, let Z3b be the cell count just prior to d 3 and let Z3a be the cell count just after d3. So if Z3a >/1 the crypt is scored, but if Z3a = 0 then the crypt is not scored. The five Z s and K are related in the following way: the distribution o f Z~ is binomial (K, P~ ). The values of Zzb and Z 1 are related by the equation Z2b = alZ~, where al >f 1, with al > 1 if repopulation has occurred between first and second dose. Because al will typically depend on the value of Z~, it was more convenient to use the approximate equation Zzb = gl + hlZ~, where gl and h~ do not
190 depend on Z1. Conditional on the value of Zzb, the distribution of Z=, is binomial (Z2b, /2). The values of Z3b and Z2~ are related by the equation Z3b = a2Z2a, where a2 t> 1, with a2 > 1 if repopulation has occurred between the second and third dose. Again, it was more convenient to use the approximate equation Z3b = g2 + hzZza" Conditional on the value of Z3b, the distribution of Z3~ is binomial (Zsb , P3)" The moment generating function G(U) of the random variable Z3a is defined as G(U) = E (UZ3~), where the expectation is taken with respect to the distribution of Z3a- After lengthy algebra it can be shown that G(U) = Q I~ + Q gl {(QI + elQzh~)K _ Q r } + (Q3 + e 3 u ) g~ [ - Q2gl {(Q1 + e , Q = h , : _ Q r } + (Q2 + e2 (Q3 + e 3 u ) h 2 ) ~' {(Q1 + el (02 + e2 (03 + e 3 u ) h 2 ) h l ) g -- Q,K}I
A property of moment generating functions is that G(0) equals the probability that Z3~ = 0. So the crypt will be counted with probability 1 - G(0) and will not be counted with probability G(0). Thus, the expected number of crypts which will be counted is: E ( C ) = 160(1 - G(0)), where G(O) = Q1x + (I - Q3g2)Qg ' {(Q1 + ? l Q 2 h l ) K -- Q1K} + Q3g2 (Q2 + P2Q3h~)gl {(Q, + ? , (Q2 + ?2Q3h:)hl) K - Qar} •
To validate Eqn. 1 consider some special cases. (i) No repopulation at all during the experiment, then a l = a 2 = 1, g i = g 2 = 0 , h i = h 2 = 1 then E ( C ) becomes 160(1 - G(0)) = 160{1 - [1 - e l e 2 e 3 ] r } , which is the correct form for the expected number of crypts from 3 doses acting independently and cumulatively. (ii) There are very long times between all three fractions so that al and a 2 are both large, gl = g2 = K and hi - - h 2 = 0, and that P2 and P3 a r e both large enough so that either the second or third dose cannot completely empty any full crypts of cells. Then E ( C ) = 1 6 0 { 1 - ( 1 - P 1 ) r } , which is the expected crypt count from a single dose of size d~. (iii) If there is a short time between the first and second doses but a long time between the second and third dose, so that a 1 ---- 1, gl = 0, h I = 1, but a 2 is large and g2 = K, h 2 = 0 and the third dose is not too large, then E ( C ) = 1 6 0 ( 1 - ( 1 - P 1 P 2 ) r ) , which is the expected crypt count from 2 doses of size d 1 and d2. The values orgy, g2, hi and h 2 in Eqn. 1 depend on the time interval between the doses and the kinetic
model of repopulation. If the logistic repopulation is used and let S(t) be the fraction of clonogenic cells in each crypt, then d S / d t = 2S(1 - S), with solution S(t) = Sol(So + (1 - So)e-at).
If So ( = Z ~ / K ) is the fraction of cells just after d~ and S(t) ( = Z2b/K ) is the fraction of cells just before d2 then Z2b = Z 1 ( g / ( z I + ( g - Z i ) e - a t ) ) , where t is the time available for repopulation between dl and dE. Thus, the factor a~ is equal to K/(Z1 + ( K - Z 1 ) e - ~ t ) . Notice that al depends on Z~, but we have assumed that a~ is a constant, but if we replace Z1 by its expected value (= K e l ) then al = 1/(P1 + (1 - el)e-~t). In a similar way we can show that a2 -- 1/(aiPiP2 + (1 - alP1P2)e - at) where t is the time available for repopulation between the second and third doses. We can also show that Z2b = g~ + h~Z~ is a good approximation where gl = Kp12al 2 (1 - e -at) and hl = (a~ - P~ai 2 (1 - e-at));
similarly Z3b = g2 + h2Z2a
where g2 = Ka12PlZp22az2 (1 - e -at) and h2 = (az - alPiP2a22 (1 - e - a t ) )
The validity of these approximations was checked by comparing the expected crypt score using the approximation to the average of some Monte Carlo simulations of Z3a using the exact binomial distributions for Z1, Z2a and Z3a described above. The approximations were found to be very good. Slightly different expressions for gi, gz, hl and h 2 are needed if Gompertz growth or growth based on the equation dS/dt = 2 J . S 0 - S ~/2) is used instead of logistic growth. Another approximation in the derivation of Eqn. 1 is that all the Z's must be integers, but the relationships ZZb = gl + hlZ~ and Z3b = gz + haZza do not necessarily guarantee this because gl, g2, hi, h2 are continuous functions. We will assume that the problems associated with this lack of discreteness are of minor importance. The approach described in this appendix can be adapted to obtain estimates of the probability of cure in growing tumors [15].