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Repopulation in the SCCVII squamous cell carcinoma assessed by an in vivo-in vitro excision assay
R
ADIOTHERAPY %N~~L~GY
ELSEVIER
Radiotherapy and Oncology 39 (1996) 137-144
Repopulation in the SCCVII squamous cell carcinoma assessedby an in vivo-in vitro excision assay Olfred Hansen*a*b, Cai Graua, SBren M. Bentzena, Jens Overgaard” ‘Danish
Cancer Society, Department bDepartment of Oncology,
of Experimental Oabse University
Clinical Oncology. Aarhus, Hospital, Odense. Denmark
Denmark
Received10 December 1994;revised 6 February 1996;accepted12 February 1996
An in vivo-in vitro excision assay was used to study repopulation after a single dose of clamped irradiation (40 Gy) in the SCCVII tumour implanted in the foot of C3H/ICm mice. The growth pattern of clonogenic cells was analysed by two different mathematical models: the logistic model and the Gompertz model. The logistic model described the data better than the Gomperta model. Accelerated repopulation was found when the regrowth rate after irradiation was compared to the growth rate at the time of treatment, and when it was compared to the growth rate in untreated tumours with a number of cells equivalent to the number that was found after irradiation. The clonogenic doubling time (CDT) wasestimated at 15.1 h (95% c.i.: 14.2; 16.0) after irradiation, and 27.8 h (95% c.i.: 16.7; 43.5) in untreated controls of matching size. However, the estimate relies on the mathematical model chosenand on extrapolation below actually measured data. A small CDT points to shortening of the cell cycle time and recruitment of non-cycling clonogenic tumour cells to be the main mechanism behind the accelerated repopulation. Keywords:
Radiotherapy; Accelerated repopulation; Murine tumour; Growth models
experiments. Two or three mice were kept per cage and they were supplied with food and water ad libitum. Tumours for experiments were obtained by subcutaneous injection of 5 ~1 minced tumour material from a donor mouse equivalent to about 2.5 x lo5 viable cells. Tumours were grown in the back of the right rear foot since this location allows irradiation of the tumour without irradiating critical normal tissue. Three orthogonal tumour diameters were measured by callipers, and the tumour volume was calculated using the formula: r/6 x D, x Dz x Ds. The SCCVIYSt carcinoma originated spontaneously in the abdominal wall of a C3H mouse in the laboratory of Dr. H. Suit, Massachusetts General Hospital [12]. Our specimen was obtained from the Medical Biophysics Unit, B.C. Cancer Research Centre, Vancouver, Canada (courtesy of Dr. D.J. Chaplin) in 1989. The tumour has been maintained by inoculation into the flank of inbred C3HN.m mice. The tumours were kept at -80°C and for every live passes a fresh turnout was inoculated into the flank. Tumours for experiments reached the treatment size of 200 mm3 in 9-l 1 days.
1. Introduction
Accelerated repopulation of tumour cells during radiotherapy has been suggested as an important cause of treatment failure in cancers of the head and neck, and other sites [13,25]. The clinical data are largely derived
from retrospective analyses,and they may be biasedfor many reasons as discussed by Bentzen and Thames [2]. In experimental studies, repopulation after single dose irradiation seems to be faster than clonogenic growth at the time of treatment, but patterns and time courses are variable and differences may exist between tumour systems [ 10,111. The present work studies the repopulation in an SCCVII squamous cell carcinoma after single doses of irradiation under clamped conditions. 2. Materials
and metbods
An SCCVII squamous cell carcinoma was used in the * Corresponding author. 0167-8140/96/$15.00 0 1996 Elsevier PIX: SO167-8140(96)01728-8
Science
Ireland
Ltd.
All rights
reserved
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et al. /Radiotherapy
2.I. Irradiation and clamping
Irradiation was given as single doses with 250 kV Xrays (10 mA) with a dose rate of 2.3 Gylmin. The focus skin distance was 20 cm. The only external filtration was an intensity flattening Al-filter with a maximum thickness of 6 mm. The HVL was 2.2 mm Cu. Dosimetry checks were done weekly using an integrated dosimeter (Dosismentor SN4) in an arrangement simulating the treatment set-up. The mice were placed in a lucite jig with the tumour-bearing leg exposed, taped to the jig and immersed in a water bath to improve the homegeneity of the dose distribution in the tumour. During irradiation, the remaining part of the body was shielded by 1 cm of lead [22,23]. All tumours were irradiated under 100% local hypoxia, which was achieved by constriction of blood flow using a rubber tube that was tightened around the leg 5 min before and during irradiation. The validity of this procedure for generating complete radiobiological hypoxia has been documented previously [7]. Clamping of tumours during irradiation was used to limit the confounding influence of possible variations in the hypoxic fraction in the analysis of tumour response. 2.2. The in vivo-in vitro excision assay
Tumours were treated at day 0 at a fixed size of 200 mm3 (range 198-205 mm3) 10 days (range 9-11 days) after inoculation and left in situ until assay time. Then the tumour volume was measured, and the mouse killed by cervical dislocation. The tumour was excised, and all tumour material was carefully transferred to a sterile petri dish. After weighing, the tumour material was flnely chopped with a scalpel and placed in an enzyme cocktail containing DNAse (80 kU/ml), pronase (6 kU/ml), and collagenase (1.32 kU/ml) for 30 min at 37°C in a water bath using a magnetic stirrer. The cell suspension was filtered through a line nylon mesh (27 am) to remove any remaining clumps, centrifuged and the cell pellet resuspended in the medium. After counting the cell density, the tumour was seeded in soft agar in plastic culture tubes at a density appropriate to the expected level of survival [5]. Heat-inactivated red blood cells from female August rats were added to the medium, which otherwise consisted of Waymouth medium enriched with 15% foetal calf serum, 1% L-glutamine, and 1% penicillin streptomycin. Two dilutions, each containing three tubes, were made from each tumour suspension 151. The tubes were incubated at 37”C, in an atmosphere of 5% O,, 5% CO*, and 90% N2 for 20-24 days. Fresh medium was added every 7 days. Colonies with more than 50 cells were counted in a low power inverted microscope. Flow-cytometric analyses have proved the cells in colonies to be aneuploid ensuring the colonies to consist of tumour cells, not fibroblasts. For
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evaluation of results, the total number of clonogenic cells per tumour was calculated as described by Stephens (201. Essentially, the absolute number of colonies counted per tumour was multiplied by the dilution factor, i.e. the percentage of cells that were plated for survival assay. 2.3. Analysis of number of clonogenic cells
The number of clonogens obtained from the excision assay was estimated by fitting two growth models to the data, the logistic and the Gompertz model. In both models, the growth rate decreases as the tumour size increases. The number of cells S(t) at time t is asymptotically approaching a maximum value S,. The logistic growth model [ 171 may be expressed as:
S(t) = s,, . (1
+ exp[- N . (a . t + c)])-‘IN
e.-+.ln[(*)N-
1]
(1)
where S0 is the initial number of cells at t = 0, N is a parameter characterizing the steepness of the curve, and a is a growth rate parameter. In the logistic model, the tumour growth is nearly exponential for small tumours with growth constant. In untreated controls, S0 was measured and after irradiation, Se was determined by linear regression. S-, N, and a were then estimated by non-linear regression analysis. The cionogenic doubling time, CDT may be calculated from Eq. (1) (see Appendix for details). In addition to the logistic model, the Gompertz model was used in a linearized form [ 151. In@(t)) = ln(Smax) - exp[(ln@/a) - a . t]
(2)
a and /3 are constants that determine the course of the growth curve. S-, a, and /3 were estimated by non-
SCCVII
mm’
0
5
tumour
volume
20
10
25
30
Tim~~doys)
Fig. 1. Median tumour volumes in irradiated and untreated tumours.
0. Hansen et al. /Radiotherapy and Oncology 39 (19%) 137-144
139
Fig. 2. The number of clonogenic cells fitted by the Gompertz, and the logistic model. Lower: The tumour volume (log scale),
linear regression analysis. The CDT may be calculated from Eq. (2) (see Appendix for details). 2.4. Statistics
Results are generally given with 95% confidence intervals in brackets. A significance level of 5% was applied for all tests. In the analysis of tumour growth data a non-parametric test (Mann-Whitney U-test) was used [ 141, and confidence intervals were calculated according to the methods given by Campbell and Gardner [4]. Statistical analyses were performed in BMDP (3R, 1990 version). The best lit of the models was determined by the least squares method. The 95% confidence intervals of the parameters in the non-linear regression analyses were estimated whenever possible by the Weisberg confidence curve technique by the BMDP program. Otherwise the 95% confidence limits were estimated as 1.96 times the asymptotic standard deviation.
3. Results
Tumour volume as a function of time after irradiation is shows in Fig. 1 and in semilogarithmic coordinates in the bottom panel of Fig. 2. After a dose of 40 Gy (clamped) the tumour growth was slowed down, but no reduction in tumour volume was seen. Fig. 2 shows all measured numbers of clonogenic cells in controls and after irradiation. The data are fitted by a logistic and a Gompertz model. In controls, an increase in cells was seen from the initial number of 3.5 x lo6 cells to a maximum of about 10’ cells in 6-7 days. After 40 Gy (clamped), dose-survival curves showed a surviving fraction of 2 x lo-’ at day 1. A regrowth of cells could be detected after 2 days, and the pretreatment number of cells at the time of irradiation was reached after 12 days. 3.1. In vitro assay
The cell yield, i.e. the number of tumour cells (in-
Table 1 Estimated parameters in the logistic growth model 3-parameter model
Control MGy
2-parameter model (N = 1)
a (days-‘)
N
0.65 (0.31; 1.30) 1.10 (0.92; 1.29)
0.95 (0.73; 1.30) 1.00 (0.97; 1.04)
a (days-‘) 7.81 (7.55; 11.7) 7.52 (7.18; 7.88)
95% confidence intervals in brackets from the Weisberg confidence curve.
0.60 (0.39; 0.99) * 1.10 (1.05; 1.16)
bc%ld 7.83 (7.57; 8.&t) 7.52 (7.20; 7.86)
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Fig. 3. Clonogenic doubling time calculated by the logistic and the Gompertz model. Error bars: 95% confidence intervals.
eluding non-clonogenic tumour cells) per mg, was counted to 1.5 x 10’ at day 0, the average weight of a 200 mm3 tumour was 135 mg yielding an average number of cells per tumour of 20.1 x 106. The number of clonogenic cells was determined by the assay to 3.5 x lo6 cells per tumour. This is a ratio of clonogenic cells to tumour cells of 17.4%.
The number of clonogens in the untreated controls was extrapolated backwards by the logistic model (Fig. 4). The logistic equation (2-parameter model) predicted the inoculated number of clonogenic cells (at day -10) to 9100 clonogenic cells. Using the above calculated tumour cell/clonogenic cell ratio of 17.4%, this is equivalent to a prediction of 5.2 x lo4 viable cells. The logistic model therefore predicted that 20% of the inoculated viable cells grew in the foot immediately after inoculation without any lag phase, or as an alternative, all ce118 grew after a lag phase of 2.9 days.
3.2. Logistic models After a single dose of 40 Gy (clamped), So was estimated at 10 cells by linear regression. This value was used for estimation of N, S-, and Q in the logistic models (formula 2) fitted by non-linear regression. The curve steepness, N, was estimated to a value of 0.95 (0.73; 1.30) in untreated controls. To simplify the model, N was fixed at the value of 1, which was contained in the 95% confidence interval and the two remaining parameters a! and S,, were estimated. Table 1 shows the estimated parameters in the 2-parameter model and the 3-parameter model. The tested models resulted in good statistical fits measured as the mean residual sum of square. The goodness of tit was 5 times better in the 3parameter logistic model compared with 2-parameter model (Table 3). In the 2-parameter model, S,, was calculated to be 0.33 x lo* cells in the controls and 0.68 x IO8 cells in the irradiated tumours with overlapping confidence intervals.
3.3. Gompertz model Estimates of the parameters, S-, (r, and 0 in the Gompertz model are shown in Table 2. Although a statistically satisfactory fit was achieved, the goodness of tit was inferior to the logistic models by a factor 5.4-6.5 S,, of 1.1 x lo8 cells was calculated in controls and 7.59 x lo8 cells in irradiated tumours with overlapping confidence intervals. The Gompertz model was not able to give any prediction of the inoculated number of cells at day -10 (Fig. 4). 3.4. Clonogenic doubling times With all models, the estimated CDT after irradiation was smaller than it was at the time of irradiation (Table
Table 2 Gompertz model with three parameters (a, 8, Smu) and the derived value of log initial number of clonogenic cells (So)
Controls 40 Gy Cl.
u (days-‘)
B (days)
0.28 (0.16; 0.54) 0.10 (0.10; 0.11)
0.78 (0.58; 1.68) 2.08 (2.04; 3.00)
8.04 (7.60; 38.39) 8.88 (7.93; 10.65)
95% confidence intervals in brackets from the We&erg confidence curve.
6.5 (6.2; 6.8) O.l(-1.1; 1.2)
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Table 3 Estimated clonogenic doubling times (in hours) as a function of tumour size
L.ogistic
at 10 clonogenss
at 3.5x lo6
19 19
27.8 (16.7; 43.5) 15.1 (14.2; 16.0)
30.0 (18.7; 44.4) 17.8 (16.0; 22.4)
19 19
4.8 (2.1; 45.1) 9.1 (7.3; 12.0)
24.7 (13.6; 54.5) 32.3 (26.4; 43.5)
model (2-parameter)
Control 3.5 x IO6 40 Gy clamped Gompertz
Animals
moa’el
Controls 3.5 x 10” 40 Gy clamped
95% confidence intervals in brackets. ‘Nadir value after irradiation at 40 Gy clamped.
3). The CDT in controls was 25-30 h. The estimates of CDT depended on the number of clonogenic cells in the tumour. This dependency was most pronounced in the Gompertz model (Fig. 3). When treated and untreated tumours were compared using the logistic models, the estimated CDTS were lower in the irradiated tumours compared with the untreated controls of similar sizes. In the 2-parameter model, the CDTS in the treated tumours were calculated to be in the range from 15.1 to 15.8 h for sizes from 10 to lo6 cells compared with 27.8 to 28.4 h in the untreated tumour of similar sizes. The differences were all statistically significant, while no statistical difference was found between treated and untreated tumours of 3.5 x lo6 cells (the number at treatment size). Using the Gompertz model, no differences between treated and untreated tumours of any sizes were detected. 4. Discussion The repopulation has been studied by two mathematical models. Both the Gompertz and the logistic model CONTROL
j 01....~..,..~....~...~~.~~.‘..~“.~~” -15 -10 -5 0 5 10 15
20
Fig. 4. The number of clonogenic cells in the untreated controls in the Gompertz and the logistic model. The curves have been extrapolated beyond the observed range of data.
showed accelerated repopulation when the early regrowth rate of irradiated tumours was compared to the growth at treatment time, but only the logistic model suggested that the repopulation rate was accelerated also when it was compared to the regrowth rate in small untreated tumours. Several authors have used the Gompertz model in tumour growth studies [ 18,241. Jung et al. [lo] using the Gompertz model concluded that growth of small tumours (controls and irradiated) could be described by the same Gompertz function after a range of single doses. For the present set of data, the Gompertz model was less attractive than the logistic model. The statistical fit was poorer, and the prediction of the inoculated number of cells was unrealistic when the growth curve was extrapolated to very small tumour volumes (Fig. 4). The validity of the extrapolation depends on how accurately the model describes the data. The growth rate in a Gompertz function approaches infinity as number of cells tends to zero, and this is not a reasonable biological behaviour. In contrast, the logistic model seemed suitable to describe the data for small untreated tumours. The model worked in treated tumours, and the prediction of the inoculated cells was reasonable. The log-linear behaviour of the models for small unirradiated tumours and tumours in an early regrowth phase implies that the CDT increases very little with increasing number of cells until a certain tumor size. Simple linear regression analysis of small tumours yielded a CDT very close to the values estimated by the logistic model, but the latter offers smaller confidence intervals since it uses the complete range of data, both the slow and the fast phase of a growth curve. Another advantage of the logistic model compared with simple linear regression analysis is that it estimates the maximum possible number of cells in the tumour. This is also the case in the Gompertz model. In both models, this number should not necessarily be regarded as a biologically meaningful number, but merely as a mathematical abstraction to give the right sort of curve for large tumours as pointed out by Wheldon 1241. The potential doubling time, T,,,,r, is the theoretical doubling time of the tumour when no cell loss is present.
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The CDT may be shorter than the TPt since some clonogenic cells may rest in the cell cycle, and not all tumour cells are clonogenic. The cDTs that were obtained in the present study tended to be smaller than the Tpt of 39.9 (32.0; 47.7) h that were determined by flow cytometric analysis at our institute [8] and the Tpt of 42-45 h determined in the SCCVII by Speke and Hill [16]. The Tpot is just within the 95% confidence intervals of the estimated cDTs in the controls. The estimated cDTs after 40 Gy were considerably shorter than the Tpt for small tumours in both mathematical models. Nonclonogenic tumour cells may have a doubling time different from the clonogenic cells. In addition, flow cytometry may estimate too high a Tpot. This has been suggested by the data of Bennet et al. [1] who examined human squamous cell carcinoma of the head and neck. They found that Tpot values, which were determined by histological evaluation, were shorter than Tpt determined by flow cytometry. This was especially evident in diploid tumours, but the histological method determined very short values, less than 24 h, in aneuploid as well as in diploid tumours, in contrast to that no Tpot lower than 2 days was determined by flow cytometry. The CDT after 40 Gy in small tumours was estimated to be even shorter than the pretreatment cell cycle time, T,, which has been measured to 28 h. The T, may not be of a constant duration, since especially the G1 may vary after treatment [19]. The CDT that was estimated by the logistic model to 15.1 h was close to the T,, which has been determined to 16.2 (13.1; 19.3) h by flow cytometry [8]. The CDT is not expected to be shorter than the T,, and therefore, the CDT that was determined to be about 9 h by the Gompertz model for small tumours after irradiation is too small. This also suggests that the Gompertz model may only be valid for larger tumours, and is in accordance with the views that have been expressed by Wheldon [24]. Little is known about the dependency of repopulation on tumour type, tumour differentiation, level of clonogenic cell depletion, and on tumour growth parameters in general. Three main mechanisms for accelerated repopulation have been proposed: faster cell cycle, increased growth fraction, and reduced cell loss. Jones 191 suggested that fractionated irradiation selects for subsets of cells with faster cycling times since they will have a greater chance to multiply between fractions. This leads to an increase in the number produced per unit time. Jung et al. [lo] did not demonstrate a shorter cell cycle time after irradiating tumours in vivo or in vitro. The results in the present study support this idea since the CDT after irradiation was faster than the cell cycle time that has been demonstrated in the untreated SCCVII tumour. Trott and Kummermehr [21] suggested that, during accelerated repopulation, the spontaneous stem cell loss due to differentiation is reduced. This may be due to tumour cells that have retained some
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of the homeostatic control mechanisms that are present in normal tissues. This suggestion was supported experimentally by Jung et al. [lo] and Brammer et al. [3], and the present finding of an acceleration in the small tumours compared with untreated controls of the same size is also in agreement with this hypothesis. Fowler [6] has argued that reduced cell loss results in ‘unmasking’ the true tumour cell production. This view was sup ported by the observations by Jung et al. [lo] who found that the cell loss decreased by at least a factor of 2 at the onset of repopulation after irradiation. If ‘unmasking’ was the main mechanism behind the accelerated growth in the present study, no difference would be expected between small irradiated tumours and similar sized untreated tumours. In conclusion, the study shows that the regrowth of SCCVII tumour cells after clamped irradiation is accelerated compared to the growth rate in untreated controls when a logistic model is used for analysis. However, great caution should be observed since the conclusion relies on extrapolation of data to small tumours, and the models analysed may crucially rely on the current set data. Shortening of the cell cycle time and recruitment of non-cycling clonogenic tumour cells is suggested as the main mechanism behind the acceleration.
A number of growth models have been proposed to describe the growth of tumours in experimental studies [ 18,241. The growth curve of a tumour will approximate an exponential relationship if the rates of cell production and of cell loss are proportional to the number of cells present in the tumour. The differential equation that describes the tumour growth is then given by
S(r) is the number of cells at time t. As the tumour size increases, the rate of tumour growth tends to slow down [ 181. This leads to a generalized model
wheref(s) is a term that tends to 0 as S(t) tends to its asymptotic maximum value, S,, The logistic models and the Gompertz models are two classes of growth models that have this characteristic. In the logistic model the term&) is defined as f(s) = 1 -
+ (
N ->
(5)
0. Hansen
In the Gompertz model the termf(s)
et al. /Radiotherapy
is defined as
f(s) = In *
(6)
The solution for the logistic model for N greater than 0 is given in Eq. (1). The constant N determines the shape of the curve. For decreasing values of N, the decelerating portion of the curve becomes more pronounced. For increasing values of N, the growth approximates the exponential growth curves [17]. From Eq. (l), it follows that the clonogenic doubling time may be expressed as
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39 (19%)
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143
In both models, S(t) - S,, as t - 00. The qualitative difference between the models is the asymptotic behaviour of S(t) as t tends to zero. In the logistic model, f(s) approaches 1 for small tumours (Eq. (5)), and this means that the growth curve for the logistic model approaches simple exponential growth for small tumours. In the Gompertz equationf(s) tends to infinity for small tumours (Eq. (6)) equivalent to that the growth curve approximates a vertical line. The growth rate becomes infinitely fast (Eq. (12)). References 111 Bennet, M.H., Wilson, G.D., Dische, S., Saunders, MI., Mar-
If N equals 1, Eq. (1) simplifies to S(t) =
GUlX 1 + exp(- (cf. 2) - c)
c=-ln[(?)
-11
03)
and the doubling time may be expressed as
CDT=--
1 a!
In
(
&ax ( SW > -
(9)
&ax 2 ( * S(t) >
snax ( >I
The solution to the Gompertz equation is S(r) = So . exp ( 1 -exp(-a.?)) [
.ln
-
so
The decrease in the growth rate with increasing tumour size is more pronounced with the Gompertz than with the logistic model. The Gompertz equation is often reparameterized using the relation between S,, and So S-
= So . exp(@cf)
(11)
and a convenient linear transformation is given in Eq. (2). The doubling time varies with number of clonogenic cells in the tumour, and may be expressed as CDT=-L.ln
a
ln2 ln(S(t)lS&
1
(12)
tindale, C.A., Robinson, B.M., O’Halloran, A.E., Leslie, M.D. and Laing, J.H.E. Tumour proliferation assessedby combined histological and flow cytometric analysis: implications for therapy in squamous cell carcinoma in the head and neck. Br. J. Cancer 65: 870-878, 1992. 121 Bentzen, SM. and Thames, H.D. Clinical evidence for tumor clonogen regeneration: Interpretations of the data. Radiother. Oncol. 22: 161-166, 1991. I31 Brammer, I., Zywietx, F., Beck-Bomholdt, H.-P. and Jung, H. Kinetics of depopulation, repopulation and host cell infiltration in the rhabdomyosarcoma RlH after 14 MeV neutron irradiation. Int. J. Radiat. Biol. 61: 703-711, 1992. 141 Campbell, M.J., Gardner, M.J. Calculating confidence intervals for some non-parametric analyses. In: Statistics with Contidence. Confidence Intervals and Statistical Guidelines, pp. 71-79. Editors: M.J. Gardner and D.G. Altman. British Medical Journal, London, 1989. PI Courtenay, V.D. A soft agar colony assay for Lewis lung tumour and Bl6 melanoma taken directly from the mouse. Br. J. Cancer 34: 39-45, 1976. R Fowler, J.F. Rapid repopulation in radiotherapy: Debate on mechanism (letter). Radiother. Oncol. 24: 126, 1992. 171 Grau, C., Horsman, M.R. and Overgaard, J. Influence of carboxyhemoglobin level on tumor growth, blood flow, and radiation response in an experimental model. Int. J. Radiat. Oncol. Biol. Phys. 22: 421-424, 1992. 181 Heyer, M., Bentzen, S.M., Sailing, L.N. and Overgaard, 1. Influence of sampling time on assessmentof potential doubling time. Cytometry 16: 144-151, 1994. [91 Jones, B. Rapid repopulation in radiotherapy: A debate on mechanism (letter). Radiother. Oncol. 24: 124-125, 1992. 1101Jung, H., Kruger, H.-J., Brammer, I., Zywietz, F. and BeckBomholdt, H.-P. Cell population kinetics of the rhabdomyosarcoma RlH of the rat after single doses of X-rays. Int. J. Radiat. Biol. 57: 567-589, 1990. 1111Milas, L., Yamada, S., Hunter, N., Guttenberger, R. and Thames, H.D. Changes in TCDSO as a measure of clonogen doubling time in irradiated and unirradiated tumors. Int. J. Radiat. Oncol. Biol. Phys. 21: 1195-1202, 1991. WI Olive, P.L., Chaplin, D. J. and Durand, R.E. Pharmacokinetics, binding and distribution of Hoechst 33342 in spheroids and murine turnouts. Br. J. Cancer 52: 739-746, 1985. iI31 Overgaard, J., Hjelm-Hansen, M., Johansen, L.V. and Andersen, A.P. Comparison of conventional and split-course radiotherapy as primary treatment in carcinoma of the larynx. Acta Oncol. 27: 147-152, 1988. iI41 Siegel, S. Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill, 1956. I151 Spang-Thomsen, M., Rygaard, K., Hansen, L., Halvorsen, A.C., Vindelev, L.L. and Briinner, N. Growth kinetics of four
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0. Hansen et al. /Radiotherapy and Oncology human breast carcinomas grown in nude mice. Breast Cancer Res. Treat. 14: 235-243, 1989. Speke, A.K. and Hill, R.P. Repopulation kinetics during fractionated irradiation and the relationship to the potential doubling time, Tpot. Int. J. Radiat. Oncol. Biol. Phys. 31: 847-856, 1995. Spratt, J.A., von Fournier, D., Spratt, J.S. and Weber, E.E. Decelerating growth and human breast cancer. Cancer 71: 2013-2019, 1992. Steel, G.G. Growth Kinetics of Tumours. Clarendon Press, Oxford, 1977. Steel, G.G. The heyday of cell population kinetics: Insights from the 1960s and 1970s. Semin. Radiat. Oncol. 3: 78-83, 1993. Stephens, T.C. Measurement of tumor cell surviving fraction and absolute number of clonogens per tumor in excision assays. In: Rodent Tumor Models in Experimental Cancer Therapy, pp. 90-94. Editor: R.F. Kallman. Pergamon Press, New York, 1987.
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[21] Trott, K.R. and Kummermehr, J. Accelerated repopulation in tumours and normal tissues. Radiother. Oncol. 22: 159-160, 1991. [22] von der Maase, H. Effect of cancer chemotherapeutic drugs on the radiation-induced skin reactions in mouse feet. Br. J. Radiol. 57: 697-707, 1984. [23] von der Maase, H. and Overgaard, 1. Interactions of radiation and cancer chemotherapeutic drugs in a C3H mouse mammary carcinoma. Acta Radiol. Oncol. 24: 181-187, 1985. [24] Wbeldon, T.E. Models of tumour growth. In: Mathematical Models in Cancer Research, pp. 63-90. Adam Hilger, Bristol, 1988. [25] Withers, H.R., Taylor, J.M.G. and Maciejewski, B. The hazard of accelerated tumor clonogen repopulation during radiotherapy. Acta Oncol. 27: 131-146, 1988.
ADIOTHERAPY %N~~L~GY
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Radiotherapy and Oncology 39 (1996) 137-144
Repopulation in the SCCVII squamous cell carcinoma assessedby an in vivo-in vitro excision assay Olfred Hansen*a*b, Cai Graua, SBren M. Bentzena, Jens Overgaard” ‘Danish
Cancer Society, Department bDepartment of Oncology,
of Experimental Oabse University
Clinical Oncology. Aarhus, Hospital, Odense. Denmark
Denmark
Received10 December 1994;revised 6 February 1996;accepted12 February 1996
An in vivo-in vitro excision assay was used to study repopulation after a single dose of clamped irradiation (40 Gy) in the SCCVII tumour implanted in the foot of C3H/ICm mice. The growth pattern of clonogenic cells was analysed by two different mathematical models: the logistic model and the Gompertz model. The logistic model described the data better than the Gomperta model. Accelerated repopulation was found when the regrowth rate after irradiation was compared to the growth rate at the time of treatment, and when it was compared to the growth rate in untreated tumours with a number of cells equivalent to the number that was found after irradiation. The clonogenic doubling time (CDT) wasestimated at 15.1 h (95% c.i.: 14.2; 16.0) after irradiation, and 27.8 h (95% c.i.: 16.7; 43.5) in untreated controls of matching size. However, the estimate relies on the mathematical model chosenand on extrapolation below actually measured data. A small CDT points to shortening of the cell cycle time and recruitment of non-cycling clonogenic tumour cells to be the main mechanism behind the accelerated repopulation. Keywords:
Radiotherapy; Accelerated repopulation; Murine tumour; Growth models
experiments. Two or three mice were kept per cage and they were supplied with food and water ad libitum. Tumours for experiments were obtained by subcutaneous injection of 5 ~1 minced tumour material from a donor mouse equivalent to about 2.5 x lo5 viable cells. Tumours were grown in the back of the right rear foot since this location allows irradiation of the tumour without irradiating critical normal tissue. Three orthogonal tumour diameters were measured by callipers, and the tumour volume was calculated using the formula: r/6 x D, x Dz x Ds. The SCCVIYSt carcinoma originated spontaneously in the abdominal wall of a C3H mouse in the laboratory of Dr. H. Suit, Massachusetts General Hospital [12]. Our specimen was obtained from the Medical Biophysics Unit, B.C. Cancer Research Centre, Vancouver, Canada (courtesy of Dr. D.J. Chaplin) in 1989. The tumour has been maintained by inoculation into the flank of inbred C3HN.m mice. The tumours were kept at -80°C and for every live passes a fresh turnout was inoculated into the flank. Tumours for experiments reached the treatment size of 200 mm3 in 9-l 1 days.
1. Introduction
Accelerated repopulation of tumour cells during radiotherapy has been suggested as an important cause of treatment failure in cancers of the head and neck, and other sites [13,25]. The clinical data are largely derived
from retrospective analyses,and they may be biasedfor many reasons as discussed by Bentzen and Thames [2]. In experimental studies, repopulation after single dose irradiation seems to be faster than clonogenic growth at the time of treatment, but patterns and time courses are variable and differences may exist between tumour systems [ 10,111. The present work studies the repopulation in an SCCVII squamous cell carcinoma after single doses of irradiation under clamped conditions. 2. Materials
and metbods
An SCCVII squamous cell carcinoma was used in the * Corresponding author. 0167-8140/96/$15.00 0 1996 Elsevier PIX: SO167-8140(96)01728-8
Science
Ireland
Ltd.
All rights
reserved
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2.I. Irradiation and clamping
Irradiation was given as single doses with 250 kV Xrays (10 mA) with a dose rate of 2.3 Gylmin. The focus skin distance was 20 cm. The only external filtration was an intensity flattening Al-filter with a maximum thickness of 6 mm. The HVL was 2.2 mm Cu. Dosimetry checks were done weekly using an integrated dosimeter (Dosismentor SN4) in an arrangement simulating the treatment set-up. The mice were placed in a lucite jig with the tumour-bearing leg exposed, taped to the jig and immersed in a water bath to improve the homegeneity of the dose distribution in the tumour. During irradiation, the remaining part of the body was shielded by 1 cm of lead [22,23]. All tumours were irradiated under 100% local hypoxia, which was achieved by constriction of blood flow using a rubber tube that was tightened around the leg 5 min before and during irradiation. The validity of this procedure for generating complete radiobiological hypoxia has been documented previously [7]. Clamping of tumours during irradiation was used to limit the confounding influence of possible variations in the hypoxic fraction in the analysis of tumour response. 2.2. The in vivo-in vitro excision assay
Tumours were treated at day 0 at a fixed size of 200 mm3 (range 198-205 mm3) 10 days (range 9-11 days) after inoculation and left in situ until assay time. Then the tumour volume was measured, and the mouse killed by cervical dislocation. The tumour was excised, and all tumour material was carefully transferred to a sterile petri dish. After weighing, the tumour material was flnely chopped with a scalpel and placed in an enzyme cocktail containing DNAse (80 kU/ml), pronase (6 kU/ml), and collagenase (1.32 kU/ml) for 30 min at 37°C in a water bath using a magnetic stirrer. The cell suspension was filtered through a line nylon mesh (27 am) to remove any remaining clumps, centrifuged and the cell pellet resuspended in the medium. After counting the cell density, the tumour was seeded in soft agar in plastic culture tubes at a density appropriate to the expected level of survival [5]. Heat-inactivated red blood cells from female August rats were added to the medium, which otherwise consisted of Waymouth medium enriched with 15% foetal calf serum, 1% L-glutamine, and 1% penicillin streptomycin. Two dilutions, each containing three tubes, were made from each tumour suspension 151. The tubes were incubated at 37”C, in an atmosphere of 5% O,, 5% CO*, and 90% N2 for 20-24 days. Fresh medium was added every 7 days. Colonies with more than 50 cells were counted in a low power inverted microscope. Flow-cytometric analyses have proved the cells in colonies to be aneuploid ensuring the colonies to consist of tumour cells, not fibroblasts. For
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evaluation of results, the total number of clonogenic cells per tumour was calculated as described by Stephens (201. Essentially, the absolute number of colonies counted per tumour was multiplied by the dilution factor, i.e. the percentage of cells that were plated for survival assay. 2.3. Analysis of number of clonogenic cells
The number of clonogens obtained from the excision assay was estimated by fitting two growth models to the data, the logistic and the Gompertz model. In both models, the growth rate decreases as the tumour size increases. The number of cells S(t) at time t is asymptotically approaching a maximum value S,. The logistic growth model [ 171 may be expressed as:
S(t) = s,, . (1
+ exp[- N . (a . t + c)])-‘IN
e.-+.ln[(*)N-
1]
(1)
where S0 is the initial number of cells at t = 0, N is a parameter characterizing the steepness of the curve, and a is a growth rate parameter. In the logistic model, the tumour growth is nearly exponential for small tumours with growth constant. In untreated controls, S0 was measured and after irradiation, Se was determined by linear regression. S-, N, and a were then estimated by non-linear regression analysis. The cionogenic doubling time, CDT may be calculated from Eq. (1) (see Appendix for details). In addition to the logistic model, the Gompertz model was used in a linearized form [ 151. In@(t)) = ln(Smax) - exp[(ln@/a) - a . t]
(2)
a and /3 are constants that determine the course of the growth curve. S-, a, and /3 were estimated by non-
SCCVII
mm’
0
5
tumour
volume
20
10
25
30
Tim~~doys)
Fig. 1. Median tumour volumes in irradiated and untreated tumours.
0. Hansen et al. /Radiotherapy and Oncology 39 (19%) 137-144
139
Fig. 2. The number of clonogenic cells fitted by the Gompertz, and the logistic model. Lower: The tumour volume (log scale),
linear regression analysis. The CDT may be calculated from Eq. (2) (see Appendix for details). 2.4. Statistics
Results are generally given with 95% confidence intervals in brackets. A significance level of 5% was applied for all tests. In the analysis of tumour growth data a non-parametric test (Mann-Whitney U-test) was used [ 141, and confidence intervals were calculated according to the methods given by Campbell and Gardner [4]. Statistical analyses were performed in BMDP (3R, 1990 version). The best lit of the models was determined by the least squares method. The 95% confidence intervals of the parameters in the non-linear regression analyses were estimated whenever possible by the Weisberg confidence curve technique by the BMDP program. Otherwise the 95% confidence limits were estimated as 1.96 times the asymptotic standard deviation.
3. Results
Tumour volume as a function of time after irradiation is shows in Fig. 1 and in semilogarithmic coordinates in the bottom panel of Fig. 2. After a dose of 40 Gy (clamped) the tumour growth was slowed down, but no reduction in tumour volume was seen. Fig. 2 shows all measured numbers of clonogenic cells in controls and after irradiation. The data are fitted by a logistic and a Gompertz model. In controls, an increase in cells was seen from the initial number of 3.5 x lo6 cells to a maximum of about 10’ cells in 6-7 days. After 40 Gy (clamped), dose-survival curves showed a surviving fraction of 2 x lo-’ at day 1. A regrowth of cells could be detected after 2 days, and the pretreatment number of cells at the time of irradiation was reached after 12 days. 3.1. In vitro assay
The cell yield, i.e. the number of tumour cells (in-
Table 1 Estimated parameters in the logistic growth model 3-parameter model
Control MGy
2-parameter model (N = 1)
a (days-‘)
N
0.65 (0.31; 1.30) 1.10 (0.92; 1.29)
0.95 (0.73; 1.30) 1.00 (0.97; 1.04)
a (days-‘) 7.81 (7.55; 11.7) 7.52 (7.18; 7.88)
95% confidence intervals in brackets from the Weisberg confidence curve.
0.60 (0.39; 0.99) * 1.10 (1.05; 1.16)
bc%ld 7.83 (7.57; 8.&t) 7.52 (7.20; 7.86)
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Fig. 3. Clonogenic doubling time calculated by the logistic and the Gompertz model. Error bars: 95% confidence intervals.
eluding non-clonogenic tumour cells) per mg, was counted to 1.5 x 10’ at day 0, the average weight of a 200 mm3 tumour was 135 mg yielding an average number of cells per tumour of 20.1 x 106. The number of clonogenic cells was determined by the assay to 3.5 x lo6 cells per tumour. This is a ratio of clonogenic cells to tumour cells of 17.4%.
The number of clonogens in the untreated controls was extrapolated backwards by the logistic model (Fig. 4). The logistic equation (2-parameter model) predicted the inoculated number of clonogenic cells (at day -10) to 9100 clonogenic cells. Using the above calculated tumour cell/clonogenic cell ratio of 17.4%, this is equivalent to a prediction of 5.2 x lo4 viable cells. The logistic model therefore predicted that 20% of the inoculated viable cells grew in the foot immediately after inoculation without any lag phase, or as an alternative, all ce118 grew after a lag phase of 2.9 days.
3.2. Logistic models After a single dose of 40 Gy (clamped), So was estimated at 10 cells by linear regression. This value was used for estimation of N, S-, and Q in the logistic models (formula 2) fitted by non-linear regression. The curve steepness, N, was estimated to a value of 0.95 (0.73; 1.30) in untreated controls. To simplify the model, N was fixed at the value of 1, which was contained in the 95% confidence interval and the two remaining parameters a! and S,, were estimated. Table 1 shows the estimated parameters in the 2-parameter model and the 3-parameter model. The tested models resulted in good statistical fits measured as the mean residual sum of square. The goodness of tit was 5 times better in the 3parameter logistic model compared with 2-parameter model (Table 3). In the 2-parameter model, S,, was calculated to be 0.33 x lo* cells in the controls and 0.68 x IO8 cells in the irradiated tumours with overlapping confidence intervals.
3.3. Gompertz model Estimates of the parameters, S-, (r, and 0 in the Gompertz model are shown in Table 2. Although a statistically satisfactory fit was achieved, the goodness of tit was inferior to the logistic models by a factor 5.4-6.5 S,, of 1.1 x lo8 cells was calculated in controls and 7.59 x lo8 cells in irradiated tumours with overlapping confidence intervals. The Gompertz model was not able to give any prediction of the inoculated number of cells at day -10 (Fig. 4). 3.4. Clonogenic doubling times With all models, the estimated CDT after irradiation was smaller than it was at the time of irradiation (Table
Table 2 Gompertz model with three parameters (a, 8, Smu) and the derived value of log initial number of clonogenic cells (So)
Controls 40 Gy Cl.
u (days-‘)
B (days)
0.28 (0.16; 0.54) 0.10 (0.10; 0.11)
0.78 (0.58; 1.68) 2.08 (2.04; 3.00)
8.04 (7.60; 38.39) 8.88 (7.93; 10.65)
95% confidence intervals in brackets from the We&erg confidence curve.
6.5 (6.2; 6.8) O.l(-1.1; 1.2)
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Table 3 Estimated clonogenic doubling times (in hours) as a function of tumour size
L.ogistic
at 10 clonogenss
at 3.5x lo6
19 19
27.8 (16.7; 43.5) 15.1 (14.2; 16.0)
30.0 (18.7; 44.4) 17.8 (16.0; 22.4)
19 19
4.8 (2.1; 45.1) 9.1 (7.3; 12.0)
24.7 (13.6; 54.5) 32.3 (26.4; 43.5)
model (2-parameter)
Control 3.5 x IO6 40 Gy clamped Gompertz
Animals
moa’el
Controls 3.5 x 10” 40 Gy clamped
95% confidence intervals in brackets. ‘Nadir value after irradiation at 40 Gy clamped.
3). The CDT in controls was 25-30 h. The estimates of CDT depended on the number of clonogenic cells in the tumour. This dependency was most pronounced in the Gompertz model (Fig. 3). When treated and untreated tumours were compared using the logistic models, the estimated CDTS were lower in the irradiated tumours compared with the untreated controls of similar sizes. In the 2-parameter model, the CDTS in the treated tumours were calculated to be in the range from 15.1 to 15.8 h for sizes from 10 to lo6 cells compared with 27.8 to 28.4 h in the untreated tumour of similar sizes. The differences were all statistically significant, while no statistical difference was found between treated and untreated tumours of 3.5 x lo6 cells (the number at treatment size). Using the Gompertz model, no differences between treated and untreated tumours of any sizes were detected. 4. Discussion The repopulation has been studied by two mathematical models. Both the Gompertz and the logistic model CONTROL
j 01....~..,..~....~...~~.~~.‘..~“.~~” -15 -10 -5 0 5 10 15
20
Fig. 4. The number of clonogenic cells in the untreated controls in the Gompertz and the logistic model. The curves have been extrapolated beyond the observed range of data.
showed accelerated repopulation when the early regrowth rate of irradiated tumours was compared to the growth at treatment time, but only the logistic model suggested that the repopulation rate was accelerated also when it was compared to the regrowth rate in small untreated tumours. Several authors have used the Gompertz model in tumour growth studies [ 18,241. Jung et al. [lo] using the Gompertz model concluded that growth of small tumours (controls and irradiated) could be described by the same Gompertz function after a range of single doses. For the present set of data, the Gompertz model was less attractive than the logistic model. The statistical fit was poorer, and the prediction of the inoculated number of cells was unrealistic when the growth curve was extrapolated to very small tumour volumes (Fig. 4). The validity of the extrapolation depends on how accurately the model describes the data. The growth rate in a Gompertz function approaches infinity as number of cells tends to zero, and this is not a reasonable biological behaviour. In contrast, the logistic model seemed suitable to describe the data for small untreated tumours. The model worked in treated tumours, and the prediction of the inoculated cells was reasonable. The log-linear behaviour of the models for small unirradiated tumours and tumours in an early regrowth phase implies that the CDT increases very little with increasing number of cells until a certain tumor size. Simple linear regression analysis of small tumours yielded a CDT very close to the values estimated by the logistic model, but the latter offers smaller confidence intervals since it uses the complete range of data, both the slow and the fast phase of a growth curve. Another advantage of the logistic model compared with simple linear regression analysis is that it estimates the maximum possible number of cells in the tumour. This is also the case in the Gompertz model. In both models, this number should not necessarily be regarded as a biologically meaningful number, but merely as a mathematical abstraction to give the right sort of curve for large tumours as pointed out by Wheldon 1241. The potential doubling time, T,,,,r, is the theoretical doubling time of the tumour when no cell loss is present.
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The CDT may be shorter than the TPt since some clonogenic cells may rest in the cell cycle, and not all tumour cells are clonogenic. The cDTs that were obtained in the present study tended to be smaller than the Tpt of 39.9 (32.0; 47.7) h that were determined by flow cytometric analysis at our institute [8] and the Tpt of 42-45 h determined in the SCCVII by Speke and Hill [16]. The Tpot is just within the 95% confidence intervals of the estimated cDTs in the controls. The estimated cDTs after 40 Gy were considerably shorter than the Tpt for small tumours in both mathematical models. Nonclonogenic tumour cells may have a doubling time different from the clonogenic cells. In addition, flow cytometry may estimate too high a Tpot. This has been suggested by the data of Bennet et al. [1] who examined human squamous cell carcinoma of the head and neck. They found that Tpot values, which were determined by histological evaluation, were shorter than Tpt determined by flow cytometry. This was especially evident in diploid tumours, but the histological method determined very short values, less than 24 h, in aneuploid as well as in diploid tumours, in contrast to that no Tpot lower than 2 days was determined by flow cytometry. The CDT after 40 Gy in small tumours was estimated to be even shorter than the pretreatment cell cycle time, T,, which has been measured to 28 h. The T, may not be of a constant duration, since especially the G1 may vary after treatment [19]. The CDT that was estimated by the logistic model to 15.1 h was close to the T,, which has been determined to 16.2 (13.1; 19.3) h by flow cytometry [8]. The CDT is not expected to be shorter than the T,, and therefore, the CDT that was determined to be about 9 h by the Gompertz model for small tumours after irradiation is too small. This also suggests that the Gompertz model may only be valid for larger tumours, and is in accordance with the views that have been expressed by Wheldon [24]. Little is known about the dependency of repopulation on tumour type, tumour differentiation, level of clonogenic cell depletion, and on tumour growth parameters in general. Three main mechanisms for accelerated repopulation have been proposed: faster cell cycle, increased growth fraction, and reduced cell loss. Jones 191 suggested that fractionated irradiation selects for subsets of cells with faster cycling times since they will have a greater chance to multiply between fractions. This leads to an increase in the number produced per unit time. Jung et al. [lo] did not demonstrate a shorter cell cycle time after irradiating tumours in vivo or in vitro. The results in the present study support this idea since the CDT after irradiation was faster than the cell cycle time that has been demonstrated in the untreated SCCVII tumour. Trott and Kummermehr [21] suggested that, during accelerated repopulation, the spontaneous stem cell loss due to differentiation is reduced. This may be due to tumour cells that have retained some
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of the homeostatic control mechanisms that are present in normal tissues. This suggestion was supported experimentally by Jung et al. [lo] and Brammer et al. [3], and the present finding of an acceleration in the small tumours compared with untreated controls of the same size is also in agreement with this hypothesis. Fowler [6] has argued that reduced cell loss results in ‘unmasking’ the true tumour cell production. This view was sup ported by the observations by Jung et al. [lo] who found that the cell loss decreased by at least a factor of 2 at the onset of repopulation after irradiation. If ‘unmasking’ was the main mechanism behind the accelerated growth in the present study, no difference would be expected between small irradiated tumours and similar sized untreated tumours. In conclusion, the study shows that the regrowth of SCCVII tumour cells after clamped irradiation is accelerated compared to the growth rate in untreated controls when a logistic model is used for analysis. However, great caution should be observed since the conclusion relies on extrapolation of data to small tumours, and the models analysed may crucially rely on the current set data. Shortening of the cell cycle time and recruitment of non-cycling clonogenic tumour cells is suggested as the main mechanism behind the acceleration.
A number of growth models have been proposed to describe the growth of tumours in experimental studies [ 18,241. The growth curve of a tumour will approximate an exponential relationship if the rates of cell production and of cell loss are proportional to the number of cells present in the tumour. The differential equation that describes the tumour growth is then given by
S(r) is the number of cells at time t. As the tumour size increases, the rate of tumour growth tends to slow down [ 181. This leads to a generalized model
wheref(s) is a term that tends to 0 as S(t) tends to its asymptotic maximum value, S,, The logistic models and the Gompertz models are two classes of growth models that have this characteristic. In the logistic model the term&) is defined as f(s) = 1 -
+ (
N ->
(5)
0. Hansen
In the Gompertz model the termf(s)
et al. /Radiotherapy
is defined as
f(s) = In *
(6)
The solution for the logistic model for N greater than 0 is given in Eq. (1). The constant N determines the shape of the curve. For decreasing values of N, the decelerating portion of the curve becomes more pronounced. For increasing values of N, the growth approximates the exponential growth curves [17]. From Eq. (l), it follows that the clonogenic doubling time may be expressed as
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143
In both models, S(t) - S,, as t - 00. The qualitative difference between the models is the asymptotic behaviour of S(t) as t tends to zero. In the logistic model, f(s) approaches 1 for small tumours (Eq. (5)), and this means that the growth curve for the logistic model approaches simple exponential growth for small tumours. In the Gompertz equationf(s) tends to infinity for small tumours (Eq. (6)) equivalent to that the growth curve approximates a vertical line. The growth rate becomes infinitely fast (Eq. (12)). References 111 Bennet, M.H., Wilson, G.D., Dische, S., Saunders, MI., Mar-
If N equals 1, Eq. (1) simplifies to S(t) =
GUlX 1 + exp(- (cf. 2) - c)
c=-ln[(?)
-11
03)
and the doubling time may be expressed as
CDT=--
1 a!
In
(
&ax ( SW > -
(9)
&ax 2 ( * S(t) >
snax ( >I
The solution to the Gompertz equation is S(r) = So . exp ( 1 -exp(-a.?)) [
.ln
-
so
The decrease in the growth rate with increasing tumour size is more pronounced with the Gompertz than with the logistic model. The Gompertz equation is often reparameterized using the relation between S,, and So S-
= So . exp(@cf)
(11)
and a convenient linear transformation is given in Eq. (2). The doubling time varies with number of clonogenic cells in the tumour, and may be expressed as CDT=-L.ln
a
ln2 ln(S(t)lS&
1
(12)
tindale, C.A., Robinson, B.M., O’Halloran, A.E., Leslie, M.D. and Laing, J.H.E. Tumour proliferation assessedby combined histological and flow cytometric analysis: implications for therapy in squamous cell carcinoma in the head and neck. Br. J. Cancer 65: 870-878, 1992. 121 Bentzen, SM. and Thames, H.D. Clinical evidence for tumor clonogen regeneration: Interpretations of the data. Radiother. Oncol. 22: 161-166, 1991. I31 Brammer, I., Zywietx, F., Beck-Bomholdt, H.-P. and Jung, H. Kinetics of depopulation, repopulation and host cell infiltration in the rhabdomyosarcoma RlH after 14 MeV neutron irradiation. Int. J. Radiat. Biol. 61: 703-711, 1992. 141 Campbell, M.J., Gardner, M.J. Calculating confidence intervals for some non-parametric analyses. In: Statistics with Contidence. Confidence Intervals and Statistical Guidelines, pp. 71-79. Editors: M.J. Gardner and D.G. Altman. British Medical Journal, London, 1989. PI Courtenay, V.D. A soft agar colony assay for Lewis lung tumour and Bl6 melanoma taken directly from the mouse. Br. J. Cancer 34: 39-45, 1976. R Fowler, J.F. Rapid repopulation in radiotherapy: Debate on mechanism (letter). Radiother. Oncol. 24: 126, 1992. 171 Grau, C., Horsman, M.R. and Overgaard, J. Influence of carboxyhemoglobin level on tumor growth, blood flow, and radiation response in an experimental model. Int. J. Radiat. Oncol. Biol. Phys. 22: 421-424, 1992. 181 Heyer, M., Bentzen, S.M., Sailing, L.N. and Overgaard, 1. Influence of sampling time on assessmentof potential doubling time. Cytometry 16: 144-151, 1994. [91 Jones, B. Rapid repopulation in radiotherapy: A debate on mechanism (letter). Radiother. Oncol. 24: 124-125, 1992. 1101Jung, H., Kruger, H.-J., Brammer, I., Zywietz, F. and BeckBomholdt, H.-P. Cell population kinetics of the rhabdomyosarcoma RlH of the rat after single doses of X-rays. Int. J. Radiat. Biol. 57: 567-589, 1990. 1111Milas, L., Yamada, S., Hunter, N., Guttenberger, R. and Thames, H.D. Changes in TCDSO as a measure of clonogen doubling time in irradiated and unirradiated tumors. Int. J. Radiat. Oncol. Biol. Phys. 21: 1195-1202, 1991. WI Olive, P.L., Chaplin, D. J. and Durand, R.E. Pharmacokinetics, binding and distribution of Hoechst 33342 in spheroids and murine turnouts. Br. J. Cancer 52: 739-746, 1985. iI31 Overgaard, J., Hjelm-Hansen, M., Johansen, L.V. and Andersen, A.P. Comparison of conventional and split-course radiotherapy as primary treatment in carcinoma of the larynx. Acta Oncol. 27: 147-152, 1988. iI41 Siegel, S. Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill, 1956. I151 Spang-Thomsen, M., Rygaard, K., Hansen, L., Halvorsen, A.C., Vindelev, L.L. and Briinner, N. Growth kinetics of four
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0. Hansen et al. /Radiotherapy and Oncology human breast carcinomas grown in nude mice. Breast Cancer Res. Treat. 14: 235-243, 1989. Speke, A.K. and Hill, R.P. Repopulation kinetics during fractionated irradiation and the relationship to the potential doubling time, Tpot. Int. J. Radiat. Oncol. Biol. Phys. 31: 847-856, 1995. Spratt, J.A., von Fournier, D., Spratt, J.S. and Weber, E.E. Decelerating growth and human breast cancer. Cancer 71: 2013-2019, 1992. Steel, G.G. Growth Kinetics of Tumours. Clarendon Press, Oxford, 1977. Steel, G.G. The heyday of cell population kinetics: Insights from the 1960s and 1970s. Semin. Radiat. Oncol. 3: 78-83, 1993. Stephens, T.C. Measurement of tumor cell surviving fraction and absolute number of clonogens per tumor in excision assays. In: Rodent Tumor Models in Experimental Cancer Therapy, pp. 90-94. Editor: R.F. Kallman. Pergamon Press, New York, 1987.
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[21] Trott, K.R. and Kummermehr, J. Accelerated repopulation in tumours and normal tissues. Radiother. Oncol. 22: 159-160, 1991. [22] von der Maase, H. Effect of cancer chemotherapeutic drugs on the radiation-induced skin reactions in mouse feet. Br. J. Radiol. 57: 697-707, 1984. [23] von der Maase, H. and Overgaard, 1. Interactions of radiation and cancer chemotherapeutic drugs in a C3H mouse mammary carcinoma. Acta Radiol. Oncol. 24: 181-187, 1985. [24] Wbeldon, T.E. Models of tumour growth. In: Mathematical Models in Cancer Research, pp. 63-90. Adam Hilger, Bristol, 1988. [25] Withers, H.R., Taylor, J.M.G. and Maciejewski, B. The hazard of accelerated tumor clonogen repopulation during radiotherapy. Acta Oncol. 27: 131-146, 1988.