Determining the parameter space for effective oxygen depletion for FLASH radiation therapy.

There has been a recent revival of interest in the FLASH effect, after experiments have shown normal tissue sparing capabilities of ultra-high-dose-rate radiation with no compromise on tumour growth restraint. A model has been developed to investigate the relative importance of a number of fundamental parameters considered to be involved in the oxygen depletion paradigm of induced radioresistance. An example eight-dimensional parameter space demonstrates the conditions under which radiation may induce sufficient depletion of oxygen for a diffusion-limited hypoxic cellular response. Initial results support experimental evidence that FLASH sparing is only achieved for dose rates on the order of tens of Gy s-1or higher, for a sufficiently high dose, and only for tissue that is slightly hypoxic at the time of radiation. We show that the FLASH effect is the result of a number of biological, radiochemical and delivery parameters. Also, the threshold dose for a FLASH effect occurring would be more prominent when the parameterisation was optimised to produce the maximum effect. The model provides a framework for further FLASH-related investigation and experimental design. An understanding of the mechanistic interactions producing an optimised FLASH effect is essential for its translation into clinical practice.

Biological Benefits of Ultra-high Dose Rate FLASH Radiotherapy: Sleeping Beauty Awoken.

FLASH radiotherapy (FLASH-RT) is a technology that could modify the way radiotherapy is delivered in the future. This technique involves the ultra-fast delivery of radiotherapy at dose rates several orders of magnitude higher than those currently used in routine clinical practice. This very short time of exposure leads to the striking observation of relative protection of normal tissues that are exposed to FLASH-RT as compared with conventional dose rate radiotherapy. Here we summarise the current knowledge about the FLASH effect and provide a synthesis of the observations that have been reported on various experimental animal models (mice, zebrafish, pig, cats), various organs (lung, gut, brain, skin) and by various groups across 40 years of research. We also propose possible mechanisms for the FLASH effect, as well as possible paths for clinical application.

Ribozyme minigene-mediated RAD51 down-regulation increases radiosensitivity of human prostate cancer cells.

The strand transferase RAD51 is a component of the homologous recombination repair pathway. To examine the contribution of RAD51 to the genotoxic effects of ionising radiation, we have used a novel ribozyme strategy. A reporter gene vector was constructed so that expression of an inserted synthetic double-stranded ribozyme-encoding oligonucleotide would be under the control of the cytomegalovirus immediate-early gene enhancer/promoter system. The prostate tumour cell line LNCaP was transfected with this vector or a control vector, and a neomycin resistance gene on the vector was used to create geneticin-resistant stable cell lines. Three stable cell lines were shown by western blot analysis to have significant down-regulation of RAD51 to 20-50% of the levels expressed in control cell lines. All three cell lines had a similar increased sensitivity to gamma-irradiation by 70 and 40%, respectively, compared to normal and empty vector-transfected cells, corresponding to dose-modifying factors of approximately 2.0 and 1.5 in the mid-range of the dose-response curves. The amount of RAD51 protein in transfected cell lines was shown to strongly correlate with the alpha parameter obtained from fitted survival curves. These results highlight the importance of RAD51 in cellular responses to radiation and are the first to indicate the potential use of RAD51-targeted ribozyme minigenes in tumour radiosensitisation.

Development of a novel rapid assay to assess the fidelity of DNA double-strand-break repair in human tumour cells.

Cellular survival following ionising radiation-mediated damage is primarily a function of the ability to successfully detect and repair DNA double-strand breaks (DSBs). Previous studies have demonstrated that radiosensitivity, determined as a reduction in colony forming ability in vitro, may be related to the incorrect repair (misrepair) of DSBs. The novel rapid dual fluorescence (RDF) assay is a plasmid-based reporter system that rapidly assesses the correct rejoining of a restriction-enzyme produced DSBs within transfected cells. We have utilised this novel assay to determine the fidelity of DSB repair in the prostate tumour cell line LNCaP, the bladder tumour cell line MGH-U1 and a radiosensitive subclone S40b. The two bladder cell lines have been shown in previous studies to differ in their ability to correctly repair plasmids containing a single DSB. Using the RDF assay we found that a substantial portion of LNCaP cells [80.4 +/- 5.3(standard error)%] failed to reconstitute reporter gene expression; however, there was little difference in this measure of DSB repair fidelity between the two bladder cell lines (48.3 +/- 3.5% for MGH-U1; 39.9 +/- 8.2% for S40b). The RDF assay has potential to be developed to study the relationship between DSB repair fidelity and radiosensitivity as well as the mechanisms associated with this type of repair defect.

The RBE issues in ion-beam therapy: conclusions of a joint IAEA/ICRU working group regarding quantities and units.

This paper summarises the conclusions of a working group established jointly by the International Atomic Energy Agency (IAEA) and the International Commission on Radiation Units and Measurements (ICRU) to address some of the relative biological effectiveness (RBE) issues encountered in ion-beam therapy. Special emphasis is put on the selection and definition of the involved quantities and units. The isoeffective dose, as introduced here for radiation therapy applications, is the dose that delivered under reference conditions would produce the same clinical effects as the actual treatment in a given system, all other conditions being identical. It is expressed in Gy. The reference treatment conditions are: photon irradiation, 2 Gy per fraction, 5 daily fractions a week. The isoeffective dose D(IsoE) is the product of the physical quantity absorbed dose D and a weighting factor W(IsoE). W(IsoE) is an inclusive weighting factor that takes into account all factors that could influence the clinical effects like dose per fraction, overall time, radiation quality (RQ), biological system and effects. The numerical value of W(IsoE) is selected by the radiation-oncology team for a given patient (or treatment protocol). It is part of the treatment prescription. Evaluation of the influence of RQ on W(IsoE) raises complex problems because of the clinically significant RBE variations with biological effect (late vs. early) and position in depth in the tissues which is a problem specific to ion-beam therapy. Comparison of the isoeffective dose with the equivalent dose frequently used in proton- and ion-beam therapy is discussed.

Loss of atm radiosensitizes multiple p53 null tissues.

An unusual clinical finding in ataxia-telangiectasia, a human disorder caused by mutations in atm, is exquisite sensitivity to gamma irradiation. By contrast, homozygous deletion of p53 is marked by radiation resistance in certain tissue compartments. Previous studies (A. J. Levine, Cell, 88: 323-331, 1997) have shown that, in vitro, p53-deficient bone marrow cells are resistant to gamma irradiation. Furthermore, the gastrointestinal radiosensitization engendered by the loss of atm has recently been shown (C. H. Westphal et al., Nat. Genet., 16: 397-401, 1997) to be independent of p53. Expanding on previous work, we have looked at in vivo bone marrow resistance in p53-deficient mice. Our results indicate that inbred FVB strain p53 null mice survive lethal irradiation doses because of bone marrow resistance. Moreover, the deletion of atm radiosensitizes even p53 null bone marrow and mouse embryonic fibroblast cells. The results presented here argue that the loss of atm radiosensitizes multiple tissues in a p53-independent manner. Hence, functional inhibition of atm in p53 null and p53 wild-type human tumors may be a useful adjunct to gamma irradiation-based antitumor therapy.

ICRP Publication 131: Stem cell biology with respect to carcinogenesis aspects of radiological protection.

Current knowledge of stem cell characteristics, maintenance and renewal, evolution with age, location in 'niches', and radiosensitivity to acute and protracted exposures is reviewed regarding haematopoietic tissue, mammary gland, thyroid, digestive tract, lung, skin, and bone. The identity of the target cells for carcinogenesis continues to point to the more primitive and mostly quiescent stem cell population (able to accumulate the protracted sequence of mutations necessary to result in malignancy), and, in a few tissues, to daughter progenitor cells. Several biological processes could contribute to the protection of stem cells from mutation accumulation: (1) accurate DNA repair; (2) rapid induced death of injured stem cells; (3) retention of the intact parental strand during divisions in some tissues so that mutations are passed to the daughter differentiating cells; and (4) stem cell competition, whereby undamaged stem cells outcompete damaged stem cells for residence in the vital niche. DNA repair mainly operates within a few days of irradiation, while stem cell replications and competition require weeks or many months depending on the tissue type. This foundation is used to provide a biological insight to protection issues including the linear-non-threshold and relative risk models, differences in cancer risk between tissues, dose-rate effects, and changes in the risk of radiation carcinogenesis by age at exposure and attained age.

Radiation induced hemopoiesis in adult mouse liver.

Repeated whole-body irradiation of adult mice induced hemopoiesis in liver, as shown by the presence of stem cells (CFUS), progenitor cells of granulocytes and macrophages (CFUC) and foci of granulocytic cells. The largest numbers of CFUS (up to 700) were found 24 to 47 days after four doses of 450 rad x-rays given at 24 day intervals and 15-17 days after 2310 rad gamma radiation given at a low dose-rate (70 rad per day). CFUS were still present (although in smaller numbers) up to 210 days after four doses of 375 rad x-rays or 225 rad neutrons. CFUC were also present in liver after four doses of 450 rad x-rays, but their numbers could not be calculated accurately because of the marked inhibitory effect of liver cells on in vitro colony growth. Irradiation with one dose of 450 rad x-rays did not result in the appearance of CFUS in liver, suggesting that hepatic hemopoiesis can be induced by radiation only after repeated or prolonged bone marrow injury.

Decline in cycling of granulocyte-macrophage colony-forming cells with increasing age in mice.

The cycling state of granulocyte-macrophage progenitor cells (GM-CFC) was measured using the [3H]thymidine suicide technique in individual male B6D2F1 mice. The proportion of the GM-CFC population in S phase was found to decrease from 25%-35% at 1-6 months of age to about 10% in mice greater than 2 years old.

Residual deficiencies in hemopoietic precursor cell populations after repeated irradiation of mice with x-rays or neutrons: dose-response relationships.

Persistent reductions in the femoral content of hemopoietic colony-forming cells (CFU-S and GM-CFC) were observed after four doses of irradiation, delivered with three weeks between doses. In general, the reductions were dose dependent, and similar reductions were produced by single doses and repeated doses using the same total dose. After the lowest doses investigated, 4 X 0.75 Gy 300-kVp x-rays or 4 X 0.38 Gy 14.7-MeV neutrons, the recovered levels remained at 60%-80% of control for at least one year after irradiation. The relative biological effectiveness (RBE) of neutrons is about 2 for these long-term hemopoietic deficiencies.

Standardization of procedures for ectopic marrow grafting. II. Influence on recipients of radiation dose and field size.

The ectopic implantation of mouse marrow to the kidney capsule offers considerable scope as an assay of the hemopoietic microenvironment. Our previous work has shown that whole-body irradiation of the graft recipient prior to implantation results in superior ossicle formation in the kidney of the host. Here we report that a range of irradiation doses over a 4-Gy threshold are equivalent with respect to conditioning the graft recipient. We also show that two distinct and separable influences affect graft growth in the irradiated recipient, namely, a local effect brought about in the irradiated kidney (and restricted to it) and secondly, a systemic effect resulting from irradiation of sites other than the kidney, which nevertheless affects ossicle growth in the shielded renal capsule.

Recovery of the proliferative and functional integrity of mouse bone marrow in long-term cultures established after whole-body irradiation at different doses and dose rates.

Injury inflicted upon the bone marrow stroma following whole-body irradiation and its repair over a 1-year period has been assessed in murine long-term bone marrow cultures established at increasing time intervals after irradiation. Different doses at different dose rates (10 Gy at 0.05 cGy/min, 4.5 Gy and 10 Gy at 1.6 cGy/min, and 4 x 4.5 Gy [3 weeks between doses] at 60 cGy/min) were chosen so as to maximize differences in effect in the stroma. The cellularity of the adherent layer in long-term cultures established 1 month after irradiation was reduced by 40%-90% depending on the dose and dose rate. Simultaneous with the poor ability of the marrow to form adherent layers, the cumulative spleen colony-forming unit (CFU-S) and granulocyte-macrophage colony-forming cell (GM-CFC) production over a 7-week period was reduced to 0% and 30% of control cultures, respectively. The slow recovery of the adherent layer was paralleled by an increase in the numbers of CFU-S and GM-CFC in the supernatant. Cultures established from repeatedly irradiated mice performed poorly over the entire 1-year period. Whereas the regeneration of the stroma was near complete 1 year after irradiation, the CFU-S and GM-CFC levels reached only between 50% and 80% of control cultures, respectively. Also, the concentration of CFU-S and GM-CFC in the supernatant remained persistently lower in cultures established from irradiated mice as compared to control cultures. The levels of sulfated glycosaminoglycans, which have been implicated in the establishment of the functional integrity of the microenvironment, were not reduced in the adherent layers at any time after irradiation. These results indicate that the regeneration of the stroma is accompanied by an incomplete recovery of active hemopoiesis in vitro. However, no evidence was found for persistent functional defects in the stroma after irradiation, using the present endpoints.

The radiosensitivity of populations of murine hemopoietic colony-forming cells that respond to combinations of growth factors.

Combinations of murine recombinant interleukin 3 (IL-3), purified murine macrophage colony-stimulating factor (M-CSF), and human recombinant interleukin 1 alpha (IL-1 alpha) were used to determine the effects of growth factors on the measured radiosensitivity of different populations of murine colony-forming cells (CFC). The data showed that combinations of growth factors resulted in different values of CFC radiosensitivity, being less than values observed when colony growth was stimulated using a single factor. For various combinations of growth factors, Do values ranged from 106 +/- 8 to 175 +/- 24 cGy for progenitor cells in normal bone marrow; 74 +/- 3 to 171 +/- 18 cGy for primitive multipotent CFC enriched using fluorescence-activated cell sorting; and from 46 +/- 4 to 131 +/- 10 cGy for more mature granulocyte-macrophage CFC, enriched by counterflow centrifugal elutriation. Only combinations of three factors produced the high values of Do reported in experiments using unpurified conditioned medium as a stimulus for colony formation.

Threshold doses and circulatory disease risks.

Tissue reactions (deterministic effects) become manifest either early or late after doses above a threshold dose, which is the basis for recommended dose limits for avoiding such effects. Threshold doses have been defined for comparative purposes at 1% incidence of an effect, although the choice of incidence level may be scenario-dependent in practice. Latency time before manifestation is related to cell turnover rates and tissue complexity. In general, threshold doses become lower for longer follow-up times because of the slow progression of injury before manifestation, particularly after lower doses. Radiosensitive individuals may contribute to low threshold doses, which would provide a safety margin for the majority of a population. A threshold dose of 0.5 Gy was proposed for radiation-induced circulatory disease, after acute or chronic exposures, in the International Commission on Radiological Protection Publication 118. However, more recent meta-analyses of low-dose population studies suggest that, if a linear dose-incidence is assumed, the risk of some types of circulatory disease after doses <0.5 Gy or <10 mGy day(-1) may be positive and similar to that for induced cancer. Animal studies show that doses >2 Gy induce the expression of inflammatory and thrombotic molecules in endothelial cells. This causes progressive loss of capillaries in the heart and leads to reduced perfusion, myocardial cell death, and fibrosis. However, doses <1 Gy inhibit both inflammatory cell adhesion to endothelial cells and the development of atherosclerosis in mice. Different mechanisms of injury at low and high doses preclude the simple extrapolation of risk on a linear-quadratic basis from acute to chronic exposures.

ICRP Publication 131: Stem Cell Biology with Respect to Carcinogenesis Aspects of Radiological Protection.

This report provides a review of stem cells/progenitor cells and their responses to ionising radiation in relation to issues relevant to stochastic effects of radiation that form a major part of the International Commission on Radiological Protection's system of radiological protection. Current information on stem cell characteristics, maintenance and renewal, evolution with age, location in stem cell 'niches', and radiosensitivity to acute and protracted exposures is presented in a series of substantial reviews as annexes concerning haematopoietic tissue, mammary gland, thyroid, digestive tract, lung, skin, and bone. This foundation of knowledge of stem cells is used in the main text of the report to provide a biological insight into issues such as the linear-no-threshold (LNT) model, cancer risk among tissues, dose-rate effects, and changes in the risk of radiation carcinogenesis by age at exposure and attained age. Knowledge of the biology and associated radiation biology of stem cells and progenitor cells is more developed in tissues that renew fairly rapidly, such as haematopoietic tissue, intestinal mucosa, and epidermis, although all the tissues considered here possess stem cell populations. Important features of stem cell maintenance, renewal, and response are the microenvironmental signals operating in the niche residence, for which a well-defined spatial location has been identified in some tissues. The identity of the target cell for carcinogenesis continues to point to the more primitive stem cell population that is mostly quiescent, and hence able to accumulate the protracted sequence of mutations necessary to result in malignancy. In addition, there is some potential for daughter progenitor cells to be target cells in particular cases, such as in haematopoietic tissue and in skin. Several biological processes could contribute to protecting stem cells from mutation accumulation: (a) accurate DNA repair; (b) rapidly induced death of injured stem cells; (c) retention of the DNA parental template strand during divisions in some tissue systems, so that mutations are passed to the daughter differentiating cells and not retained in the parental cell; and (d) stem cell competition, whereby undamaged stem cells outcompete damaged stem cells for residence in the niche. DNA repair mainly occurs within a few days of irradiation, while stem cell competition requires weeks or many months depending on the tissue type. The aforementioned processes may contribute to the differences in carcinogenic radiation risk values between tissues, and may help to explain why a rapidly replicating tissue such as small intestine is less prone to such risk. The processes also provide a mechanistic insight relevant to the LNT model, and the relative and absolute risk models. The radiobiological knowledge also provides a scientific insight into discussions of the dose and dose-rate effectiveness factor currently used in radiological protection guidelines. In addition, the biological information contributes potential reasons for the age-dependent sensitivity to radiation carcinogenesis, including the effects of in-utero exposure.

The distribution of colony-forming cells in the kidney.

A method is described for producing outgrowths of small nephron segments (average 24 cells) in culture. The method was used to estimate an overall colony-forming efficiency of 4.6% for cells constituting the segments. Efficiency was found to be lower for thick segments (1%) than for thin segments (6%) from Henle's loop. The latter higher level indicates that precursor cells are concentrated near the middle of the nephron. For comparison, a two-dose irradiation technique was used to calculate a mean number of 5 +/- 2 (SE) clonogens per segment producing outgrowths. This tended to be higher than the value of about 1 calculated from the 65% of segments producing outgrowths, as expected if the remaining segments contained no clonogens.

Intestinal crypt clonogens: a new interpretation of radiation survival curve shape and clonogenic cell number.

Estimates of the clonogen content (number of microcolony-forming cells) of murine intestinal crypts using microcolony assays show an apparent dependence on the radiation dose used in the assay of clonogen content. Crypt radiation survival curves often show increased curvature beyond that expected on the basis of the conventional linear-quadratic model. A novel form of crypt survival curve shape is proposed based on two contributory mechanisms of crypt killing. Six previously published sets of microcolony data were re-analysed using a dual-kill model, where target cells are killed by two contributory mechanisms, each described by a linear-quadratic function of dose. The data were analysed as two series--high-dose rate and low-dose rate irradiation. The data were fitted to the models using direct maximization of a quasi-likelihood, explicitly allowing for overdispersion. The dual-kill model can reproduce both the apparent dose-dependence of the clonogen estimates and the high-dose curvature of the dose-response curves. For both series of data the model was a significantly better fit to the data than the standard linear-quadratic model, with no evidence of any systematic lack of fit. The parameters of the clonogenic cell component of the model are consistent with other studies that suggest a low clonogen number (somewhat less than five) per crypt. The model implies that there is a secondary mechanism decreasing clonogen survival, and hence increasing clonogen number estimates, at high doses. The mechanisms underlying the modification of the dose-response are unclear, and the implied mechanisms of, for example, slow growth, induced either directly in the surviving cells or indirectly through stromal injury or bystander effects are only speculative. Nevertheless, the model fits the data well, demonstrating that there is greater kill at high doses in these experimental series than would be expected from the conventional linear-quadratic model. This alternative model, or another model with similar behaviour, needs to be considered when analysing in detail and interpreting microcolony data as a function of dose. The implied low number of < or = 5 of these regenerative and relatively radioresistant clonogenic cells is distinct from a similar number of much more radiosensitive precursor stem cells which undergo early apoptosis after doses around 1 Gy.

Residual skin injury after repeated irradiation: differences observed using healing, macrocolony, and microcolony endpoints.

Following three repeated tolerance doses to mouse tail skin, residual injury was characterized by a 35% reduction in the iso-effective dose compared to age-matched controls, using healing or macrocolony endpoints. In contrast, the reduction was only 9%, measured using microcolony formation. The colony data showed that the reduction was a constant dose, not a dose-modifying effect. The residual injury is interpreted as due to a reduced density of microcolony-forming cells in the epidermis, and these are less capable of macrocolony formation and hence of re-epithelialization in the repeatedly-irradiated epidermis.

A realistic closed-form radiobiological model of clinical tumor-control data incorporating intertumor heterogeneity.

PURPOSE: To investigate the role of intertumor heterogeneity in clinical tumor control datasets and the relationship to in vitro measurements of tumor biopsy samples. Specifically, to develop a modified linear-quadratic (LQ) model incorporating such heterogeneity that it is practical to fit to clinical tumor-control datasets. METHODS AND MATERIALS: We developed a modified version of the linear-quadratic (LQ) model for tumor control, incorporating a (lagged) time factor to allow for tumor cell repopulation. We explicitly took into account the interpatient heterogeneity in clonogen number, radiosensitivity, and repopulation rate. Using this model, we could generate realistic TCP curves using parameter estimates consistent with those reported from in vitro studies, subject to the inclusion of a radiosensitivity (or dose)-modifying factor. We then demonstrated that the model was dominated by the heterogeneity in alpha (tumor radiosensitivity) and derived an approximate simplified model incorporating this heterogeneity. This simplified model is expressible in a compact closed form, which it is practical to fit to clinical datasets. Using two previously analysed datasets, we fit the model using direct maximum-likelihood techniques and obtained parameter estimates that were, again, consistent with the experimental data on the radiosensitivity of primary human tumor cells. This heterogeneity model includes the same number of adjustable parameters as the standard LQ model. RESULTS: The modified model provides parameter estimates that can easily be reconciled with the in vitro measurements. The simplified (approximate) form of the heterogeneity model is a compact, closed-form probit function that can readily be fitted to clinical series by conventional maximum-likelihood methodology. This heterogeneity model provides a slightly better fit to the datasets than the conventional LQ model, with the same numbers of fitted parameters. The parameter estimates of the clinically important time factors and lag periods are very similar to those obtained from the conventional LQ model, but with slightly narrower confidence intervals, reflecting the better fit to the clinical data. DISCUSSION: We have demonstrated, as have others, the importance of intertumor heterogeneity in the response of patient populations to radiotherapy. With the possible inclusion of a radiosensitivity-modifying factor (in vitro/in vivo) of around 1.7, the in vivo data can be made consistent with the in vitro SF2 and Tpot data. Fitting two previously analyzed multicenter datasets indicated that previous analyses based on conventional LQ models gave results for clinically important time factors and lags periods that were not significantly biased by the failure to include intertumor heterogeneity, with slightly narrower confidence intervals, reflecting the better fit to the clinical data. The simple closed-form model we have developed allows direct estimation of the heterogeneity in radiosensitivity within clinical series, and should prove useful in the analysis of other clinical series.

Time factors in larynx tumor radiotherapy: lag times and intertumor heterogeneity in clinical datasets from four centers.

PURPOSE: To use the time-dependent linear-quadratic model, both in the standard form and in a form modified to incorporate intertumor heterogeneity, in a reanalysis of 4 datasets for larynx tumor control, to provide more representative and direct estimates of the lag period, the time factor (lambda/alpha), and the clonogen population inactivation dose ([lnk]/alpha). METHODS AND MATERIALS: The data comprised 2,225 patients treated in Edinburgh (UK), Glasgow (UK), Manchester (UK), or Toronto (Canada), with tumor control assessed after at least 2 years. Heterogeneity in each series was taken into account using the coefficient of variation (CV) of the clonogen radiosensitivity (alpha). Maximum likelihood techniques were used to provide best estimates of the parameters, and also direct estimation of the more stable parameter ratios of interest. RESULTS: The use of different heterogeneity factors for the different series allowed common dose/time parameters to be fitted across all four series in a way not possible using the standard model, enabling the inherent effect of heterogeneity in flattening dose-response curves and in reducing time factors to be separated from the underlying more-representative values. Radiosensitivity CVs were calculated to be 30% (Edinburgh), 36% (Glasgow), 40% (Manchester), and 71% (Toronto). The lag phase was 32 days (95% CL 20-38 days) which was longer than the value of 23 days (11-36 days) deduced using the standard model without the heterogeneity parameter. The time factor was 1.2 (0.8-2.2) Gy/day, again greater than the value of 0.80 (0.54-1.41) Gy/day derived using the standard model. Similar larger time factors and longer lag periods could be reproduced using the standard model either by using a parameterization based on parameter ratios, or by omitting the discordant Toronto data and refitting just the data from the three UK centers. CONCLUSION: It was concluded that the heterogeneity model provides a better representation of the time factor for tumor control when data are analyzed comprising different stages of disease treated at different centers. The model allows different amounts of heterogeneity in different series, which tend to flatten dose-responses curves and reduce time factors, to be taken in to account. Also, direct maximum likelihood estimates can be made of the lag period, the time factor (lambda/alpha), and the fractionation sensitivity (beta/alpha), as well as the clonogen population inactivation dose (lnk)/alpha. Values of these parameter ratios are more robust and stable than the individual parameter values. The results of the present analysis using a total of 2,225 patients from four centers indicate that the average lag period may be somewhat longer and the average time factor somewhat greater (and the 95% confidence limits of the time factor exclude previous estimates), than the values deduced previously using simpler models and more diverse multi-center datasets.