Application of Bayesian inference to characterize risks associated with low doses of low-LET radiation

Improved risk characterization for stochastic biological effects of low doses of low-LET radiation is important for protecting nuclear workers and the public from harm from radiation exposure. Here we present a Bayesian approach to characterize risks of stochastic effects from low doses of low-LET radiation. The stochastic effect considered is neoplastic transformation of cells because it relates closely to cancer induction. We have used a published model of neoplastic transformation called NEOTRANS1. It is based on two different classes of cellular sensitivity for asynchronous, exponentially growing populations (in vitro). One sensitivity class is the hypersensitive cell; the other is the resistant cell. NEOTRANS1 includes the effects of genomic damage accumulation, DNA repair during cell cycle arrest, and DNA misrepair (non-lethal repair errors). The model-associated differential equations are solved for conditions of in vitro irradiation at a fixed rate. Previously published solutions apply only to high dose rates and were incorrectly assumed to apply to only high-LET radiation. Solutions provided here apply to any fixed dose rate and to both high- and low-LET radiations. Markov chain Monte Carlo methods are used to carry out the Bayesian inference of the low-dose risk for neoplastic transformation of aneuploid C3H 10T1/2 cells for X-ray doses from 0 to 1000 mGy. We have assumed that for this low-dose range only the hypersensitive fraction of the cells are affected. Our results indicate that the initial slope of the risk vs dose relationship for neoplastic transformation is as follows: (1) directly proportional to the fraction, f1, of hypersensitive cells; (2) directly proportional to the radiosensitivity of the genomic target; and (3) inversely proportional to the rate at which hypersensitive cells with radiation-induced damage are committed to undergo correct repair of genomic damage. Further, our results indicate that very fast molecular events are associated with the commitment of cells to the correct repair pathway. Results also indicate a relatively large probability for misrepair that leads to genomic instability. Our results are consistent with the view that for very low doses, dose rate is not an important variable for characterizing low-LET radiation risks so long as age-related changes in sensitivity do not occur during irradiation.     

A model for radiation-induced bystander effects, with allowance for spatial position and the effects of cell turnover

Bystander effects, whereby cells that are not directly exposed to ionizing radiation exhibit adverse biological effects, have been observed in a number of experimental systems. A novel stochastic model of the radiation-induced bystander effect is developed that takes account of spatial location, cell killing and repopulation. The ionizing radiation dose- and time-responses of this model are explored, and it is shown to exhibit pronounced downward curvature in the high dose-rate region, similar to that observed in many experimental systems, reviewed in the paper. It is also shown to predict the augmentation of effect after fractionated delivery of dose that has been observed in certain experimental systems. It is shown that the generally intractable solution of the full stochastic system can be considerably simplified by assumption of pairwise conditional dependence that varies exponentially over time.     

Incorporating dose-rate effects in Markov radiation cell survival models

Markov models for the survival of cells subjected to ionizing radiation take stochastic fluctuations into account more systematically than do non-Markov counterparts. Albright's Markov RMR (repair-misrepair) model (Radiat. Res. 118, 1-20, 1989) and Curtis's Markov LPL (lethal-potentially lethal) model [in Quantitative Mathematical Models in Radiation Biology (J. Kiefer, Ed.), pp. 127-146. Springer, New York, 1989], which assume acute irradiation, are here generalized to finite dose rates. Instead of treating irradiation as an instantaneous event we introduce an irradiation period T and analyze processes during the interval T as well as afterward. Albright's RMR transition matrix is used throughout for computing the time development of repair and misrepair. During irradiation an additional matrix is added to describe the evolving radiation damage. Albright's and Curtis's Markov models are recovered as limiting cases by taking T----0 with total dose fixed; the opposite limit, of low dose rates, is also analyzed. Deviations from Poisson behavior in the statistical distributions of lesions are calculated. Other continuous-time Markov chain models ("compartmental models") are discussed briefly, for example, models which incorporate cell proliferation and saturable repair models. It is found that for low dose rates the Markov RMR and LPL models give lower survivals compared to the original non-Markov versions. For acute irradiation and high doses, the Markov models predict higher survivals. In general, theoretical extrapolations which neglect some random fluctuations have a systematic bias toward overoptimism when damage to irradiated tumors is compared with damage to surrounding tissues.     

Tissue repair and repopulation in the tumor bed effect

These experiments were designed to study the kinetics and magnitude of cell repair and repopulation in tissues whose damage results in the tumor bed effect. The right hind thighs of mice were irradiated with single doses or two equal gamma-ray fractions. Interfraction intervals ranging from 30 min to 24 h (to measure the kinetics of repair from sublethal damage) and 6 and 12 weeks (to determine the extent of repopulation) were used. One day after the second radiation dose 5 X 10(5) FSA tumor cells were inoculated into the center of the irradiated field. Radiation dose-response curves were obtained by calculating the time required for tumors to reach 12 mm diameter. No recovery occurred within 6 h of the radiation delivery as measured by this assay. Some recovery, 3.2-4.6 Gy above a single radiation dose, occurred when the interval between two fractions was 24 h. With increasing interfraction intervals of 6 and 12 weeks further dose sparing occurred in the amount of 5.0-6.9 and 7.5-8.3 Gy, respectively. The data suggest that repopulation is the major contributor to the radiation dose-sparing recovery of stromal tissue and that some proliferative response may occur as early as 1 day after the first irradiation.     

Repopulation kinetics of rat rhabdomyosarcoma tumors following single and fractionated doses of low-LET and high-LET radiation

Measurements were made of clonogenic cell survival in rat rhabdomyosarcoma tumors as a function of time following in situ irradiation with single or fractionated doses of 225-kVp X rays or with 557-MeV/u neon ions in the distal position of a 4-cm extended-peak ionization region. Single doses of 20 Gy of X rays or 7 Gy of peak neon ions reduced the initial surviving fraction to approximately 0.025 for each modality. Daily fractionated doses (four fractions in 3 days) of either peak neon ions (1.75 Gy per fraction) or X rays (6 Gy per fraction) achieved a cell survival of approximately 0.02-0.03 after the fourth dose of radiation. In the single-dose experiments, significant 5- and 10-fold decreases in the fraction of clonogenic cells were observed between the third and fourth days after irradiation with peak neon ions and X rays, respectively. After the sixth day postirradiation, the residual clonogenic cells exhibited a rapid burst of proliferation leading to doubling times for the surviving cell fractions of approximately 1.5 days. Radiation-induced growth delay was consistent with the cellular repopulation dynamics. In the fractionated-dose experiments with both radiation modalities, a large delayed decrease in cell survival was observed at 1-3 days after completion of the fractionated-dose schedule. Cellular repopulation was consistent with postirradiation tumor volume regression and regrowth for both radiation modalities. The extent of decrease in survival following the four-fraction radiation schedule was approximately two times greater in X-irradiated than in neon-ion-irradiated tumors that produced the same survival level immediately after the fourth dose. Mechanisms underlying the marked reduction in cell survival 3-4 days postirradiation are discussed, including the possible role of a toxic host cell response against the irradiated tumor cells.     

Repair, redistribution and repopulation in V79 spheroids during multifraction irradiation

Abstract. Development of predictive assays for measuring tumour radiosensitivity has generated much recent interest, particularly with the recognition that tumour cell survival at doses of about 2 Gy may correlate well with tumour curability. Clinical data, however, suggest that overall treatment time may be of considerable significance in radioresponsive tumours, especially for rapidly growing tumours capable of accelerated repopulation. Because neither factor can be repeatedly assessed in human tumours, we used cells growing as multicell spheroids to determine whether the initial radiation response would be predictive for multifraction exposures, or whether other factors including repopulation rate should be considered. Potential problems of hypoxia and reoxygenation were avoided by using small spheroids which had not yet developed radiobiologically hypoxic regions. Repair and redistribution dominated the responses in the first two or three exposures, with repopulation playing a minor role. As the fractionation schedule was extended, however, repopulation between fractions largely determined the number of viable cells per spheroid. We conclude that the radiation response of cells from untreated spheroids provides a general indication of net sensitivity, but that repair and redistribution produces considerable variation in radiosensitivity throughout a fractionation protocol. Ultimately, repopulation effects may dominate the multifraction response.     

Promoting action of radiation in the atomic bomb survivor carcinogenesis data?

The age-time patterns of risk in the atomic bomb survivor data on incidence of solid cancers suggest an action of low-LET radiation not only on the initiating event but also on promotion in a biologically motivated model that allows for both actions. The favored model indicates a decrease of radiation risks with age at exposure due to the initiating effect and with time since exposure due to the promoting effect. These result in a relative risk that depends mostly on attained age for ages at exposure above 20 years. According to the model, a dose of 100 mGy is inducing about the same number of initiating events that occur spontaneously in 1 year. Assuming that several mutations are needed to obtain intermediate cells with growth advantage does not improve the quality of fit. The estimated promoting effect could be explained if the number of intermediate cells increases by 80% at 1 Gy, e.g. due to stimulated cell repopulation.     

A new view of radiation-induced cancer: integrating short- and long-term processes. Part I: Approach

Mathematical models of radiation carcinogenesis are important for understanding mechanisms and for interpreting or extrapolating risk. There are two classes of such models: (1) long-term formalisms that track pre-malignant cell numbers throughout an entire lifetime but treat initial radiation dose–response simplistically and (2) short-term formalisms that provide a detailed initial dose–response even for complicated radiation protocols, but address its modulation during the subsequent cancer latency period only indirectly. We argue that integrating short- and long-term models is needed. As an example of this novel approach, we integrate a stochastic short-term initiation/inactivation/repopulation model with a deterministic two-stage long-term model. Within this new formalism, the following assumptions are implemented: radiation initiates, promotes, or kills pre-malignant cells; a pre-malignant cell generates a clone, which, if it survives, quickly reaches a size limitation; the clone subsequently grows more slowly and can eventually generate a malignant cell; the carcinogenic potential of pre-malignant cells decreases with age.     

Review and meta-analysis of epidemiological associations between low/moderate doses of ionizing radiation and circulatory disease risks, and their possible mechanisms

Although the link between high doses of ionizing radiation and damage to the heart and coronary arteries has been well established for some time, the association between lower-dose exposures and late occurring cardiovascular disease has only recently begun to emerge, and is still controversial. In this paper, we extend an earlier systematic review by Little et al. on the epidemiological evidence for associations between low and moderate doses of ionizing radiation exposure and late occurring blood circulatory system disease. Excess relative risks per unit dose in epidemiological studies vary over at least two orders of magnitude, possibly a result of confounding and effect modification by well-known (but unobserved) risk factors, and there is statistically significant (p < 0.00001) heterogeneity between the risks. This heterogeneity is reduced, but remains significant, if adjustments are made for the effects of fractionated delivery or if there is stratification by endpoint (cardiovascular disease vs. stroke, morbidity vs. mortality). One possible biological mechanism is damage to endothelial cells and subsequent induction of an inflammatory response, although it seems unlikely that this would extend to low-dose and low-dose-rate exposure. A recent paper of Little et al. proposed an arguably more plausible mechanism for fractionated low-dose effects, based on monocyte cell killing in the intima. Although the predictions of the model are consistent with the epidemiological data, the experimental predictions made have yet to be tested. Further epidemiological and biological evidence will allow a firmer conclusion to be drawn.     

The characterization and transmissibility of chromosome aberrations induced in peripheral blood lymphocytes by in vitro alpha-particle radiation

Peripheral blood lymphocytes were irradiated in vitro with (213)Bi alpha particles at doses of 0, 10, 20, 50, 100, 200 and 500 mGy. Chromosome analysis was performed on 47-h cultures using single-color fluorescence in situ hybridization (FISH) to paint chromosomes 1, 3 and 5. The whole genome was analyzed for unstable aberrations to derive aberration frequencies and determine cell stability. The dose response for dicentrics was 33.60 +/- 0.47 x 10(-2) per Gy. A more detailed analysis revealed that the majority of aberrations scored as dicentrics were part of complex/multiple aberrations, with the proportion of cells containing complexes increasing with dose. Cells containing aberrations involving painted chromosomes (FISH aberrations) were further classified according to cell stability and complexity. The majority of cells with FISH aberrations were unstable. The proportion of aberrant FISH cells with complex/multiple aberrations ranged from 56% at 10 mGy to 89% at 500 mGy. A linear dose response for genomic frequencies of translocations in stable cells fitted the data from 0 to 200 mGy with a dose response of 7.90 +/- 0.98 x 10(-2) per Gy, thus indicating that they are likely to be observed in peripheral blood lymphocytes from individuals with past or chronic exposure to high-LET radiation. Comparisons with the dose response for low-LET radiation suggest an RBE of 13.6 for dicentrics in all cells and 3.2 for translocations in stable cells. Since stochastic effects of radiation are attributable to genetic changes in viable cells, translocations in stable cells may be a better measure when considering the comparative risks of different qualities of radiation.     

Expansion or extinction: deterministic and stochastic two-patch models with Allee effects

We investigate the impact of Allee effect and dispersal on the long-term evolution of a population in a patchy environment. Our main focus is on whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global extinction. Our study is based on the combination of analytical and numerical results for both a deterministic two-patch model and a stochastic counterpart. The deterministic model has either two, three or four attractors. The existence of a regime with exactly three attractors only appears when patches have distinct Allee thresholds. In the presence of weak dispersal, the analysis of the deterministic model shows that a high-density and a low-density populations can coexist at equilibrium in nearby patches, whereas the analysis of the stochastic model indicates that this equilibrium is metastable, thus leading after a large random time to either a global expansion or a global extinction. Up to some critical dispersal, increasing the intensity of the interactions leads to an increase of both the basin of attraction of the global extinction and the basin of attraction of the global expansion. Above this threshold, for both the deterministic and the stochastic models, the patches tend to synchronize as the intensity of the dispersal increases. This results in either a global expansion or a global extinction. For the deterministic model, there are only two attractors, while the stochastic model no longer exhibits a metastable behavior. In the presence of strong dispersal, the limiting behavior is entirely determined by the value of the Allee thresholds as the global population size in the deterministic and the stochastic models evolves as dictated by their single-patch counterparts. For all values of the dispersal parameter, Allee effects promote global extinction in terms of an expansion of the basin of attraction of the extinction equilibrium for the deterministic model and an increase of the probability of extinction for the stochastic model.     

Cell kinetics and radiation biology

The cell cycle, the growth fraction and cell loss influence the response of cells to radiation in many ways. The variation in radiosensitivity around the cell cycle, and the extent of radiation-induced delay in cell cycle progression have both been clearly demonstrated in vitro. This translates into a variable time of expression of radiation injury in different normal tissues, ranging from a few days in intestine to weeks, months or even years in slowly proliferating tissues like lung, kidney, bladder and spinal cord. The radiosensitivity of tumours, to single doses, is dominated by hypoxic cells which arise from the imbalance between tumour cell production and the proliferation and branching of the blood vessels needed to bring oxygen and other nutrients to each cell. The response to fractionated radiation schedules is also influenced by the cell kinetic parameters of the cells comprising each tissue or tumour. This is described in terms of repair, redistribution, reoxygenation and repopulation. Slowly cycling cells show much more curved underlying cell survival curves, leading to more dramatic changes with fractionation, dose rate or l.e.t. Rapidly cycling cells redistribute around the cell cycle when the cells in sensitive phases have been killed, and experience less mitotic delay than slowly proliferating cells. Reoxygenation seems more effective in tumours with rapidly cycling cells and high natural cell loss rates. Compensatory repopulation within a treatment schedule may spare skin and mucosa but does not spare slowly proliferating tissues. Furthermore, tumour cell proliferation during fractionated radiotherapy may be an important factor limiting the overall success of treatment.     

Model of accelerated carcinogenesis based on proliferative stress and inflammation for doses relevant to radiotherapy

Recent findings demonstrate that accelerated carcinogenesis following liver regeneration is associated with chronic inflammation-induced double-strand DNA breaks in cells, which escaped apoptosis due to proliferative stress. In this work, proliferative stress and inflammation-based carcinogenesis at large dose were included in a cancer induction model considering fractionation. At large dose, tissue injury due to irradiation could be so severe that under the regenerative proliferative stress induced by cell loss, the genomic unstable cells generated during irradiation and/or inflammation escape senescence or apoptosis and reenter the cell cycle, triggering enhanced carcinogenesis. This acceleration—modeled to be proportional to the number of repopulated cells—is only significant, however, when tissue injury is severe and thus proportional to the cell loss in the tissue. The general solutions to the resulting differential equations for carcinoma induction were computed. In case of full repopulation or acute low-dose irradiation, the acceleration term disappears from the equation describing cancer induction. The acceleration term is affecting the dose–response curve for carcinogenesis only at large doses. An example for bladder cancer is shown. An existing model for cancer induction after fractionated radiotherapy which is based on cell mutations was extended here by including the effects of inflammation and proliferative stress, and an additional model parameter was established which describes acceleration. The new acceleration parameter affects the dose–response model only at large dose and is only effective when the tissue is not capable of fully repopulating between dose fractions.     

Migration of Langerhans cells and gammadelta dendritic cells from UV-B-irradiated sheep skin

Depletion of dendritic cells from UV-B-irradiated sheep skin was investigated by monitoring migration of these cells towards regional lymph nodes. By creating and cannulating pseudoafferent lymphatic vessels draining a defined region of skin, migrating cells were collected and enumerated throughout the response to UV-B irradiation. In the present study, the effects of exposing sheep flank skin to UV-B radiation clearly demonstrated a dose-dependent increase in the migration of Langerhans cells (LC) from the UV-B-exposed area to the draining lymph node. The range of UV-B doses assessed in this study included 2.7 kJ/m2, a suberythemal dose; 8 kJ/m2, 1 minimal erythemal dose (MED); 20.1 kJ/m2; 40.2 kJ/m2; and 80.4 kJ/m2, 10 MED. The LC were the cells most sensitive to UV-B treatment, with exposure to 8 kJ/m2 or greater reproducibly causing a significant increase in migration. Migration of gammadelta+ dendritic cells (gammadelta+ DC) from irradiated skin was also triggered by exposure to UV-B radiation, but dose dependency was not evident within the range of UV-B doses examined. This, in conjunction with the lack of any consistent correlation between either the timing or magnitude of migration peaks of these two cell types, suggests that different mechanisms govern the egress of LC and gammadelta+ DC from the skin. It is concluded that the depression of normal immune function in the skin after exposure to erythemal doses of UV-B radiation is associated with changes in the migration patterns of epidermal dendritic cells to local lymph nodes.     

Spatial patterns of coexistence of competing species in patchy habitat

The single-species spatially realistic patch occupancy metapopulation model is, in this study, extended to a metacommunity of many competing species. Competition is assumed to reduce the local carrying capacity (effective patch area), which in turn increases local extinction rates and reduces colonization rates because of smaller population sizes. Each species is described by three parameters: pre-competitive abundance (equilibrium incidence of patch occupancy, which reflects the rate of colonization in relation to extinction rate), the spatial range of migration, and competitive ability. The model ignores spatio–temporal correlations caused by interspecific interactions, because in metacommunities of unequal competitors inhabiting heterogeneous landscapes, correlations in the occurrence of species are driven more by patch heterogeneity than by competition. The model allows the calculation of multispecies equilibria in patchy habitats without simulations. In general, the number of coexisting species in the metacommunity increases with decreasing strength of competition, increasing rate of colonization, and decreasing range of migration. Habitat heterogeneity in the form of spatial variation in patch areas tends to facilitate coexistence. Poor competitors may coexist with superior competitors in the patch network if the former have higher colonization rates (competition–colonization trade-off). When migration distances are short, competition leads to spatial pattern formation: Species tend to have restricted spatial distributions in the network, but contrary to intuitive expectations, often the distributions of many species are nested. Having more dispersive species enhances both local and global diversity, whereas more local migration decreases local but increases global diversity.     

Stochastic dynamics in exotic and native blowflies: an analysis combining laboratory experiments and a two-patch metapopulation model

In this study we explored the stochastic population dynamics of three exotic blowfly species, Chrysomya albiceps, Chrysomya megacephala and Chrysomya putoria, and two native species, Cochliomyia macellaria and Lucilia eximia, by combining a density-dependent growth model with a two-patch metapopulation model. Stochastic fecundity, survival and migration were investigated by permitting random variations between predetermined demographic boundary values based on experimental data. Lucilia eximia and Chrysomya albiceps were the species most susceptible to the risk of local extinction. Cochliomyia macellaria, C. megacephala and C. putoria exhibited lower risks of extinction when compared to the other species. The simultaneous analysis of stochastic fecundity and survival revealed an increase in the extinction risk for all species. When stochastic fecundity, survival and migration were simulated together, the coupled populations were synchronized in the five species. These results are discussed, emphasizing biological invasion and interspecific interaction dynamics.     

The dynamics of gene amplification described as a multitype compartmental model and as a branching process.

The present work is aimed at developing the mathematical tools by which the dynamics of gene amplification (GA) can be described in detail. Some discrete compartmental models of GA by disproportionate replication and a general model for other putative GA mechanisms are presented and analyzed. The dynamical distribution of gene copy number in the cell population is calculated with the loss of cells taken either as constant or as copy-number-dependent. Our analysis shows that for a one-copy GA process with constant loss of cells, the relative frequency of single-gene-copy cells (sensitive cells) converges to zero, with the rate of convergence depending on the amplification probability. In contrast, for a one-copy GA process with copy-number-dependent loss of cells, the relative frequency of single-copy cells is bounded, implying a bounded compartment of many-gene-copy cells. Using branching processes theory we calculate the dynamical distribution of the single-gene-copy compartment as well as its extinction probability. Our models are used for estimating treatment prognosis as affected by drug resistance due to GA, showing significant differences in prognosis resulting from small changes in drug dose.     

Secondary carcinogenesis in patients treated with radiation: a review of data on radiation-induced cancers in human, non-human primate, canine and rodent subjects

Concern for risk of radiation-induced cancer is growing with the increasing number of cancer patients surviving long term. This study examined data on radiation transformation of mammalian cells in vitro and on the risk of an increased cancer incidence after irradiation of mice, dogs, monkeys, atomic bomb survivors, occupationally exposed persons, and patients treated with radiation. Transformation of cells lines in vitro increased linearly with dose from approximately 1 to approximately 4-5 Gy. At <0.1 Gy, transformation was not increased in all studies. Dose-response relationships for cancer incidence varied with mouse strain, gender and tissue/organ. Risk of cancer in Macaca mulatta was not raised at 0.25-2.8 Gy. From the atomic bomb survivor study, risk is accepted as increasing linearly to 2 Sv for establishing exposure standards. In irradiated patients, risk of cancer increased significantly from 1 to 45 Gy (a low to a high dose level) for stomach and pancreas, but not for bladder and rectum (1-60 Gy) or kidney (1-15 Gy). Risk for several organs/tissues increased substantially at doses far above 2 Gy. There is great heterogeneity in risk of radiation-associated cancer between species, strains of a species, and organs within a species. At present, the heterogeneity between and within patient populations of virtually every parameter considered in risk estimation results in substantial uncertainty in quantification of a general risk factor. An implication of this review is that reduced risks of secondary cancer should be achieved by any technique that achieved a dose reduction down to approximately [corrected] 0.1 Gy, i.e. dose to tissues distant from the target. The proportionate gain should be greatest for dose decrement to less than 2 Gy.     

The impact of phenotypic heterogeneity of tumour cells on treatment and relapse dynamics

Intratumour heterogeneity is increasingly recognized as a frequent problem for cancer treatment as it allows for the evolution of resistance against treatment. While cancer genotyping becomes more and more established and allows to determine the genetic heterogeneity, less is known about the phenotypic heterogeneity among cancer cells. We investigate how phenotypic differences can impact the efficiency of therapy options that select on this diversity, compared to therapy options that are independent of the phenotype. We employ the ecological concept of trait distributions and characterize the cancer cell population as a collection of subpopulations that differ in their growth rate. We show in a deterministic model that growth rate-dependent treatment types alter the trait distribution of the cell population, resulting in a delayed relapse compared to a growth rate-independent treatment. Whether the cancer cell population goes extinct or relapse occurs is determined by stochastic dynamics, which we investigate using a stochastic model. Again, we find that relapse is delayed for the growth rate-dependent treatment type, albeit an increased relapse probability, suggesting that slowly growing subpopulations are shielded from extinction. Sequential application of growth rate-dependent and growth rate-independent treatment types can largely increase treatment efficiency and delay relapse. Interestingly, even longer intervals between decisions to change the treatment type may achieve close-to-optimal efficiencies and relapse times. Monitoring patients at regular check-ups may thus provide the temporally resolved guidance to tailor treatments to the changing cancer cell trait distribution and allow clinicians to cope with this dynamic heterogeneity.     

A reanalysis of liver cancer incidence in Danish patients administered thorotrast using a two-mutation carcinogenesis model

In recent years, a two-mutation carcinogenesis (TMC) model has been used to analyze epidemiological data and estimate the radiation risks at low doses for the organs affected. Here the TMC model was used to reanalyze the liver cancer incidence in the Danish population in general and in patients administered Thorotrast, and to estimate the radiation risks for the liver. The data for 807 patients for whom sufficient data on the injected volumes of Thorotrast were available were used in this reanalysis. These data were combined with data on liver cancer incidence in the Danish population as the baseline or background incidence. Because males and females show different baseline liver cancer incidences, separate fits were made for males and females. The fits showed that the radiation effect could be ascribed entirely to the radiation dependence of the first mutation rate of the TMC model, which was higher for females than for males. The second mutation rate was not significantly dependent on dose. The radiation risks for the liver were calculated on the basis of the model parameters. These risks for lifetime exposures are about the same for males and females and are between a factor of 2 and 10 higher than current estimates. The discrepancy between the model results and previous risk estimates probably arises because the model calculations give more complete lifetime radiation risk estimates. For short-term exposures of the liver to ionizing radiation, the maximum radiation-induced excess liver cancer risk per unit dose applies to exposures at the age of about 10; exposures at ages above 35 have a radiation effect of less than approximately 15% of this maximum.