British Journal of Radiology 74 (2001),529-536 © 2001 The British Institute of Radiology
Radiation carcinogenesis modelling for risk of treatment-related second tumours following radiotherapy
K A Lindsay1,
E G Wheldon1,
C Deehan2 and
T E Wheldon3,4
1Department of Mathematics, University Gardens, University of Glasgow, Glasgow G12 8QW, 2Department of Physics, Royal Marsden Hospital, Fulham Road, London, 3Department of Radiation Oncology, Glasgow University, CRC Beatson Labs, Glasgow G61 1BD, and 4Department of Clinical Physics, Beatson Oncology Centre, Western Infirmary, Glasgow G11 6NT, UK
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Abstract
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Radiobiological modelling of the risk of radiation-induced tumours following high dose radiation implies a general form for the dose–response relationship. Generally, risk will rise with radiation dose at low doses, reach a maximum value and then decline with further increase in dose. The magnitude of risk and the dose at which this risk is maximum are strongly dependent on the kinetics of repopulation by surviving normal and mutant cells and on genetic factors likely to differ between tissues and between individuals. The most reliable way to reduce the risk of second tumours is to reduce radiation dose further at sites where the dose is already low. These sites are usually distant from the primary treatment volume. For illustrative purposes, we have compared the predicted relative risks of second tumours at "distant sites" for treatment plans giving similar dose distributions (dose volume histograms) at the primary site. We suggest that dose reduction to distant sites could be of significant benefit in reducing the risk of second tumours. Further improvement will require more detailed knowledge of the radiation sensitivities and mutagenicities, together with the repopulation kinetics of the various cell lineages within the treatment volume.
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Introduction
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Ionizing radiation is a known carcinogen for a wide variety of tumours in animals and man. Human populations are exposed to radiation from natural and industrial background sources and from medical diagnostic examinations, but by far the highest doses are received by radiotherapy patients. In round figures, about 40% of the population is expected to develop cancer at some time in their lives and radiotherapy is likely to be used as part of the treatment management for about one-third of these. Therefore about 12% of the population will be exposed to high dose radiation, receiving tens of Gy to tissues within the treatment volume and lesser doses elsewhere. These numbers of patients greatly exceed the numbers of individuals considered to be at risk from cancers caused by environmental radiation such as radon exposure. Of course, many radiotherapy patients are elderly and will have reduced life expectancy as a result of their initial cancer. In many cases, cells which have been mutated by radiotherapy and are progressing down a malignant pathway will fail to materialize as macroscopic tumours during the remaining lifetime of the patient. However, there are increasing cure rates amongst children and younger patients, placing them at risk of developing radiation-induced second tumours [1]. Moreover, there is growing evidence that second tumours are more likely following combined modality treatment, which is increasingly common. In some cases, younger patients may be especially vulnerable at particular ages, providing developmental "windows of opportunity" for second tumour development. For example, Bhatia et al [2] have reportedhigh vulnerability of adolescent girls receiving curative radiotherapy for Hodgkin's disease, leading to the projection of very high rates of breast cancer in later life. There are as yet unresolved questions about the vulnerability of genetically predisposed individuals who may be at markedly increased risk. It seems likely that there will in future be increasing clinical concern about young good-prognosis patients for whom there may be non-trivial risks of radiotherapy-induced second tumours. In this paper, mathematical modelling will be used to consider the form of the dose–response relationship for radiation-induced second tumours and to assess whether there are opportunities to minimize such risks without impairing the primary treatment.
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Dose–response relationship for radiation carcinogenesis
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Despite the large body of experimental and clinical experience which now exists, the general form of the dose–response curve for radiation carcinogenesis following high dose radiation is not completely clear. Experimental studies have yielded an array of dose–response relationships described by Hall [3] as "bewildering". Clinical studies in human patients have variously shown a rising rate of carcinogenic risk over a wide dose range (i.e. 60 Gy fractionated), a plateauing of risk at lower doses or phenomena suggesting adecline of risk with increasing dose once a critical dose level has been reached. For example, some studies have reported highest incidences of radiation-induced tumours as occurring at field peripheries where doses would be less than at fieldcenters [1]. Observation of high cancer risk at field peripheries or penumbrae, implying the existence of an "optimal dose" for radiation carcinogenesis, is consistent with the popular idea that radiation cell kill becomes dominant over carcinogenic mutation as radiation dose increases. This concept will be discussed further below. In considering the risk of second tumours following radiotherapy, and how this risk might be reduced, it is important to know what the dose–response relationship is in real situations.
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Radiation carcinogenesis, cell kill and repopulation kinetics
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In the multistage model of carcinogenesis, cells progress towards malignant change by accumulating a series of mutations (possibly no more thantwo) which results eventually in full malignancy. Radiation both mutates cells, causing them to advance down the pathway, and sterilizes cells, whether existing mutants or newly created mutants. Biologically it is axiomatic that lethal mutations causing cell sterilization eliminate pre-malignant mutant cells from the pathway of malignant change. No cancers can arise from cells that have received very high doses of radiation, since all of them will have been sterilized. However, it is less obvious what happens with doses which leave a few cells, or even a single cell, reproductively viable. Most proliferative normal tissues, especially acute-responding tissues like the epithelial and haemopoietic tissues, respond to irradiation by accelerated repopulation, a triggering of surviving cells into the proliferative cycle, which contributes to tissue sparing during fractionated radiotherapy. It is not known how pre-malignant mutant cells will respond to the signals for increased proliferation. In the simplest model, such cells continue to behave like normal tissue cells until sufficient mutations have been accumulated to initiate malignant change. Therefore pre-malignant cells and normal cells might fully compensate for cell sterilization by enhanced proliferation of survivors. In principle, compensation could be complete within each compartment provided at least one cell was left viable after irradiation. At higher doses than this, proliferative compensation would no longer be possible within each compartment. There have been few studies of the role of repopulation in offsetting radiation kill and the expected effect on the dose–response relationship for cancer incidence. In what follows we shall try to reach some general conclusions using a simple model, and assess whether such information could be used to reduce risk.
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Radiation carcinogenesis model
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The two-stage mutation model used in this article consists of three compartments and is illustrated in Figure 1
. Cells which are mutated by dose D have, by default, also survived irradiation. This dual event occurs with probability p:
where 
and ß are the intrinsic cellular radiosensitivities, µ is the spontaneous mutation rate, and 
and 
are the mutatonal radiosensitivities. The functions P0(t) and P1(t) are, respectively, the number of non-mutated stem cells (i.e. normal stem cells) and one-mutant stem cells at time t, while Z(t) is the probability that at least one malignant transformation has occurred in the time interval [0,t] when irradiation was administered at time t=0. Normal stem cells are subjected to ongoing spontaneous mutation at rate µ, causing an accumulation of one-stage pre-malignant mutant cells. These one-stage mutants are also subjected to the same risk of mutation and at the same rate µ. When a cell has received a second mutation, this results in a malignant transformation. Dashed lines in Figure 1
show that the probability of a cell being killed by a radiation dose D is a linear–quadratic function of the dose, while the probability of a cell being mutated by a dose is a (different) linear–quadratic function of the radiation dose. Surviving normal stem cells and surviving one-hit mutants (i.e. one-stage pre-malignant mutant cells) are subject to the same growth control law. Thus, when needed, both types of cell repopulate at the same rate. This makes for rapid repopulation when the stem cell population (including the one-mutant cells) is at or near depletion.
Surviving cells repopulate in accordance with a "Gomp-ex" function, which depends on the total population. It is assumed that total cell populations below 106 cells grow exponentially while total cell populations exceeding this threshold level grow progressively slowly in accordance with Gompertz kinetics and a maximum sustainable population of 108 cells. A further complication arises when severe doses of radiation are administered, since in this case the expected number of cells in the one-mutant compartment can reach single figures, thereby raising the serious possibility that no one-mutant cells survive irradiation. This situation requires the more careful analysis described in detail in Wheldon et al [4]. Stem cells which have undergone two mutations are considered to have undergone malignant transformation, i.e. they are no longer subject to growth control but continue to grow exponentially to form a malignant tumour after some latent period.
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Results of computer simulation
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The dose–response relationship (i.e. cancer incidence vs radiation dose) has been modelled for a wide range of parameter values. The dose–response relationship curve shows an early rise of cancer incidence until it reaches a peak value after which the cancer incidence declines with dose. However, the maximum value of cancer incidence and the dose at which this maximum value occurs are strongly dependent on the model parameters. This peak value and where it occurs are particularly sensitive to repopulation kinetics after irradiation, but are also affected by intrinsic and mutational radiation sensitivities as well as differential growth rates between normal stem cells and one-mutant cells. It can be demonstrated that the incidence of cancer is an increasing function of mutational sensitivity but a decreasing function of intrinsic radiosensitivity. Therefore details of the dose–response relationship may differ between the various tissues and organs. Table 1 gives standard settings for the level of cellular repopulation, the intrinsic cellular radiosensitivities and ß, the spontaneous mutation rate µ and the mutational radiosensitivities and .
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Table 1. Standard values for the level of cellular repopulation, intrinsic radiosensitivities, mutation rate and mutational radiosensitivities
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In the case of a two-stage model of tumorigenesis, Figure 2 illustrates how the incidence of carcinogenesis 20 years after a single homogeneous dose D is influenced by changing the level of cellular repopulation in graph (A), the intrinsic cellular radiosensitivity in graph (B), the spontaneous mutation rate in graph (C) and the mutational radiosensitivity in graph (D). Each graph shows the effect of changing one variable while the others take their values from Table 1. Wheldon et al [4] give a detailed discussion of the influence of model parameters on the incidence of tumorigenesis.
If DM is the dose for which the incidence of cancer is maximum, then for radiation doses above DM a dose reduction may increase rather than decrease cancer incidence. This remark may appear counterintuitive at first sight since it is commonly assumed that the best policy is to administer the smallest dose that meets the requirements of the treatment plan. Since DM is likely to vary between tissues, the optimal policy is most problematic for tissues near the site of the primary tumour and which therefore receive the high doses. Only within low dose regions, i.e. regions that receive a dose less than DM, could it be guaranteed that dose reduction would lead to a lowered incidence of cancer. Thus, in tissues which are far from the primary treatment volume and which receive lower doses of radiation it would seem beneficial to give a lower dose of radiation.
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Second tumour avoidance
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The main conclusion of the modelling study described and of the clinical and experimental data available is that dose–response relationships showing a tissue-specific turning point will be the norm for high dose radiation carcinogenesis. Clinical and experimental evidence for this is reviewed in an earlier paper [4]. Therefore it cannot be guaranteed that dose reduction is an appropriate strategy for second tumour avoidance when the dose received is so high that it lies above the dose at which cancer incidence is maximum. A dose reduction in this case would lead to an increase in the incidence of second tumours. This will primarily apply to tissues within or close to the designated treatment volume. This suggests that doses to tissues close to the treatment volume ought not to be modified to reduce a perceived risk of second cancers. Such an approach may be counterproductive in terms of second tumours, as well as possibly compromising the efficacy of treatment at the primary site. Of course, there are good reasons for limiting the dose to all normal tissues to minimize the risk of deterministic side effects. However, situations may arise in which a reduction in dose to prevent deterministic side effects (i.e. treatment-related normal tissue morbidity) leads to an increase in the probability of carcinogenesis in a particular tissue. This will be discussed later in the conclusions. Conversely, tissues and organs which are relatively far away from the treatment volume and receiving much lower doses (e.g. owing to an exit beam or to internal scatter) are more likely to be on the "ascending limb" of the dose–response curve, so that dose reduction results in a corresponding reduction in the incidence of second tumours. From modelling studies using biologically realistic parameters, tissues receiving less than about 2 Gy (if not repopulative) are likely to be in the ascending limb dose region and should therefore benefit by dose reduction. Tissues which repopulate are likely to have a turning point located at higher doses. These considerations suggest that conventional treatment plans in radiotherapy which achieve similar dose distributions within the treatment volume could be ranked for doses delivered to "distant sites", perhaps those receiving 0–2 Gy, and the treatment plan giving the least distant doses selected. In practical terms, if all distant sites are included, this would be a major undertaking.
The results of a pilot study which was undertaken for illustrative purposes are now described. This study aimed to investigate variations in the predicted incidence of second tumours arising from treatment plans which were indistinguishable in respect of the delivery of dose to the target region, but differed significantly in respect of the dose distribution to distant sites. Uncertainty in parameter values means that model predictions should only be used to assess the scope that might exist for reducing the incidence of second tumours by a judicious choice of treatment plan among a portfolio of competing plans which all satisfy the dose requirements of the primary site.
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Alternative treatment plans
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For this exercise, three different treatment plans for a radical treatment of lung carcinoma were used. Planning was done from a set of CT images to provide an accurate patient outline as well as satisfactory delineation of the organ of interest, in this case the left and right lung and the spinal cord. Individual plans were based on a coplanar, three-field set-up each differing in wedge angle, beam weighting and gantry angle. To produce the dose–volume histograms (DVHs) in Figure 3, a standard dose fractionation pattern was assumed of 60 Gy delivered in 30 fractions each of 2 Gy. In all three cases the aim was to achieve an acceptable dose distribution to the planning target volume (PTV), which formed a cylindrical volume of roughly 210 cm3, and at the same time to limit the maximum spinal cord dose to 40 Gy. Plans were optimized manually and were generated using beam data for a 6 MeV linear accelerator. All stages of planning from patient outlining to final production of DVHs were performed on a SUN ULTRA 170E workstation using a software package written by Dr A T Redpath (Western General Hospital, Edinburgh). DVH data files were then exported for subsequent analysis. Table 2 provides the main statistical properties of each plan.
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Figure 3. (a), (b) and (c) each illustrate a dose–volume histogram (DVH) on the left with its respective treatment plan on the right. In each case, the DVH for the tumour (T) is shown on the far right of the graph, along with the DVHs for the spinal cord (S, dashed line), left lung (L, solid line on left) and right lung (R, dotted line).
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Table 2. Mean and standard deviation of the dose distribution for plans (a), (b) and (c) at the tumour site, spinal cord and each lung
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Modelling the relative incidence of second tumours
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To compute incidences of second tumours, the radiation carcinogenesis model described was modified for dose distributions rather than for single doses (see Appendix). 2000 simulations of the model were performed for each DVH and each site. In each simulation, the intrinsic cellular radiosensitivity parameters and ß, the spontaneous mutation rate µ and the mutational radiosensitivity parameters and were drawn from normal distributions with respective means =0.35, ß=0.035, µ=10-8, =2 x 10-5 and =2 x 10-6 and with standard deviations set at 10% of the corresponding mean. The choice of mean values for , ß and µ was motivated by the findings of Wheldon et al [4]. In combination with the parameter choices for and , the model predicted approximately a 1% incidence of second tumours at the right lung for treatment plan (a). While this figure is in line with second tumour incidences reported for typical radiotherapy patients [5, 6], its primary role is to provide a baseline for comparison. Table 3 reports the mean and standard deviation of the anticipated incidence of a second tumour at the site of the existing tumour, at the right and left lung and at the spinal cord. The calculations show that significant differences exist between these treatment plans when judged for incidence of second tumours at just a few distant sites. These findings provide encouragement that this approach could lead to a reduction in the incidence of second tumours following radiotherapy.
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Table 3. Mean and standard deviation of the incidence of second tumours at the site of the original tumour, in the left and right lungs, and in the spinal cord are shown for treatment plans (a), (b) and (c)
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Conclusions
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Despite a large body of experimental and clinical studies, radiation carcinogenesis is not yet fully understood. From our analysis, it seems possible to determine some general features. Very likely, the incidence of radiation-induced tumours first rises then falls with increasing dose as radiation cell kill becomes the predominant effect. However, the dose at which the turning point occurs depends on the interplay between the level of cell repopulation and other factors, and cannot be predicted for individual tissues without further information. We have considered an approach to the avoidance of second tumours by focusing attention on tissues and organs at distant sites rather than at the treatment volume. Pilot modelling studies suggest that alternative conventional treatment plans may differ appreciably in the second cancer risk at distant sites (see Table 3) and scope may exist for risk reduction at these sites by appropriate choice of treatment plan. Potential treatment plans could be ranked for least risk to distant rather than adjacent organs. To develop this approach further would require a more comprehensive evaluation of dose distributions at a range of body sites resulting from alternative treatment plans at different primary sites. This is a large task which involves computations going beyond those available from present treatment planning packages. Situations may arise where a reduction in dose to prevent deterministic side effects (i.e. treatment-related normal tissue morbidity) leads to an increase in the probability of carcinogenesis in a particular tissue. The priority in such cases will probably be a matter for clinical judgement at present until robust strategies for dealing with such dilemmas are developed. This will undoubtedly be a consideration with some of the newer techniques such as intensity modulated radiotherapy treatment (IMRT). IMRT tends to involve a greater number of treatment fields than usual to help discriminate between tumour and organs at risk, and these modulated beams are often set at equi-angled spacing around the patient. Because of this, the dose to tissues distant from the tumour can be more than it would be in the case of more conventional treatment. Moreover, dose escalation may tend to make these matters worse and so it may be that carcinogenesis will have to be considered along with normal tissue morbidity when inverse planning optimization techniques are used. The treatment planning process does not routinely involve the evaluation of dose distributions at organ sites remote from the intended treatment volume. It may be necessary to develop Monte Carlo methods to provide dose estimation at distant locations. Further progress probably requires input on kinetic and other biological features of the carcinogenesis process in individual tissues. For example, it would be very important to know the potential of stem cells and pre-malignant mutant cells in individual tissues to repopulate, since this strongly affects the position of the turning point of the dose–response curve aswell as the magnitude of the maximum cancer incidence. In future, such information may be forthcoming from experimental and clinical studies over a sufficiently wide dose range.
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Appendix
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Differential equation model
The differential equation model is a two-stage radiation carcinogenesis model comprising three compartments: P0(t)—the non-mutated stem cells; P1(t)—the one-mutant stem cells, which in this instance are also the precursor cells of malignant transformation; Z(t), the probability that an individual cell population has received at least one malignant transformation in the time interval [0,t] after irradiation. Stem cells are subjected to a spontaneous mutation rate of µ, causing an accumulation of one-mutant cells. These cells are likewise subjected to mutation at the same rate µ. Cells experiencing two such mutations can become ancestors of a malignant clone after successful division, resulting in a tumour after a latent period. The evolution of P0, P1 and Z is governed by the differential equations
where P(t)=P0(t) + P1(t) is the total cell population and f(P) is its instantaneous growth rate. These equations assume that the normal and one-mutant stem cell populations are homeostatically self-regulating, with compensatory proliferation whenever the total cell population is depleted (e.g. by radiation).
Equations (1) require initial values for P0, P1 and Z immediately after radiation is administered. These initial values depend on the corresponding value of P0, P1 and Z immediately before irradiation and the shape of the dose–volume histogram (DVH), say F(d),of the applied radiation. (Figure 3 illustrates the DVHs for three different treatment plans.) Similarly, the level of cell kill and induced-mutation resulting from the radiation will depend on the shape of F(d). Since the negative derivative of F is the density of the dose distribution, the fraction of cells surviving irradiation is
Similarly, the fraction of cells surviving irradiation and not mutated is
Therefore if P0(-) and P1(-) are the expected P0 and P1 cell populations immediately before irradiation and P0(+) and P1(+) are the corresponding expected populations immediately after irradiation then
Similarly, the expected number of malignant transformations occurring as a result of the radiation dose is
The quantity M(+) is now interpreted as the mean of a Poisson process in which the random variable is the number of malignant transformations occurring as the result of the radiation dose. Thus 1-exp[-M(+)] represents the cancer incidence attributable directly to the administration of the radiation. Hence Equations (1) are to be solved with initial conditions
Let M(t) denote the expected number of malignant transformations to have occurred by time t after irradiation, then
and the final equation in (1) has solution Z(t)=1-e-M(t). If the incidence of carcinogenesis is small (perhaps a small value for µ) then Z and M are approximately equal.
Gomp-ex model for cell repopulation
The function f(P) appearing in Equations (1) was chosen in accordance with Gomp-ex kinetics. This model of cell growth assumes that cell populations less than Pcrit (106 cells here) grow exponentially while those above Pcrit follow a Gompertz growth curve with the cell population growing increasingly slowly towards the maximum population Pmax (108 cells here). Specifically
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Radiation kill and mutation factors
The probability of surviving cell kill and cell mutation as a result of dose D is modelled by the usual linear–quadratic expression e-aD-bD2 in which the parameters a and b are tissue dependent and take different values for cell kill and cell mutation. For high doses, it can happen that the expected number of one-mutant cells surviving radiation is measured in single figures. In this instance the continuum analysis is unreliable and is replaced by a stochastic analysis in which the number of surviving one-mutant cells is regarded as a Poisson statistic with prescribed mean value. There is now a non-trivial probability that no one-mutant cells survive irradiation. The analysis of this scenario is described in more detail in Wheldon et al [4]. The conclusion of this paper is that there is a fundamental distinction between no surviving one-mutant cells and at least one surviving one-mutant cell.
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Acknowledgments
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E G Wheldon acknowledges the support of the Department of Health (project award 22116) in this work. C Deehan acknowledges the support of the Scottish Hospitals Endowments Research Trust (SHERT) grant number 1497 in this work.
Received for publication April 5, 2000.
Revision received November 16, 2000.
Accepted for publication December 21, 2000.
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Abstract
Introduction
