A new view of radiationinduced cancer: integrating short and longterm processes. Part I: Approach
Authors
Open AccessArticle
 First Online:
 Received:
 Accepted:
DOI: 10.1007/s0041100902303
Abstract
Mathematical models of radiation carcinogenesis are important for understanding mechanisms and for interpreting or extrapolating risk. There are two classes of such models: (1) longterm formalisms that track premalignant cell numbers throughout an entire lifetime but treat initial radiation dose–response simplistically and (2) shortterm 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 longterm models is needed. As an example of this novel approach, we integrate a stochastic shortterm initiation/inactivation/repopulation model with a deterministic twostage longterm model. Within this new formalism, the following assumptions are implemented: radiation initiates, promotes, or kills premalignant cells; a premalignant 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 premalignant cells decreases with age.
Introduction
Short and longterm biologically based models
Biologically motivated mathematical modeling of background and ionizing radiationinduced carcinogenesis has a history spanning more than 50 years (Nordling 1953). Many of the models can be characterized as shortterm, in that they focus on processes occurring during and shortly after irradiation (Hahnfeldt and Hlatky 1998; Radivoyevitch et al. 2001; Mebust et al. 2002; Schollnberger et al. 2002; Sachs and Brenner 2005; Hofmann et al. 2006; Shuryak et al. 2006; Little 2007; Sachs et al. 2007; Schneider and Walsh 2008). The main advantage of such models is that they provide a detailed initial dose–response relation for shortterm endpoints, which are used as surrogates for carcinogenesis. The main disadvantage is that the possibly substantial modulations of the magnitude and shape of this initial dose–response during the lengthy period (multiple yearsdecades) between irradiation and manifestation of typical solid tumors are not considered mechanistically.
In contrast, another class of biologically motivated models can be characterized as longterm, in the sense that they track carcinogenesis rates throughout the entire life span, e.g., the ArmitageDoll model (Armitage and Doll 1954; Armitage 1985), the MoolgavkarVenzonKnudson twostage clonal expansion (TSCE) model (Moolgavkar 1978, 1980; Moolgavkar and Knudson 1981), the twostage logistic model (Sachs et al. 2005), and many others (Yakovlev and Polig 1996; Pierce and Mendelsohn 1999; Wheldon et al. 2000; Little and Wright 2003; Pierce and Vaeth 2003; Ritter et al. 2003; Little and Li 2007). The main advantage of longterm models is the more detailed treatment of slow carcinogenesis processes, including the modulation of the radiation dose–response during the long latency period. The main disadvantage is that the initial dose–response is typically treated in a nonmechanistic, phenomenological manner.
The need for a new approach: integration of long and shortterm models
Motivations for the specific unification used here
Here, we provide a specific example of integrating short and longterm radiation carcinogenesis models. In that a variety of short and longterm models have been proposed, our approach should be considered as illustrative, intended to investigate the practicality of integrating the two model types. The goal was to produce a novel formalism, which can describe typical patterns of background and radiogenic carcinogenesis with the smallest possible number of adjustable parameters. This goal required multiple simplifying assumptions about the complex multistep carcinogenesis process. We suggest that for the purposes of generating a preliminary integrated model, the consequent reduction in biological realism is compensated by the reduction in model complexity and number of parameters. More complicated examples of unifying long and shortterm models are certainly possible, e.g., models analyzing multiple premalignant cell stages, analyzing genomic instability, and/or analyzing stochastic effects at both time scales, rather than just at shorttime scales as we shall do.
Shortterm model
The shortterm part of our model (Fig. 1a) is the more mathematically intensive part of the integrated formalism. It is based on existing initiation, inactivation and repopulation (iir) models originally designed for analyzing second cancers induced by radiotherapy (Lindsay et al. 2001; Sachs and Brenner 2005; Shuryak et al. 2006; Little 2007; Sachs et al. 2007). This shortterm part analyzes normal and premalignant stem cells during complex radiation exposure regimens and during the following weeks of tissue recovery. It tracks individual premalignant cell clones rather than just the total number of premalignant cells. The probability that a premalignant clone becomes extinct during radiotherapy can be substantial, so a fully stochastic formalism is used for the population dynamics of premalignant cells.
Longterm model

Normal and premalignant stem cells are located in specific tissue niches (Slack 2000; Borthwick et al. 2001; Potten and Booth 2002; Bennett et al. 2003; Fuchs et al. 2004; Ghazizadeh and Taichman 2005; Li and Xie 2005). We here have in mind a general concept of a niche, including, e.g., a clone whose further growth is prevented or slowed by any kind of microenvironmental constraints. The number of niches per organ, and the number of normal or premalignant stem cells per niche, is homeostatically regulated (Fuchs et al. 2004; Li and Xie 2005). We allow for the possibility that premalignant stem cells are regulated somewhat less stringently than their normal counterparts, so that a niche filled with premalignant cells can contain more such cells on average than a niche filled with normal cells.

An initiated stem cell either dies out, or grows into a premalignant clone, which quickly (e.g., within a year, Campbell et al. 1996) fills the entire stem cell niche in which it originated, i.e., “initiates” the whole niche. To expand beyond the first niche, the premalignant clone needs to invade an adjacent niche; alternatively, the niche containing the clone can divide into two daughter niches, e.g., colon crypt fission (Greaves et al. 2006; Johnston et al. 2007; Edwards and Chapman 2007). These processes require years or decades, and may involve acquisition of new mutations (Spencer et al. 2006).

Premalignant cells in all niches are assumed to gradually lose their carcinogenic potential with age, so that at old age they have a progressively smaller probability of being transformed to malignant cells. This assumption is suggested by a decrease in incidence of most cancers at very old age, e.g., >80 years in humans [Surveillance Epidemiology and End Results (SEER) database, http://seer.cancer.gov] and >800 days in mice (Pompei et al. 2001). The likely mechanism is senescence of stem cells and/or deterioration of stem cell function, or niche function, with age (Brunet and Rando 2007; Carlson and Conboy 2007; Sharpless and DePinho 2007).

To reduce the number of model parameters, clonal expansion of premalignant stem cell niches is treated deterministically, using a net proliferation rate. In other words, it is estimated by a deterministic “exponential model”, instead of the stochastic TSCE model. Given realistic parameter values, the deterministic and stochastic versions of the twostage model are numerically very similar from birth until old age (Heidenreich and Hoogenveen 2001). In our analyses, by the time irradiation is over and cell repopulation in the exposed organ has occurred, surviving premalignant clones have grown to substantial sizes, making extinction very unlikely. This reduces the need for stochastic treatment of clone dynamics during the decades following radiation exposure. At old age, the deterministic approximation predicts an increasing hazard, whereas the stochastic TSCE model predicts a plateau. Considering the evidence cited above about a decrease in cancer incidence at old age, neither behavior is fully realistic. We here instead model the downturn in incidence at old age explicitly, according to the assumption above.
Unification of short and longterm models
The integration of long and shortterm models will have two features typical of multitimescale modeling (Engquist and Runborg 2005): (1) information is passed in both directions between the two components; and (2) a formally infinite time interval in the shortterm model represents a short time interval, here typically several months, in the longterm model (Fig. 1).
The deterministic longterm equations provide the mean number of niches filled with premalignant stem cells and the mean number of premalignant cells per niche just before irradiation. The stochastic shortterm equations then provide the number of these niches that are eradicated by radiation as well as the number of premalignant clones that are induced by, and survive, the radiation exposure. Each of these clones is assumed to be independent (i.e., located in a different part of the organ), and to be capable of quickly filling a niche with premalignant cells. Therefore, the total number of premalignant niches soon after irradiation can be calculated by the stochastic shortterm model. The mean of this number is the initial condition for the deterministic longterm equations, which are applied from this point onwards until old age (Fig. 1). Cancer incidence is assumed to be proportional to the total number of premalignant cells in all niches, shifted by a lag time. A mathematical description of the model is provided below.
Materials and methods
Summary of model parameters
Parameter 
Units 
Interpretation 

a 
time^{−2} 
Spontaneous stem cell initiation and transformation 
b 
time^{−1} 
Premalignant niche replication 
c 
time^{−2} 
Agedependent premalignant cell senescence 
δ 
time^{−1} 
Homeostatic regulation of premalignant cell number per niche 
Z 
cells/niche 
Carrying capacity for premalignant cells per niche 
X 
time/dose 
Radiationinduced initiation 
Y 
dose^{−1} 
Radiationinduced promotion 
α, β 
dose^{−1}, dose^{−2} 
Stem cell inactivation by radiation 
λ 
time^{−1} 
Maximum net stem cell proliferation (repopulation) rate 
L 
time 
Lag period between the first malignant cell and cancer 
Longterm model for background cancers
The longterm model was intended mainly to place the results of the stochastic shortterm calculations in an appropriate context, enabling estimation of the effects of age at exposure and time since exposure on predicted cancer risk. Simplicity and parsimony were emphasized. The longterm model in the absence of radiation consists just of a few rather simple deterministic equations, as follows.
The functions A and H are numerically very similar for realistic parameter values; they diverge substantially only if cancer incidence is high. We use the exact hazard H for data fitting in the companion paper, but use the simpler expression for A in the equations below, keeping in mind its interpretation and limitations.
Effects of radiation
Shortterm processes
The exposure scenario analyzed here is radiotherapy for an existing malignancy, where there are K daily dosefractions, all of equal size d in some nearby organ, with treatment gaps during the weekends. Straightforward generalizations to variable doses per fraction, to other temporal patterns, and/or to cases where one must consider dosevolume histograms are omitted for brevity. A single acute dose exposure is a simple special case, where the number of fractions is K = 1.
The shortterm part of the model considers initiation, inactivation, and repopulation (iir) of normal and premalignant stem cells during the radiation regimen and a recovery period of about a month (Fig. 1). Niche takeover by premalignant cells that survive the irradiation and modulation of niche size by radiogenic promotion (discussed below) are here also considered shortterm processes. They are assumed to be essentially completed by a few months following exposure, before the longterm processes of spontaneous initiation and niche replication, which operate on the time scale of multiple years, start to have an appreciable effect (Fig. 1).
Normal stem cells are treated deterministically because their number per organ is assumed to be large. In contrast, the number of premalignant stem cells per clone is much smaller, and extinction of some premalignant clones is a real possibility. Consequently, a stochastic approach described below is used to estimate the average number of live premalignant niches when postirradiation longterm processes begin.
Equations (7–9) allow the calculation of the normal stem cell number n(t) in the entire organ at all relevant times t throughout the radiation exposure and recovery periods. Explicit results for n(t) are provided in the Appendix.
We assume that the longterm growth advantage of premalignant cells manifests itself only on the scale of years and decades, and is negligible on the much shorter time scale of a few weeks of radiotherapy. Consequently, the maximum net proliferation rates for normal and premalignant stem cells are assumed to be equal, described by the parameter λ.
Unifying short and longterm processes
Calculation of absolute and relative cancer risks
Note that the senescence parameter (c) cancels out of this ERR expression; due to the way we have defined our radiation initiation parameter (X), the spontaneous initiation parameter (a) also cancels out. The term Q _{1} can be interpreted as the normalized size of old premalignant stem cell niches. Q _{2} is proportional to the number of such niches. Q _{3} is proportional to the number of new premalignant niches, and Q _{4} is proportional to the total number of premalignant niches under background conditions.
Results
Figure 4 shows an example of a modelgenerated agedependent background incidence curve for an adultonset solid cancer, using the four relevant parameters a, b, c, and L. The shape of this curve agrees well with the data for many cancers (e.g., SEER database, http://seer.cancer.gov), including the downturn in incidence at old age, which is not described as well by standard models.
Promotiondriven ERR can also be modulated by time after exposure. This occurs due to the assumption that the number of premalignant cells per niche is homeostatically regulated (by parameter δ), so that radiationinduced hyperproliferation of cells within their niches can be reversed, as preirradiation cell birth/death rates, and hence niche sizes, are restored. If δ > 0, ERR due to promotion will decrease over time following exposure. This effect is seen in Figs. 6, 7: If the risk is measured at some constant age (e.g., 70 years), which is a sum of age at exposure and time since exposure, a decrease in ERR with time since exposure due to a δ > 0 will appear as an increase in the ERR with age at exposure (Fig. 6). A decrease in promotiondriven ERR with time since exposure also can have a conceptually important effect on the dose–response—at longer times after exposure, not only the magnitude, but also the shape, of the dose–response can change (Fig. 7).
Discussion
The formalism presented here is the first comprehensive attempt to unify short and longterm modeling approaches. The shortterm part of the model belongs to the previously discussed iir category. It tracks the numbers of premalignant cells throughout irradiation stochastically. The longterm part of the model builds on the concepts developed in previous twostage formalisms by adding an analysis of some aspects of tissue architecture (i.e., stem cell niches/compartments) and aging of premalignant stem cells. The particular short and longterm models that we have chosen to use are not crucial—the real issue is the integration. Certainly other longterm models could perfectly well be integrated into this short/longterm framework—and we hope they will be.
The unified formalism has a number of advantages. The shortterm part can generate reasonable predictions even at high doses, such as those in cancer radiotherapy. The longterm part analyzes the entire lifetime of the individual, putting the shortterm predictions in an appropriate context by estimating the effects of age at exposure and time since exposure. The combined approach therefore allows the dose–response for the number of premalignant cells to be examined at any time point, from the start of irradiation until development of cancer years to decades later, which is not possible using either short or longterm models alone.
Our model can be used for estimating risks of second malignancies induced by radiotherapy. This issue is growing in importance (Brenner et al. 2000; Ron 2006) as patients are treated earlier in life and the number of cancer survivors increases (Editorial 2004); the lifetime risk of radiationinduced second cancers is not negligible (Brenner et al. 2007). Direct measurement of second cancer incidence requires decades of followup because the latency period for radiogenic solid tumors is long (Tokunaga et al. 1979; Brenner et al. 2000; Ivanov et al. 2004, 2009). Meanwhile, radiotherapy protocols are rapidly changing. Our model, calibrated using data from older protocols, can produce risk predictions for any modern or prospective radiotherapeutic protocol. This application of the model is discussed in the accompanying paper (Shuryak et al. 2009).
Acknowledgments
Research supported by National Cancer Institute grant 5T32CA009529 (IS), Department Of Energy DEFG0203ER63668 and National Institutes of Health CA78496 (PH), National Aeronautics and Space Administration 03OBPR0700590065(LH), National Aeronautics and Space Administration NSCOR NNJ04HJ12G/NNJ06HA28G (RKS), National Institutes of Health grants P41 EB00203309 and P01 CA49062 (DJB).
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Appendix
This appendix gives some details on the equations for the shortterm calculation and on their derivation. We first analyze repopulation effects for normal stem cells deterministically (16–18); then we analyze survival probabilities for premalignant clones stochastically. The equations derived are extensions of results given previously (Sachs et al. 2007).
Normal stem cell numbers
Denote the time of the kth dosefraction by T(k). We will derive recursive equations valid for k = 1,…, K, where the number of fractions is K as in the main text. We formally define T(K + 1) ≡ ∞; this infinite value represents the end of the recovery period (Fig. 1). It reflects a standard procedure in multitimescale analyses (Engquist and Runborg 2005), an infinite time interval in a shorttimescale model represents a short time interval in the next larger timescale model (here several months in our longtimescale model). We correspondingly set n ^{−}(K + 1) = ν, the set point value attained at the end of the recovery period.
Premalignant stem cell clones
In (20) n(t) is a known function of time, determined as discussed above.
Here, T ^{+}(k) denotes the time just after the kth fraction. Because of the way a Dirac δ function behaves, using T ^{+} rather than T is important in the expressions for ψ and ξ.
Equation (22) is valid for k = 0, 1,…, K, with k = 0 referring to premalignant cells present before radiation starts and F(K) = 1. It is the primary mathematical result needed for the data analysis discussed in the main text.
We have some extensions, not needed in the present paper. Generalizing to situations where the spontaneous death rate is nonzero and/or one needs the probability that a clone has a given number of cells at the final time, can readily be done by using results given by Tan (2002). In addition, it can be shown that (22) holds even if the recovery equation for normal cells is different from the logistic equation used in this paper, e.g., is Gompertzian.