J. Math. Biol. (1992) 30:493-511 dournalof Mathematical 61ology © Springer-Verlag1992 Evolution of DNA damage in irradiated cells P. Hahnfeldt 1, R. K. Sachs 2, and L. R. Hlatky 1 i Joint Center for Radiation Therapy, Havard Medical School, 50 BinneyStreet, Boston, MA 02115, USA 2 Departments of Mathematics and Physics, UCB, Berkeley,CA 94720, USA Received November 11, 1990; receivedin revised form May 5, 1991 Abstract. Ionizing radiation damage to a mammalian genome is modeled using continuous time Markov chains. Models are given for the initial infliction of DNA double strand breaks by radiation and for the enzymatic processing of this initial damage. Damage processing pathways include DNA double strand break repair and chromosome exchanges. Linear, saturable, or inducible repair is considered, competing kinetically with pairwise interactions of the DNA double strand breaks. As endpoints, both chromosome aberrations and the inability of cells to form clones are analyzed. For the post-irradiation behavior, using the discrete time Markov chain embedded at transitions gives the ultimate distribution of damage more simply than does integrating the Kolmogorov forward equations. In a representative special case explicit expressions for the probability distribution of damage at large times are given in the form used for numerical computations and comparisons with experiments on human lymphocytes. A principle of branching ratios, that late assays can only measure appropriate ratios of repair and interaction functions, not the functions themselves, is derived and discussed. Key words: Chromosome aberrations- Ionizing radiation damage to D N A Markov chain m o d e l s - Chemical kinetics master e q u a t i o n - Radiation cell survival 1 Introduction When mammalian cells are subjected to ionizing radiation, their DNA is damaged. The outcome may be to "kill" the cells by rendering them incapable of forming a clone. Cancer radiation therapy aims to kill tumor cells by this method. An alternate outcome is to cause mutations in the cells, possibly rendering them malignant. Background ionizing radiation is of concern partly for this reason. The damage to the DNA can sometimes be observed at the next mitosis, in the form of chromosome aberrations, as discussed in many references, e.g. Lea (1955), Bedford and Cornforth (1987), Bender et al. (1988) and Savage (1989). This work was supported in ~ DMS-9025103 494 P. Hahnfeldtet al. Radiation dose Ionizing radiation consists of high energy particles (photons, electrons, protons, neutrons, helium nuclei, etc.). These deposit energy in matter by ionizing molecules and by other mechanisms. Macroscopically, the main parameter which determines ionizing radiation damage to cells is radiation dose, defined as the amount of energy deposited in a region divided by the mass of that region. Thus the units of dose are Joules/kilogram, and 1 Joule/kilogram is called a Gray, after a pioneering worker in the field. An x-ray dose of several Gray delivered in a short time is usually enough to severely damage many of the exposed cells. Typical human environmental exposures from all kinds of ionizing radiation are considerably less than 10 -3 Gray per year x-ray equivalent dose and such doses are not known to cause major health problems. The damage to a cell is due to individual particle tracks, randomly intersecting or coming near DNA. Sometimes there are only a few tracks per cell nucleus. Always each track makes a complicated, stochastic pattern of ionizations. Therefore it is often essential to introduce the specific energy, a stochastic quantity whose expected value is dose (Kellerer 1985). Time scales In this paper we will be concerned with time intervals of seconds or more, not milliseconds or less. The time it takes for an individual ionizing radiation track to cross a cell nucleus is very short < 10-13 seconds. The damage produced by one such track has developed chemically into comparatively long-lived DNA lesions within about 10 -3 seconds (Ward 1990). In the models of this paper a time interval of 10 - 3 seconds will be regarded as a single instant. We will refer to immediate radiation damage, meaning the results of a chain of reactions starting with the crossing of a cell nucleus by a track and ending some 10 -3 seconds later. The entire process of irradiating a cell may take less than a minute or it can be prolonged for many days. For our purposes, irradiation consists of a period during which radiation tracks arrive at random times and then instantaneously cause immediate damage. Cellular enzymes, operating during periods of seconds to days (FrankenbergSchwager 1989), play an important role in modulating the immediate radiation damage during irradiation and/or after irradiation ends. It is these longer time scale processes which will here be the center of attention. Some of the biological mechanisms involved are discussed by Friedberg and Hanawalt (1988) and by Wallace and Painter (1990). The immediate damage may be partially or completely repaired, but one can also have "misrepair", which changes the immediate damage into an unrepairable lesion. Mathematical models Many models have been developed to analyze ionizing radiation damage to DNA and to mammalian cells, as summarized briefly by Weber (1988). Some of the models deal with averages only. However, stochastic effects play an important role throughout the development of the radiation damage. Apart from the Evolution of DNA damage in irradiated cells 495 stochastic physical input, many of the biological responses are also stochastic. For example, the efficiency of a few repair enzymes attempting to repair a few initial lesions may determine the fate of a cell. Consequently many radiation damage models use random variables and stochastic processes. In particular, continuous time Markov chain models have been considered. Important early ones are due to Hug and Kellerer (1966) and Obaturov et al. (1980). Some more recent Markov chain models are summarized by Sachs et al. (1990). A motivation for using Markov processes is that radiation tracks arrive in a Poisson process (Kellerer 1985) and the immediate DNA damage they cause can plausibly be modeled as a batch Poisson arrival process (see Sect. 5 below). In addition, many of the biological responses, such as the total amount of repairable damage still remaining at a given time during a post-irradiation period, seem to show decaying exponential time dependence (FrankenbergSchwager 1989; Hagen 1989), perhaps indicative of Markov behavior. Most of the Markov models correspond to combining a batch Poisson arrival process with the master equation (McQuarrie 1967) of stochastic chemical mass-action kinetics. The master equation is appropriate when the number n of reacting molecules is so small that Poisson fluctuations in the initial number may be important, fluctuations in the final yield may be measurable, the number of pairs must be written as n(n - 1)/2, not approximated by n2/2, etc. (Doraiswamy and Kulkarni 1987; Erdi and Toth 1989). We shall consider a rather general class of continuous time Markov models for radiation cell damage and prove some basic results useful in applications. General Markov chain formalism It will be assumed that the possible states of a cell form a denumerable set E. In our models, a state N E E is an r-tuple of non-negative integers, corresponding to various kinds of DNA lesions. In our main example N is a pair: N = (n, m), where n is the number of DNA double strand breaks and m is the number of irreparable chromosome aberrations in the genome. In some of the other examples the elements of E are triplets or sextuplets. We will be concerned with the probability PN(t) a cell is state N at time t. Unless specifically denied we will assume PN(O is a non-defective mass distribution, i.e. Pu(t)>>.O, NEE; ~PK(t)=l. (1.1) K Summation will always be over all the elements of E unless explicitly specified otherwise. For t/> to we will have the standard equations of a continuous time Markov chain (Hou and Guo 1988; Wolff 1989): PN(t) = ~ PK(to)PKN(t, to) VN E E. (1.2) K Here PKw(to, to) = 6KN and PK~(t, to) is a solution of the Kolmogorov forward equations: dPKN(t, to)/dt = ~ PKj(t, to)QgN, K, N E E, J (1.3) 496 P. Hahnfeldtet al. where O<<,QKN<~ (a) QK~v=RKlv+ Smv, (K CN); ~QKs=O. (1.4) s Thus IQ/a<l is the fraction of cells in state K expected to exit state K per unit time (could be > 1); similarly QKN is the rate of transitions from state K to state N. Both have dimensions of inverse time. The generator Q of the chain is the matrix with elements QKN. When (1.2)-(1.4) hold we will also have dPN(t)/dt= ~KPK(t)QKN and will also refer to Ply(t) as a solution of the Kolmogorov forward equations with generator Q. Q will later be subjected to regularity conditions which guarantee the consistency of (1.1) and (1.2). In accordance with our above remarks on irradiation, the Markov chain will be piecewise time-homogeneous. Specifically, with T > 0, there will be an irradiation period - T ~< t < 0 and there will be a post-irradiation period 0 <<,t < oo such that -aD < QKK ~<0; - T <<.t<O; (b) QKN~-SKN, 0 ~ < t < ~ . (1.5) Here RKN, SKN are constant VK, N u E. RKN is a damage generator, modeling the immediate effect of the radiation and operative only during irradiation. Taking R~CN constant involves the assumption, appropriate in most experiments, that irradiation is at a constant dose rate. SKN will model the effect of enzymatic cellular repair and misrepair processes, operative throughout and idealized as time-homogeneous. In the present paper we will be primarily concerned with Eq. (1.5b) and briefly concerned with that special case of Eq. (1.5a) where SKu is negligible compared to RKN. Analyzing (1.5a) when neither SKN nor RKU are negligible requires extensive numerical calculations and a number of analytical techniques different from those used in the present paper (Sachs et al. 1990; Sachs et al. 1991). 2 Examples Some examples of using (1.5) to describe immediate damage and damage evolution pathways will now be given. We henceforth consider only normal diploid mammalian cells and assume irradiation takes place prior to the cells replicating their DNA. For humans, the DNA is in a genome consisting of 46 chromosomes. When intact, a chromosome has a telomere at each end and a centromere somewhere between the ends (Fig. 1). A chromosome has, very roughly, several thousand genes. The cell may be able to tolerate loss or alteration of a few genes, but apparently cannot tolerate absence of a large fraction of a chromosome (Revell 1983). A centromere is required for proper inheritance of a chromosome when a cell divides (Savage 1989). DNA double strand breaks In an intact chromosome there are two helically intertwined strands of DNA which are susceptible to being broken by radiation. In the examples of this paper Evolution of D N A damage in irradiated cells Start: • A 0 tl ~ 497 0 chromosome eentromere • chromosome telomere DNA double strand break X Some Possible Reactions and Resulting States: A. Repair ~ i Or It v B. Complete Symmetric Exchange C. Complete Asymmetric Exchange D. Incomplete Exchange (Makes One of the Following) n0~ m m • Fig. 1. Damage processing steps: Suppose that at t = 0, just after irradiation, the only damage to a cell's genome consists of two D N A double strand breaks as at the top. Some alternative next steps in the evolution o f this damage are shown. If the next step is A one or th~ other double strand break is repaired. B - D represent the interaction of two double strand breaks. B is a complete reciprocal exchange, called symmetric because each of the altered chromosomes remains with one centromere. C and D are other possible forms of exchange. In a Markov chain model for post-irradiation damage processing, one step (such as A, B, C or D) is one transition of the chain. Subsequent steps can follow and if there is a large amount of damage at t = 0 many different pathways can occur. The various alternative products may have different biological and/or experimental properties. For example in C the altered chromosome with two centromeres may cause mechanical difficulties at the next mitosis and the one without any centromere may become lost, reducing the likelihood the cell will produce a clone. In B the cell may well be able to produce a clone, In some models one keeps track in detail of particular kinds of damage; in others one merely counts how many alterations of the genome have occurred the double stranded nature of DNA will not complicate the mathematics: a break, repair, or interaction will always be regarded as acting simultaneously on both strands (Fig. 1). In particular the breaks we shall consider in this paper involve breakage of the covalent bonds in both of the two intertwined strands, not just one. These are called DNA double strand breaks. There is considerable evidence that such double strand breaks, rather than other kinds of DNA damage (e.g. single strand breaks, base deletions, etc.), are the most important immediate lesions caused by ionizing radiation (Iliakis 1988; Ward 1988; Radford 1988; Frankenberg-Schwager 1989). There are various kinds of double strand breaks, differing in their detailed structure (Charlton et al. 1989; Ward 1990), but we shall here regard all kinds as equivalent. 498 P. Hahnfeldt et al. Restitution or complete reciprocal exchange model Equation (1.5b), used to describe damage evolution during the post-irradiation period 0 ~< t < o% is simpler than Eq. (1.5a) which describes processes occurring during irradiation. We therefore start by discussing some representative pathways for the processing of damage previously incurred. We assume, for the time being, that the only relevant damage made during irradiation consists of DNA double strand breaks and that the state of a cell at t = 0 is fully determined by the number n of such breaks in its genome, regardless of how these breaks are distributed among the chromosomes. In the present example the intuitive picture is that when a double strand break occurs the broken ends do not fly apart and diffuse independently throughout the cell nucleus. Rather they are held in close proximity to each other by DNA associated nucleoproteins. More specifically, suppose that after a double strand break occurs one of two competing events eventually happens: either the cell repairs the break by rejoining as in Fig. 1A or two different breaks interact (for example as shown in 1B). In the present model we assume, as an idealization, that repair by rejoining (Fig. 1A) leaves that portion of the chromosome as good as new. We also assume that the rate at which such repairs take place depends only on the number n of double strand breaks still present, e.g. is independent of how many repairs or pairwise interactions have taken place. One thus has a repair rate function f(n). This function can have various forms. Depending on the convexity, there are three main alternatives: Linear Repair: f = 2n=*.f(n + 2) -- 2f(n + 1) + f(n) = 0; Inducible Repair: f(n + 2) - 2f(n + 1) +f(n) > 0, Saturable Repair: f(n + 2) - 2 f ( n + 1) +f(n) < 0, e.g. f = 2n2; (2.2a) (2.2b) e.g. f = 2n/J1 + (n/K)]. (2.2c) In Eq. (2.2) 2 is a positive constant with dimensions of inverse time, the repair rate constant, and in (2.2c) K is a dimensionless positive constant, analogous to a Michaelis-Menten constant. Assuming linear repair, 2 is approximately 5 × 10 -4 repairs per break per second (Iliakis 1988; Frankenberg-Schwager 1989). Linear repair (2.2a) appears to be the normal case. Roughly speaking, inducible repair (2.2b) corresponds to a situation where the cell mobilizes its repair resources when it senses radiation damage; and saturable repair (2.2c) corresponds to a situation where the repair enzymes can become overloaded. For recent empirical evidence of saturable repair in some circumstances see, e.g., Reddy et al. (1990). For empirical evidence of inducible repair see Shadley et al. (1987). In our analysis we shall leavef(n) general, subject only to the restrictions that it is a non-decreasing function of n which obeys f(0) = 0,f(1) > 0 (compare (2.2)). In the present model one further assumes that break-break interactions are always complete reciprocal exchanges (reciprocal translocations), i.e. one end of one break is joined with one end of the other break and the remaining two ends are also joined. Figures 1B and 1C show the possible results when the breaks are on different chromosomes. One actually has 8 different injured DNA strand termini at the start and these are joined pairwise (respecting polarity and intertwining). However, as mentioned above, the picture of a transition from Fig. 1 start to Fig. 1B or C suffices throughout the present paper. In many cases 499 Evolution of D N A damage in irradiated cells complete reciprocal exchange appears to be the most common type of break-break interaction (Bedford and Cornforth 1987; Savage 1989). If the complete reciprocal exchange rate is taken to be the same for all pairs of breaks, independent of the number of breaks present, and independent of the number of complete reciprocal exchanges that have already occurred, then the rate is given by g(n) = ~cn(n - 1), (2.3) where 2K is a non-negative constant, the pairwise interaction rate constant. The order of magnitude of 2x may be, very roughly, 10 - 6 exchanges per pair of double strand breaks per second (Hlatky et al. 1991). We call g(n) the pairwise interaction rate. A fraction p (0 ~<p < 1) of the exchanges is assumed to leave the DNA as good as new; for example, in a favorable case an exchange which results in the configuration 1B could restore full integrity. The remaining exchanges are considered to produce one irreparable aberration (e.g. the configuration 1C). The probability 1 - p that an exchange produces an irreparable aberration is taken to be the same for all exchanges. It is assumed that the biologically relevant and experimentally measurable quantities are the number n of unrepaired breaks and the number m of irreparable aberrations present in a cell's genome. In the biological literature a repair by rejoining (Fig. 1A) is sometimes called more specifically a "restitution", to distinguish it from a reaction such as that shown in 1B, which alters chromosomes but in some cases also leaves the genome fully competent. Hence we call the model under discussion the "restitution or complete reciprocal exchange model". Its assumptions are oversimplifications of a complex situation (compare Revell 1983; Bedford and Cornforth 1987; Savage 1989; Hlatky et al. 1991). Markov chain for the restitution or complete reciprocal exchange model Let enm(t) be the probability that at time t >~0 a cell has n remaining double strand breaks and m irreparable, measurable aberrations have been formed. Thus our set E of states is E = {(n, m) [ n, m = 0, 1 , . . . } and we have abbreviated PK = P(,,m) by Pnm" Set P , _ 1 = 0 as a notational device. Our verbal description of the model is made precise by the following explicit form of the Kolmogorov forward equation, which determines the coefficients SNN" in (1.5b): dP,m/dt = --s(n)Pnm + f ( n + 1)P,+ 1,,, +g(n + 2)[PP,+ 2,m + (1 -P)Pn+2,m-1]; (2.4) f(0)=0, f(1)>0, f(n+l)>>,f(n); s(n)=f(n)+g(n); 0~<p<l; where g(n) is given by (2.3). Note that the matrix elements Quu" = Q(,.m)(n,,m,) determined by (2.4) meet the requirements in (1.4), in particular ~(j,k) Q(,,,,)(j,k)= 0 for each pair (n, m)E E. Henceforth we regard Eqs. (1.1), (2.3) and (2.4), not our verbal description, as the restitution or complete reciprocal exchange model. The case p = i1, f ( n ) = 2n has been compared to published experiments by Hlatky et al. (1991). One can solve (2.4) explicitly, given Pnm(0). Let us regard P,,, for fixed m and t as a row vector Pro(t). Then (2.4) can be rewritten in the form em = Pm A "~ Pm--1a; t >>.0 (2.5) 500 P. Hahnfeldt et al. where P - 1 ~ 0 and A, H are lower triangular matrices. Specifically A has as its only non-zero matrix elements Ann = - [ f ( n ) +g(n)], An+l,, = f ( n + 1), A, + 2, = pg(n + 2), i.e. A= f ( 01 ) pg(2) 0 0 --f(1) f(2) pg(3) 00 --s(2) f(3) 00 0 -s(3) "1 • (2.6) We have used f ( 0 ) = 0 and used g(0) = g(1) = 0 from (2.3). The non-zero matrix elements of H are H, + 2,, = (1 - p ) g ( n + 2). (2.7) Equation (2.5) can be integrated by diagonalizing A and then iterating in m (Hahnfeldt 1991). From (2.6) the eigenvalues of A are 0, - f ( 1 ) , - f ( 2 ) - g ( 2 ) . . . . . The solutions approach equilibrium, with the most slowly decaying transient governed by the factor e-f(1)t. We shall not give the details, since in many situations the resulting solution goes to its equilibrium at a much faster rate than the damage assays are performed (Hlatky et al. 1991; compare Obaturov et al. 1980 and Albright 1989). Then the t ~ oo limit is the case of interest. This limit can be obtained more transparently for a more general case by introducing an embedded discrete time Markov chain, as in Sect. 3 below. Other pathways Sometimes incomplete chromosome exchanges take place (Fig. 1D). A different model, which takes this possibility into account, is the following. Let n denote the number of double strand breaks which have not yet participated in repair or exchange. For example n = 2 at the start in Fig. 1, n = 1 in each subcase of 1A and n = 0 in each subcase of 1B-D. Let d denote half the number of "orphaned" ends, whose correct partners have already participated in an exchange. For example d = 1 in each subcase of 1D and d = 0 in all other cases of Fig. 1. Let m denote the number of incomplete exchanges that have already occurred. For example, rn --- 0 in 1A and at the start, rn = 1 in each subcase of 1D, and m = 2 in 1B and 1C. Assume all exchanges are equally likely. Then appropriate differential equations are dP,am/dt = - [ f ( n ) + ltc((n + d) 2 - n -½d)]P, am + f ( n + 1)P, + 1,din +x[(n + 2)(n + 1)P,+ 2,d- 1,m-1 + 2d(n + 1)P,+ 1,a,m-1 +½(a + 1)(2d + 1)P,,a+ ,,m- 1]. (2.8) for example the term 2d(n + 1)P, + ~,a,,,- 1 refers to a transition in which one end of one double strand break is joined with one orphaned end, thereby decreasing the number of pristine double strand breaks by one, making a new orphan to replace the old orphan, and increasing the number of incomplete exchanges by one. In the case of Eq. (2.8) N of Eq. (1.1) is a triplet (n, d, m). The appendix mentions other post-irradiation pathways that have been analyzed. It also presents a very detailed model which enables one to keep track of various biologically important features, such as the number of altered chromosomes which lack centromeres. Evolution of DNA damage in irradiated ceils 501 Immediate radiation damage We next consider a model for immediate radiation damage, corresponding to the matrix elements Rxu in Eq. (1.5a). Suppose the immediate radiation damage consists of D N A double strand breaks. The process of inflicting double strand breaks is quite complicated (Charlton et al. 1989; Warters and Lyons 1990) but for gamma irradiation it is believed that in many situations all of the following idealizations are appropriate (Kellerer 1985). The breaks are inflicted during independent events, each event corresponding to a single photon depositing energy in a cell nucleus; each event can be regarded as instantaneous on the time scales of interest; the arrival of events at a cell nucleus can be regarded as a Poiss0n process with an expected value of Q events per unit time, where Q is independent of the cell and is constant in time during the irradiation period - T ~< t < 0; the probability an event inflicts exactly one double strand break is a constant # ~ 1 independent of amount of damage already present; and the probability an event inflicts more than one double strand break is negligible, so that the probability it inflicts no double strand breaks is 1 -/~. Let P,(t) be the probability a cell has exactly n breaks in its genome. When all these assumptions hold one has during irradiation den/dt=Q]A(-en -[- e n 1) -q- Sn, - - T <<. t <<. O. (2.9) Here Sn is an abbreviation for terms modeling cellular damage processing mechanisms, corresponding to SKN in Eq. (1.5a) and exemplified by the right hand side of (2.4). It often happens that irradiation is very rapid and ~ is so large that Sn is negligible during irradiation. In that case Eq. (2.9) leads to a Poisson distribution of breaks. Specifically, suppose the cell is undamaged at time t = - T, i.e. P~( - T) = 6n0. Then if S~ is negligible, integrating (2.9) gives P,(O)=e-aa"/n!, a=~T, n=0,1,.... (2.10) The appendix gives some generalizations of (2.9). 3 Asymptotic behavior We state our remaining assumptions, suggested by the examples. Then results will be given on the behavior of the Markov chains for large times. In this section we confine attention to the post-irradiation period t ~> 0. Irreversibility In all our examples of post-irradiation behavior, the reactions "run irreversibly downhill"; a cell in state N must reach an absorbing state after a finite number of transitions. In order to formalize this behavior in a form also convenient for analyzing behavior during irradiation (Sachs et al. 1991) we henceforth assume in (1.5) that there is a function L : E ~ R with properties analogous to those of a Lyapunov function for the post-irradiation part SKN of QKNo Specifically we assume that VK, N e E, K ~ N, SKN ~ 0 ::~ L(K) > L(N) (3.1) 502 P. Hahnfeldt et al. and that the inverse image L-1(N) is a finite set VN e E. For example in (2.4), L(n,m) =n + m is such a function, and in example (5.1) of the appendix L(n, m) = 2n + m is such a function. The assumptions on L imply that in (1.5b) all states which are not absorbing are transient in the very strong sense that they can never be revisited. For example in (2.4) the states for which n = 0 are absorbing and the states for which n ~ 0 are transient. Equation (3.1) implies that, for all t, to ~ [0, ~ ) with t/> to, the minimal solution P(t, to) of the Kolmogorov backward equation which is the identity matrix for t = to is also the unique solution to the Kolmogorov forward equation (1.5b); is the unique solution of the Kolmogorov backward equation (Wolff 1989); and is stochastic, i.e. has non-negative elements and row sums unity. Since the process is homogeneous during the interval 0 ~< t < 0% P(t, to) actually depends only on the difference t - to. An embedded chain The large time limit for the solutions of interest can be obtained as follows. For t/> 0 form the embedded discrete time Markov chain at transitions (Breimann 1969; Wolff 1989) in the usual way. Thus the chain has one-step transition probabilities SKj, with 'gKK equal to 1 or 0 according as K is absorbing or not, Sxs = 0 for K absorbing and J ~ K , and Sxs = S,, /IsKKI whenever K is not absorbing and J ~ K. For example if the continuous time Markov chain is given by (2.4) the non-zero transition probabilities for the embedded chain are S(o,m)(0,m) = 1, S(n+ S(n + 2,m)(,,m) =pg(n + 2)/s(n + 2), ,,m)(n,m) = f (n + 1) /s(n + 1), ~(n+ 2,m)(,,m+ 1) = (1 --p)g(n + 2)/s(n + 2); s(n) =-f(n) +g(n); n, m = 0, 1, 2 . . . . . (3.2) In any case, regard Sxs as matrix elements of a matrix ~q; then ~ is stochastic by virtue of (1.4). By construction Eq. (3.1) holds if ~KN is substituted for SKU, SO every sample path of the embedded chain reaches an absorbing state after a finite number of steps. For any non-negative integer r the matrix power ~r exists since is stochastic (actually, in our case, each element of ~r is a finite sum by virtue of (3.1)). L e t P u ( 0 ) be the initial mass distribution and (~r)Ks denote the matrix elements of S r. Then fiN (r) = ~ PK ( O)(Sr)KN (3.3) K is the mass distribution after the rth step of the embedded discrete chain with initial condition PK(0) = PK(O). Eq. (3.1) implies (Breimann 1969) VN e E, lim PN(t)= lim Pu(r), t ---~ oO (3.4) r~oO where PN(t) is the solution of the Kolmogorov forward equations as in the discussion of (1.2). The limit in (3.4) is a non-defective mass distribution PN(oo) on the absorbing states. For the actual, continuous time chain the approach to PN(~) involves exponential time factors, e.g. e v', where --v is a non-zero eigenvalue of A in (2.6). Evolution of DNA damage in irradiated cells 503 Recursion relations In the cases of interest, one can find a recursion relation which determines PN(OO). We illustrate the procedure in the case of example (2.4). Recall that there the absorbing states are {(0, m)}. Let Pnm(O) be a non-defective mass distribution which gives the initial state of both the continuous time and the embedded Markov chains. Let f,,, be the probability that a cell in state (n, 0) at time t = 0 is found in state (0, m) at t --- oo. We setfn,_l = 0 as a convention and note f orn = 6orn =Yarn" The probability aberrations is (3.5) Pore(00) a cell's genome ultimately contains exactly m irreparable e0m((X)) = ~ ~ P . . . . j(O)fnj (3.6) n=0j=0 since the transition probabilities S(n,m)(n',m')i n (3.2) depend only on m ' - m , not m and m' separately. By using (3.4) and conditioning on the first step of the embedded chain (3.2) one obtains the desired recursion relation: f , + 2,m = [f(n + 2)/s(n + 2)]f~ + 1,m -']-[g(n + 2) /s(n + 2)][pfn m + (1 --P)fn,rn-1]" (3.7) This has a unique solution obeying (3.5). Truncation Consider the truncated vector with components fiN(to) = PN(to), N <. No; Suppose t>~t o. The error defined and obeys fiN(to) = 0, U > No; to >i 0. (3.8) APN(t ) =--~K[PK(tO)--fiK(to)]PKN(t, to) is well 0 ~ APN(t) <<.~ APK(t) = ~. Ae~(to) = ~ ~< 1. K K (3.9) In particular, for to = 0 we may use a truncation (3.8) such that e ~ 1 and then use f i N ( 0 ) / ( 1 - 5) as initial condition. Thus assuming Pw(0) is a non-defective mass distribution (1.1) with only finitely many components different from zero, as is done when comparing to experiments (Sachs et al. 1990; Hlatky et al. 1991), involves no essential loss of generality. We shall henceforth make this assumption. Matrix expressions for the limits The limiting distribution PN(OO) can be worked out explicitly in terms of matrices (Hahnfeldt 1991) and we have found the result convenient in numerical calculations (Hlatky et al. 1991). Example (2.4) can be used to show the essential ideas. With m fixed write Pro(r) for the row vector with components Pnm(r), where P,m(r) is as in (3.3), and define P-1 - 0. In this notation one has from 504 P. Hahnfeldt et al. (3.2) an equation for the embedded chain corresponding to (2.5) for the continuous time chain: Pm(r ÷ Pm(r)()? + I0) + P m +1 (r)g. 1) = (3.10) Here Io is taken to be the projection onto the zeroth state, i.e. (Io)jk = 6jorko. Consequently j?= 0 0 0 1 0 0 pg(2) f(2) 0 s(2) s(2) pg(3) f(3) 0 s(3) s(3) 0 0 g(2) ; /~ = (1 - p ) 0 0 0 0 0 0 o o g(3) s(3) 0 0 (3.11) g(4) s(4) By inspection of (3.11) each component of the matrix )7r, for r a positive integer, is a well defined non-negative number; ~ = o Jfr converges, where )?0 is taken as the identity matrix I; and l~ ~ = 0 J ? ~ ~ = 1 J~. Let A be the space of row vectors which have only finitely many components different from zero and regzard J? as a linear operator A ~ A by p ~pJ?. Then, in view of J? ~ = 0 X~= ~r~= 0 )~r __ I = I ~ = 0 )7~ --/, the matrix (I -- J~) has an inverse: ( / _ _ J ~ ) - - l = ~ £r. r=0 (3.13) Theorem. Define ( Z m -J(l -- X ) -1)n 0 . Z=(I-J~)-l/q. Then (3.12) Pom(O0) = 2?=o 2n~Z=oPnj(O) Proof Consider the embedded chain (3.2). A sample path which goes from (n, 0) at r = 0 to (0, m) at r = oe must undergo exactly m "vertical" transitions (i.e. Am = 1). Other transitions are "horizontal", i.e. Am = 0. Suppose there are ro i> 0 horizontal transitions before the first vertical transition, rl/> 0 horizontal transitions between the first and second vertical transition, etc,, where remaining in the same absorbing state is not counted as a transition. Then the probability for going from (n, 0) to (0, m) along this path is the matrix element ( £ rO I~ £ rl g "'" I~ £rm)n O. For example if r o + r l + ' " + r m >n-2m the probability is zero by (3.11). Summing over all the independent possibilities one has ro=Orl=0 Using (3.12) theorem. rm=O gives f . m = ( Z m ( I - - £ ) - l ) . O . Using (3.6) now gives the [] Extending (3.13) to our other examples is mainly a question of appropriate ordering. For example, in the case of Eq. (2.8) one can define P,,(t) to be the vector with components PF(,,d) = P, dm(t), where F is the one-one onto ("triangu- Evolution of DNA damagein irradiated cells 505 lar") function defined by F~n, d) = n + ½(n + d)(n + d + 1) e {0, 1,... }. One defines appropriate matrices A,/~, ~ by AF(n,d)F(j,k) = S(n,d,m)(j,k,m) ; X ~_. z~ _ I0; ~[F(n,d)F(j,k) = S(n,d,m + 1)(j,k,m)" (3.14) Then the result (3.13) and its proof hold essentially as above. In the case (5.3), where the state space consists of sextuplets of integers, a wholly analogous method can be used. Applications Theorem (3.13) has been applied (Hlatky et al. 1991) to extensive data on chromosome aberrations in human lymphocytes obtained at about 30 laboratories (Lloyd et al. 1987; Bender et al. 1998). In chromosome experiments, the observed quantities are the average damage as a function of dose and sometimes its statistical distribution. For example, having calculated P0m(°o) from (3.13) one can compute the expected value rh =~2=oP0m(oo)m and compare to empirical averages of observed chromosome aberrations, rh depends on the total radiation dose D received by the cells, as indicated by (2.9), (2.10) and (5.5). One thus gets theoretical curves ffffD) for comparisons with the experimental ones. The lymphocyte experiments include data on the ratio of variance to mean, which is given in this model by ~m~=0 Po,,(oo)(rn --nq)2/rh. In this connection we mention an unsolved mathematical problem. We have found by numerical methods that some post-irradiation models, such as (2.4) with p = 1 and f = 2n, give underdispersion: i.e. if the Poisson distribution (2.10) holds initially the ultimate ratio of variance to mean is less than 1 at all relevant doses. Other models give overdispersion. Using power series in the dose one can understand the numerical results at small doses: the essential difference is the nature of the relevant pathways when only a small amount of damage is present, e.g. 4 double strand breaks or fewer at the time irradiation steps. However, we do not know explicit analytic estimates for the dispersion at larger doses. Markov chain models have also been applied to analyzing experiments on cell survival (Obaturov et al. 1980; Albright and Tobias 1985; Yang and Swenberg 1986; Curtis 1988; Albright 1989; Sachs et al. 1990). In this case, the key theoretical quantity is the zero class, e.g. Po0(Oo) in (5.1) and (5.2). The principle of branching ratios A conceptually interesting application can be obtained by noting that the elements of the matrix ~ for the discrete time chain embedded at transitions are obtained as ratios of the elements in S, as exemplified in Eq. (3.2). ~ remains invariant if any row of S is rescaled by any positive factor C : S~:~v-o C(K)SKN. In view of (3.4) this implies that an experimental assay performed at t = oo, no matter how detailed, cannot measure SKN, only ratios like SICN/ISKr[. Now many semi-empirical discussions focus attention on the implications of experiments for the elements of Sx~v, i.e. repair rates, pairwise interaction rates, etc. A great deal of the literature thus centers on the detailed properties of functions like g(n) in (2.3) or alternatives like (2.2) for the repair rate function f(n). The 506 P. Hahnfeldtet al. realization that t -- ~ assays are actually measuring not g(n) or f(n) but ratios like f(n)/[f(n)+g(n)] turns out to be quite fruitful in heuristic arguments (Hlatky et al. 1991). For example an experimental result assayed at t = oo and apparently due to linear misrepair combined with linear repair, g(n)oc n, and f(n) oc n, can in principle be explained instead by a combination of quadratic misrepair and inducible repair, g oc n 2 and foc n 2. The ratios measurable by assays at t = oo are analogous to branching ratios in nuclear decay schemes. We therefore refer to the implication of (3.4) that such assays measure only ratios of elements of SKN as the principle of branching ratios. A simple, well known example is that in the most usual version of the Markov RMR model of cell killing (Albright 1989), it is only the ratio of quadratic repair constant to linear repair constant, not the separate values of these two constants, which determines the cell survival and statistical distribution of lethal lesions after large incubation times. For some assays, e.g. PCC experiments (Iliakis 1988), the solutions of (1.3) and (1.5b) at finite times are relevant. The principle of branching ratios does not apply to such experiments. 4 Discussion Models for the damage to human cells by ionizing radiation are important above all at low doses. When the dose is, e.g., 10 -2 Gray x-ray equivalent, the effects on any one cell are sufficiently small that experiments may not give unambiguous results. But even doses of this magnitude, if imparted to a sufficiently large number of people, e.g. by radon daughters, could perhaps be extremely damaging to many individuals. For such doses theoretical extrapolations are needed. Other areas of major interest include cancer radiotherapy and radioimmunotherapy. Of course there is also always the possibility that studying DNA damage and repair in cells subjected to abnormal stress may provide decisive insights into normal behavior. The results given in the present paper indicate that continuous time Markov chains provide one powerful and flexible approach to studying the time evolution of DNA damage caused by ionizing radiation. The results proved here have been applied in detail to experimental observations (Sachs et al. 1990; Hlatky et al. 1991) with some, though limited, success. The Markov models lend themselves to a combination of analytic methods with numerical methods. They are thus less CPU intensive and more transparent than outright Monte-Carlo simulations of stochastic effects. The analytic results on Markov chains give useful general insights, such as the principle of branching ratios. DNA-DNA interactions For 50 years, it has been inferred from experiments that ionizing radiation induces lesions which can interact pairwise (compare Lea's classic (1955) summary of work done during the 1940's and Cornforth (1990)). Such lesions have been given various names ("uncommitted lesions", "sublesions", etc., etc.), assumed to interact during and/or after irradiation, identified with various macromolecular species (e.g. DNA double strand breaks) and putatively endowed with a variety of kinetic or combinatorial properties. The models of this Evolution of DNA damagein irradiated cells 507 paper likewise emphasize such interactions. Oddly enough, during the entire half-century the existence and importance of pairwise interacting lesions have been clearer than their exact identification and behavior. This situation is not unique. Since the days of Einstein and Debye it has been clear that "harmonic oscillators" play a key role in solid state physics. The microphysical interpretation and assumed detailed properties (e.g. frequency spectrum) of the harmonic oscillators have changed drastically several times during this period, but harmonic oscillators have continued to be central. Limitations A serious limitation of most time-dependent DNA damage models, including the models considered in the present paper, is that in analyzing exchanges they treat the cell nucleus as spatially uniform and ignore nanometer-scale fine structure. In (2.4) the fact that breaks on the same chromosome are, on the average, closer together and therefore presumably more likely to interact than breaks on different chromosomes is completely ignored. Moreover, there must be many different kinds of double strand breaks, which differ from each other on the nanometer scale. It seems very likely that models which are more fine-grained spatially (cf. Charlton et al. 1989), and reaction-diffusion models (Brenner 1990), will ultimately be needed to understand the processes of DNA radiation damage and repair. However, the simpler spatially uniform models will presumably survive as guides to the more detailed models. 5 Appendix: More examples Other post-irradiation models Another Markov model has been used in the literature (Albright 1989; Sachs et al. 1990) to describe cell killing. It developed from a non-Markov model called the repair misrepair model (RMR) for cell killing (Tobias et al. 1980) and is referred to as the Markov RMR model (Albright and Tobias 1985; Albright 1989). In the model a cell state is described by a pair (n, m) of non-negative integers. One says a cell in state (n, m) has n uncommitted and m lethal lesions. Uncommitted and lethal lesions correspond roughly to breaks and exchanges respectively. Suppose Pnm(t) is the probability a cell is in state (n, m) at time t and again formally define Pn,-1 - 0 . The time dependence after irradiation steps is given by dPnm = - [ f ( n ) + g(n)]Pnm +f(n + 1)Pn + l,,~ + g(n + 1)P, + l,m-1, f(n)=e~2n+O~n(n--1), g(n)=(1--qb)2n+(1--O)~n(n--1). (5.1) (5.2) Here 2, ~, ~, q~ are constants, with 2, ~ > 0, 1 >~~, q~ >~0. The use of ~ and ~b is related to the question of eurepair (correct repair) vs. misrepair. For example if q~ = 0.9 one says that 90% of linear repair is eurepair and 10% of linear repair is misrepair (Albright 1989). In this model linear misrepair is capable of modeling either mistakes by the repair enzymes or a gradual loss of an uncommitted lesion's ability to undergo any reaction. A model more detailed than any we have considered is the following, which applies to incomplete exchanges at doses so low there is at most one double 508 P. Hahnfeldt et al. strand break per chromosome (this restriction can be relaxed; see Hlatky et al. 1991). Let n be the number of double strand breaks. Let dl be the number of orphaned ends on chromosome pieces that contain a centromere; see the discussion of Eq. (2.8) for the definition of an orphaned end. Let d2 be the number of orphaned ends on chromosome pieces which have no centromere. Let m0, ml, m2 be the number of exchanges which have made altered chromosomes with two telomeres and 0, 1, or 2 centromeres respectively (compare Fig. 1B and 1C). On assumptions similar to those used in (2.4) and (2.8) the appropriate chain is determined by 11 d e n d l d 2 m o m l m 2 / d t = E Gi" i=1 (5.3) Here, with f i n ) , g(n) given by (2.2) and (2.3), the Gi are: G 1 = - [ f ( n ) + 2g(n) + tcn(d 1 + de) + (1/2)g(dl + d2)]Pnala2mom in2" G2 = f ( n q- 1)en+ l,dld2mOmlm2. G3 = ½g(n + 2)P,+2,ala2-2,mo- 1,mlm 2" G 4 =½K(n + 1)(d1 + l ) e n + l , a l + l , d 2 _ l , m o _ l , m l m 2 . G5 = ( 1/2)tog(4 + 2)P.a, + 2.a2,,,o- l,m ,"2" G 6 = g(n + 2 ) P n + 2 , d l _ l , d 2 - l , m 0 m l - l,rn2" G7 = ½~c(n+ 1 ) ( d 1 + d 2 ) P , + l,dld2mom I - 1,m 2. G 8 = ½K(d 1 -~- 1)(d2 + 1)Pn,d, + 1,a2+ 1,morn , _ 1,rn2. G9 = ½g(n + 2)P. + 2.a, - 2.a2mo.,2 - 1. 61o = ½~c(n+ 1)(d2 + 1)P,+ 1,a,- 1,a2+ 1,momlm2--1" 1 p 611 ~--~g(d 2-~- 2) n d l d 2 + 2 , m o m l m 2 _ l . If one takes d = 5(dl 1 + d2) and m = mo + ml + m2 the model (5.3) becomes formally identical with (2.8). The Markov chain formalism is flexible enough to incorporate additional pathways into any given model. One example is that in a model such as (2.4) one can assume double strand breaks gradually lose their ability to either repair or undergo binary interactions leading to exchanges. The simplest way of inserting this assumption into the mathematics is to add a term proportion to - - n P n m to the right hand side of (2.4). Then 1 - ~ u P N ( t ) represents the fraction of cells which have at least one double strand break that has undergone such a reaction. Many other examples of additional or alternate pathways can be considered. It is also possible to incorporate some cell-kinetic effects into the formalism, though these are perhaps more naturally handled with semi-Markov processes. General equations for immediate damage For certain kinds of radiation the probability that one ionizing radiation track induces more than one double strand break is not negligible. For example an alpha particle (helium nucleus) emitted by a radon daughter has a high probability of causing several double strand breaks in each cell nucleus it crosses. Such situations can be modeled as follows (Yang and Swenberg 1986). Define an event Evolution of DNA damage in irradiated ceils 509 for the nucleus of one cell as all the energy depositions which are due directly or indirectly to one primary radiation particle (Kellerer 1985) and consider the specific energy defined in Sect. 1. Define a one-event distribution function F1 (z) as the probability one event imparts specific energy z or less to a cell's nucleus. F~ (z) is determined by some rather complicated geometric factors and properties of ionizing radiation tracks. For several decades an extensive experimental and theoretical program has been devoted to measuring and modeling such functions FI (z) for various kinds of radiation in cell nuclei of various sizes (Kellerer 1985; Goodhead 1987). The corresponding one-particle distribution density fl (z) = dF 1/ dz is typically continuous and obeys: fl (Z) >/0; 0 f0 fl (Z) dz = 1; fl(g)zNdz<oo, N=0,1 ..... (5.4) For example, it is usual to denote the expected value of specific energy per event as ZF ~" S~ zUI (Z) dz. With this notation (5,5) O = OeFT where 0 and T are as in (2.9) and D is the total dose (see Sect. 1). It is reasonable to assume that an ionizing radiation event immediately makes a Poisson distribution of double strand breaks whose expected value a is a function a(z) of the specific energy deposited (Kellerer 1985; Albright 1989). Then the probability a single event makes exactly n double strand breaks is I.t, = e-a(zl[a(z)]"fl(z) dz/n!. (5.6) 0 Under these circumstances, the appropriate generalization of (2.9) is dP,/dt = ~ PkRkn + S,, - T <~ t <~ O, (5.7) k where the matrix R has the Toeplitz form (Hug and Kellerer 1966) R=~ --# 0 0 0 #1 -~ 0 0 P2 ~1 -~ 0 #3 g2 ~1 --# (5.8) Here # = ~ 2 = 1 ~n SO R obeys the restrictions (1.4) for a Markov chain. As long as irradiation is so rapid S, is negligible, integration of the Kolmogorov forward equations corresponding to (5.7) is rather straightforward, due to the upper triangular form of R. The resulting distributions are batch Poisson. Numerically computed examples of the case where S, is not negligible are given in Sachs et al. (1990). In all the models mentioned up to now, the only immediate damage is double strand breaks. There are some situations where irradiation immediately makes other kinds of damage. Markov models for such processes can be constructed (Curtis 1988). For example, there may be a probability vm of making exactly m lethal lesions in one event. Then the appropriate generalization of (2.9) involves a Kronecker product of two matrices similar to (5.8). 510 P. Hahnfeldt et al. References Albright, N., Tobias, C. A.: Extension of the time-independent repair-misrepair model of cell survival to high LET and multicomponent radiation. In: LeCam, L., Olshen, R. A. (eds.) Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, pp. 397-424. Belmont CA: Wadsworth 1985 Albright, N.: A Markov formulation of the repair-misrepair model of cell survival. Radiat. Res. 118, 1-20 (1989) Bedford, J. S., Cornforth, M. N.: Relationship between recovery from sublethal x-ray damage and the rejoining of chromosome breaks in normal human fibroblasts. Radiat. Res. 111, 406-423 (1987) Bender, M. A., Awa, A, A., Brooks, A. L., Evans, H. J., Groer, P. G., Littlefield, L. G., Pereira, C., Preston, R. J., Wachholz, B. W.: Current status of cytogenetic procedures to detect and quantify previous exposures to radiation. Mutat. Res. 196(2), 103-59 (1988) Breimann, L.: Probability and stochastic processes. New York: Houghton Mifflin 1969 Brenner, D. J.: Track structure, lesion development, and cell survival. Radiat. Res. 124, $29-$37 (1990) Charlton, D. E., Nikjoo, H., Humm, J. L.: Calculation of initial yields of single and double strand breaks in cell nuclei from electrons, protons and alpha particles. Int. J. Radiat. Biol. 56(1), 1-19 (1989) Cornforth, M. N.: Testing the notion of the one-hit exchange. Radiat. Res. 121, 21-27 (1990) Curtis, S. B.: The lethal and potentially lethal model - a review and recent development. In: Kiefer, J. (ed.) Quantitative mathematical models in radiation biology, pp. 137-146. Berlin Heidelberg New York: Springer 1988 Doraiswamy, L. K., Kulkarni, B. D.: The analysis of chemically reacting systems: a stochastic approach. New York: Gordon and Breach 1987 Erdi, P., Toth, J.: Mathematical models of chemical reactions. Princeton, NJ: Princeton University Press 1989 Frankenberg-Schwager, M.: Review of repair kinetics for DNA damage induced in eukaryotic cells in vitro by ionizing radiation. Radiother Oncol 14, 307-320 (1989) Friedberg, E. C., Hanawalt, P. C.: Mechanisms and consequences of DNA damage processing. New York: Alan R. Liss 1988 Goodhead, D. T.: Relationship of microdosimetric techniques to applications in biological systems. In: Kase, K., Bjarngard, B., Attix, F. (eds.) The Dosimetry of ionizing radiation, vol. II, pp. 1 89. Orlando, FA: Academic Press 1985 Hagen, U.: Biochemical aspects of radiation biology. Experientia 45(1), 7-12 (1989) Hahnfeldt, P.: Markov models of radiation damage. Thesis, MIT, 1991 Hlatky, L., Sachs, R. K., Hahnfeldt, P.: Reaction kinetics for the development of radiation-induced chromosome aberrations. Int. J. Radiat. Biol. 59(5), 1147-1172 (1991) Hou, Z., Guo, Q.: Homogeneous denumerable Markov processes. Berlin: Springer and Beijing: Science Press 1988 Hug, O., Kellerer, M.: Stochastik der Strahlenwirkung. Berlin Heidelberg New York: Springer 1966 Iliakis, G.: Radiation-induced potentially lethal damage: DNA lesions susceptible to fixation. Int. J. Radiat. Biol. 53(4), 541-584 (1988) Kellerer, A. M.: Fundamentals of microdosimetry. In: Kase, K., Bjarngard, B., Attix, F. (eds.) The dosimetry of ionizing radiation, pp. 78-162. Orlando, FA: Academic Press 1985 Lea, D. E., Catcheside, D. G.: The mechanism of induction by radiation of chromosome aberrations in Tradescantia. J. Genet. 44, 216-245 (1942) Lea, D. E.: Actions of radiations on living cells, second edition. Cambridge: Cambridge University Press 1955 Lloyd, D. C., Edwards, A. A., Prosser, J. S., Barjaktarovic, N., Brown, J. K., Horvat, D., lsmail, S. R., Koteles, G. J., Almassy, Z., Krepinsky, A.: A collaborative exercise on cytogenetic dosimetry for simulated whole and partial body accidental irradiation. Mutat. Res. 179(2), 197-208 (1987) McQuarrie, D. A.: Stochastic approach to chemical kinetics. J. Appl. Probab. 4, 413-478 (1967) Evolution of DNA damage in irradiated cells 511 Obaturov, G. M., Mateeva, L. A., Tyatte, E. G., Yas'kova, E. K. The stochastic model of formation of chromosome aberrations and radiation inactivation of cells (in Russian). Radiobiol. 20, 803-809 (1980) Radford, I. R.: The dose-response for low-LET radiation-induced DNA double-strand breakage: methods of measurement and implications for radiation action models. Int. J. Radiat. Biol. 54(1), 1-11 (1988) Reddy, N. M. S., Mayer, P. J., Lange, C. S.: The saturated repair kinetics of Chinese hamster V79 ceils suggests a damage accumulation-interaction model of cell killing. Radiat. Res. 121, 304-311 (1990) Revell, S. H.: Relationships between chromosome damage and cell death. In: Ishihara, T., Sasaki, M. S. (eds.) Radiation-induced chromosome damage in man, pp. 215-233. New York: Alan R. Liss 1983 Sachs, R. K., Hlatky, L.: Stochastic dose-rates in radiation cell survival models. Rad. Environ. Biophys. 29, 169-184 (1990) Sachs, R. K., Hlatky, L., Hahnfeldt, P., Chen, P-L.: Incorporating dose rate effects in Markov radiation cell-survival models. Radiat. Res. 124, 216-226 (1990) Sachs, R. K., Chen, P-L., Hahnfeldt, P., Lai, D., Hlatky, L.: DNA damage in non-proliferating cells subjected to ionizing radiation at various dose rates. (In preparation, 1991) Savage, J. R. K.: The production of chromosomal structural changes by radiation: an update of Lea (1946), Chapter VI. Br. J. Radiol. 62(738), 507-520 (1989) Shadley, J. D., Afzal, V., Wolff, S.: Characterization of the adaptive response to ionizing radiation induced by low doses of X rays to human lymphocytes. Radiat. Res. 111(3), 511-517 (1987) Tobias, C. A., Blakely, E. A., Ngo, F. Q. H., Yang, T. C. Y.: The repair-misrepair model of cell survival. In: Meyn, R. E., Withers, H. R. (eds.) Radiation biology in Cancer Research, pp. 195-229. New York: Raven 1980 Virsik, R. P., Harder, D.: Recovery kinetics of radiation-induced chromosome aberrations in human Go lymphocytes. Radiat. Environ. Biophys. 18, 221-228 (1980) Wallace, S., Painter, R.: Ionizing radiation damage to DNA: molecular aspects. J. Cell. Biochem. Supplement 14A, 21-84 (1990) Ward, J. F.: DNA damage produced by ionizing radiation in mammalian cells: identities, mechanisms of formation, and repairability. In: Cohn, W., Moldave, K. (eds.) Prog. Nucleic Acid Res. Mol. Biol., vol. 35, pp. 95-125. New York: Academic Press 1988 Ward, J. F.: The yield of DNA double strand breaks produced intracellularly by ionizing radiation: a review. Int. J. Radiat. Biol. 57(6), 1141-1150 (1990) Weber, K. J.: Models of cellular radiation action- an overview. In: Kiefer, J. (ed.) Quantitative mathematical models in radiation biology, pp. 3-28. Berlin Heidelberg New York: Springer 1988 Wolff, R. W.: Stochastic modeling and the theory of queues. Englewood Cliffs, NJ: Prentice Hall 1989 Yang, G. L., Swenberg, C. E.: Stochastic models for cells exposed to ionizing radiation. In: Eisenberg, J., Witten, M. (eds) Modelling of Biomedical Systems, pp. 85-89. Amsterdam New York: Elsevier 1986