Supplemental Material for
Erratum to: A new view of radiation-induced cancer: integrating short- and long-term processes. Part I: Approach, and Part II: Second cancer risk estimation. Radiat Environ Biophys
Igor Shuryak, Philip Hahnfeldt, Lynn Hlatky, Rainer K. Sachs, David J. Brenner
As briefly summarized by Shuryak et al. 2011, there was an error in Eq. (22) of the Appendix of Shuryak et al. 2009a. The Results and Figures in this “Formalism” paper (Shuryak et al. 2009a), as well as in the subsequent “Results” paper (Shuryak et al. 2009b) remain unchanged. The conclusion of the two papers, that this mechanistically-based formalism provides a practical methodology for predicting radiation-induced cancer risks at radiotherapeutic doses, still stands.
There follows here a corrected Appendix to Shuryak et al. 2009a, which gives the derivation and validation of the corrected version of Eq. 22, as well as its biological rationale and interpretation.
Corrected Appendix to Shuryak et al. 2009a
First, repopulation effects for normal stem cells are analyzed deterministically (Eqs. 1-3); then survival probabilities for pre-malignant clones are analyzed stochastically; finally, intuitive interpretations of the stochastic equations are given. The equations derived are extensions of results given previously (Sachs et al. 2007).
Normal stem cell numbers
Denote the time of the k-th dose fraction by T(k). We will derive recursive equations valid for
k = 1,…, K, where the number of fractions is K as in Shuryak et al. 2009a. We formally define
T(K+1) =¥; this infinite value represents the end of the recovery period (Fig. 1 of Shuryak et al. 2009a). It reflects a standard procedure in multi-timescale analyses (Engquist et al. 2005): an infinite time interval in a short-timescale model represents a short time interval in the next larger timescale model (here several months in our long-timescale model). We correspondingly set
n-(K+1)=n, the set point value attained at the end of the recovery period. Solving Eq. (9) in (Shuryak et al. 2009a) gives the ratio [n(t)/n+(k)] for any time t after the k th dose fraction and before the (k+1)st fraction:
n(t)/n+(k) = n/{n+(k) + [n – n+(k)] exp[–l(t – T(k))]}, for T(k) < t < T(k + 1) (1)
In particular for the recovery factor between the k th and (k+1) st dose fractions, defined as cumulative relative proliferation n-(k + 1)/n+(k) and denoted here by R(k), Eq. (1) gives
R(k) º n-(k + 1)/n+(k) = n/(n+(k) + [n – n+(k)] exp[–l(T(k + 1) – T(k))]) (2)
Normal stem cell number n(t) can be calculated recursively by combining the proliferation equations, (1) and (2), with the survival equation discussed in (Shuryak et al. 2009a), namely:
(3)
Pre-malignant stem cell clones
To analyze pre-malignant clones stochastically, consider a clone that starts with a single live stem cell initiated by the k th dose fraction, and is followed in time from just after that fraction until just after the last dose fraction, i.e. for T+(k) £ t £ T+(K), where T+(k) is used to denote the time just after the k th fraction. The time evolution of the clone is modeled as a time inhomogeneous birth-death process with birth rate b(t) and death rate r(t) (Tan 2002). The appropriate death rate, taking the spontaneous death rate as zero to minimize the number of adjustable parameters, is the death rate due to the dose fractions, namely
, (4)
where d [t] is the Dirac delta function. Equation (4) corresponds to the statements that on average the surviving cell fraction for the k th dose-fraction is given by Eq. (3) and that pre-malignant stem cells have the same radiosensitivity to inactivation as do normal stem cells. By our assumption that during the comparatively short radiotherapy period normal and pre-malignant cells have effectively the same proliferation rate, the appropriate birth rate is
b(t) = l{1 – [n(t)/n]} (5)
In Eq. (5) n(t) is a known function of time, determined as discussed above.
It is well known [e.g. (Tan 2002), pp. 169-171] that by integrating an appropriate partial differential equation for the probability generating function one can deduce the following expression for the probability F(k) that the pre-malignant clone survives all subsequent dose fractions:
(6)
Because of the manner in which a Dirac d function behaves, using T+ rather than T is important in the expressions for y and x ; however, using T(k) and/or T(K) for the limits of integration in the integral for z would not alter the result.
Performing the integrals in Eq. (6) with the help of Eqs. (1-5) gives, after considerable algebraic manipulation, the following convenient equation for the clone survival probability F(k) in terms of cell survival probabilities [Eq. (3)] and cell recovery factors [Eq. (2)]:
(7)
Equation (7) is valid for k=0, 1,..., K, with k=0 referring to pre-malignant cells present before radiation starts and F(K) = 1. It is the primary mathematical result needed for the data analysis discussed in (Shuryak et al. 2009a). Generalizing to situations where one needs the probability that a clone has a given number of cells at the final time, and/or situations where the spontaneous death rate is non-zero, can readily be done by using results given by Tan 2002.
Interpretation of Eq. (7)
In order to facilitate interpretations, this subsection analyzes a slightly generalized situation where the dose can vary from fraction to fraction; we designate the dose in the k th fraction by d(k). Setting d(k) = d, independent of k, then gives Eq. (7) used in the actual data analysis of Shuryak et al. 2009a. In the general case, the survival S is different for different fractions, as follows:
(8)
The corresponding generalization of Eq. (7), obtained by the same calculations as Eq. (7), is
(9)
Equation (9) has a number of useful intuitive interpretations and special cases which give insight into clone eradication [compare Hanin 2004]. The j th term of the sum (S) in Eq. (9) refers to eradication of the clone by the j th dose-fraction. The products (P) refer to successive proliferation and survival events. For example, suppose for concreteness there are just six dose-fractions and we want the chance F(3) that a clone initiated as one cell in the 3rd dose-fraction survives the 4th, 5th, and 6th dose-fractions. Then the denominator D in Eq. (9) is
/ (10)Among the intuitive checks and interpretations of Eq. (9) and its special case Eq. (10) are the following:
1. Suppose there is no proliferation. Then we expect that the chance of killing a clone is just that of killing one cell, e.g. S(4)S(5)S(6) in Eq. (10). Setting R(j) = 1 in Eq. (10) gives
D = 1/[S(4)S(5)S(6)], implying F(k) = S(4)S(5)S(6), exactly as expected; the analogous result holds for the general case, Eq. (9).
2. Each term in D is ³0. Being in the denominator, each non-zero term reduces the calculated clone survival probability. For example, in Eq. (10) the term containing the cell killing fraction [1 – S(5)] for dose-fraction 5 refers to clonal extinction in dose-fraction 5. The more cells on average in the clone at the time of dose-fraction 5 (i.e. the bigger R(3)S(4)R(4), referring to proliferation R(3), then survival S(4), then proliferation R(4)) the less the chance of eradicating the whole clone in that fraction. Similarly the 1 – S(4) and 1 – S(6) terms are divided by factors referring to the proliferation and survival events that led up to the corresponding dose-fraction; analogous interpretations apply to the general equation (9).
3. Suppose cell survival in at least one relevant fraction is negligible, i.e. S(i) = 0 for some i ³ 3 in Eq. (10) or i ³ k in Eq. (9), then D is infinite. So the clone survival probability is zero, as required conceptually.
4. Suppose proliferation is enormous for some time interval. Then intuitively the only way to eradicate the clone is to wipe it out before that interval happens. Afterwards the clone contains too many cells to kill them all. As an example, setting, say, R(4) > 1 we find that Eq. (10) describes exactly such behavior, since for R(4) à ¥ all terms in D, which involve killing later (i.e. involve 1 – S(5) or 1 – S(6)), are negligible.
5. Suppose there is no proliferation within some one interval. Say R(4) = 1 for the sake of illustration. Then some algebra applied to Eq. (10) gives, elegantly,
(11)
Equation (11) can be interpreted as follows: sublethal damage repair occurs during the 4 to 5 inter-fraction interval; but there are no births or deaths since R(4) = 1; so we can imagine postponing the dose d(4) till T(5), administering d(4) simultaneously with d(5), but allowing for full sublethal damage repair between d(4) and d(5) in the b term. Then survival in that combined dose fraction at time T(5) is S' = S(4)S(5). Equation (11) shows that this collapsing of two dose-fractions into one does give back Eq. (9) for five dose-fractions total (1, 2, 3, 4 and 5 together, and 6), instead of six. So the equation and intuition do indeed match.
6. Suppose there is no killing in the i th dose fraction. Say, for example, d(5) = 0 and thus
S(5) =1 in Eq. (10). Then we get, also elegantly,
(12)
Equation (12) is intuitively correct because if S(5) = 1, proliferation between the 4th and 6th fractions proceeds smoothly, as if we had only a single, extra long, time interval. One can show that then R' = R(4)R(5) is exactly the right recovery factor. One can give this interpretation to R' = R(4)R(5) by detailed algebra in Eq. (2) for recovery factors, taking into account n+(5) = n-(5).
References
Engquist B, Lötstedt P, Runborg O (eds) (2005) Multiscale Methods in Science and Engineering Lecture Notes in Computational Science and Engineering, Vol. 44 Springer, New York
Hanin LG (2004) A stochastic model of tumor response to fractionated radiation: limit theorems and rate of convergence. Math Biosci 191 (1):1-17
Sachs RK, Shuryak I, Brenner D, Fakir H, Hlatky L, Hahnfeldt P (2007) Second cancers after fractionated radiotherapy: Stochastic population dynamics effects. J Theor Biol 249:518-531
Shuryak I, Hahnfeldt P, Hlatky L, Sachs RK, Brenner DJ (2009a) A new view of radiation-induced cancer: integrating short- and long-term processes. Part I: approach. Radiat Environ Biophys 48:263-274
Shuryak I, Hahnfeldt P, Hlatky L, Sachs RK, Brenner DJ (2009b) A new view of radiation-induced cancer: integrating short- and long-term processes. Part II: second cancer risk estimation. Radiat Environ Biophys 48:275-286
Shuryak I, Hahnfeldt P, Hlatky L, Sachs RK, Brenner DJ (2011) Erratum to: A new view of radiation-induced cancer: integrating short- and long-term processes, Parts I and II. Radiat Environ Biophys. Radiat Environ Biophys, DOI xxx
Tan W (2002) Stochastic models with applications to genetics, cancers, AIDS, and other biomedical systems. World Scientific, London
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