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The priming dose effect, called also the Raper–Yonezawa effect or simply the Yonezawa effect, is a special case of the radia- tion adaptive response phenomenon (radioadaptation), which refers to: (a) faster repair of direct DNA lesions (damage), and (b) DNA mutation frequency reduction after irradiation, by applying a small priming (conditioning) dose prior to the high detrimental (challenging) one. This effect is observed in many (but not all) radiobiological experiments which present the reduction of lesion, mutation or even mortality frequency of the irradiated cells or species. Additionally, the multi-parameter model created by Dr. Yonezawa and collaborators tried to explain it theoretically based on experimental data on the mortal- ity of mice with chronic internal irradiation. The presented paper proposes a new theoretical approach to understanding and explaining the priming dose effect: it starts from the radiation adaptive response theory and moves to the three-parameter model, separately for two previously mentioned situations: creation of fast (lesions) and delayed damage (mutations). The proposed biophysical model was applied to experimental data—lesions in human lymphocytes and chromosomal inversions in mice—and was shown to be able to predict the Yonezawa effect for future investigations. It was also found that the strong- est radioadaptation is correlated with the weakest cellular radiosensitivity. Additional discussions were focussed on more general situations where many small priming doses are used. Keywords Adaptive response · Radiation · Yonezawa effect · Priming dose · Challenging dose · Radioadaptation · Radiosensitivity · Radiation biophysics · Cancer physics · Lymphocytes Introduction adaptation to radiation, which causes faster DNA lesion repair and reduction of DNA mutation creation (Olivieri The radiation adaptive response effect (called also a radio- et al. 1984; Azzam et al. 1994; Wolff 1998 ; Feinendegen adaptation) is a biophysical phenomenon, which may occur 1999; Tapio and Jacob 2007; Dimova et al. 2008; Mitchel in the organism irradiated to low doses of ionizing radia- 2009; Guéguen et al. 2019). However, because not all irra- tion. This effect connects the irradiation process with the diation conditions induce radioadaptation (Mortazavi et al. 2003; Wójcik et al. 1996), and the effect shows strong indi- vidual variability (Bosi and Olivieri 1989), it can be very Ludwik Dobrzyński: Deceased. difficult to predict. Therefore, many scientific investigations in radiobiology and radiation biophysics aimed at under- * Krzysztof W. Fornalski krzysztof.fornalski@ncbj.gov.pl standing this phenomenon are still going on. Till recently, it has been assumed that the radiation adap- National Centre for Nuclear Research (NCBJ), tive response effect occurred when enhanced repair mecha- ul. A. Sołtana 7, 05-400 Otwock-Świerk, Poland nisms started after the creation of damage within the DNA Faculty of Physics, Warsaw University of Technology, chain (Mitchel 2009). Nowadays, new findings show that ul. Koszykowa 75, 00-662 Warsaw, Poland small radiation dose(s) can affect the expression of gene Department of Biophysics, Physiology and Pathophysiology, transcription (Sokolov and Neumann 2015). Exposure in Faculty of Health Sciences, Medical University of Warsaw proper conditions can produce “an alert, triggering and (WUM), ul. T. Chałubińskiego 5, 02-004 Warsaw, Poland Vol.:(0123456789) 1 3 222 Radiation and Environmental Biophysics (2022) 61:221–239 parameter, which is the main quantification of the general- ized Yonezawa effect. One has to note, however, that the first observation of this effect (without deeper explanation) was conducted by Prof. Raper in 1940s during the Manhattan Project (Raper 1947; Cronkite et al. 1950). The first theoretical explanation of this phenomenon was given by Dr Yonezawa and collaborators, who created the phenomenological multi-parameter model describing the effect (Smirnova and Yonezawa 2003, 2004). The approach proposed by Smirnova and Yonezawa is a deterministic model, describing the effect of ionizing radiation on the cells of four major hematopoietic lines: thrombocytes, lym- phocytes, erythrocytes, and granulocythes. It divides cells into categories depending on the stages of development Fig. 1 The scheme of the Yonezawa effect (also called the prim- (bone marrow precursor cells, nondividing maturing cells, ing dose effect or the Raper-Yonezawa effect): the single small dose and mature blood cells) as well as on the level of damage (D ) generates much less mutations than the single high dose (D ). 1 2 (undamaged cells, damaged cells, and heavily damaged However, when D follows D with some time distance between 2 1 them (Δt), the mutation (or lesion) frequency for that D + D total cells), and describes the relationships between those cat- 1 2 dose is lower than for single D by the exemplary value of δ = 0.73. egories. To describe the damaging effect of ionizing radia- This exemplary result was obtained using the input data: D = 1 UAD tion on the cells, the one-target-one-hit theory is used. The (Unit of Absorbed Dose), D = 5 UAD, Δt = 3 UT (Unit of Time), −3 −2 −1 −1 model consists of a system of differential equations with α = 1 [UT UAD ], α = 1 [UAD ], α = 0.7 [UT ]. The parame- 0 1 2 ter δ is therefore showing the percentage difference between the num- many unknown (free) parameters that must be determined ber of mutations (or lesions) generated by the single dose D (without based on experimental data. It allows the simulation of the the priming dose) and the combination of D + D 1 2 effect of either chronic irradiation or pulse single doses. It should be noted that this model has limited application to altering cellular responses to defend against subsequent certain types of cells (namely thrombocytes, lymphocytes, high dose-induced damages, and accelerating the cell repair erythrocytes, and granulocythes only) and that the large process. Moreover, the p53 signalling pathway was found number of equations and unknown (free) parameters make to play critical roles in regulating DNA damage responses” it hard to apply in practice. (Hou et al. 2015) because “the p53 pathway is composed The presented paper proposes a novel theoretical of a network of genes and their products that are targeted approach to the radiation adaptive response phenomenon, to respond to a variety of intrinsic and extrinsic stress sig- leading to the Yonezawa effect. This results in a three- nals that impact upon cellular homeostatic mechanisms that parameter model, which works well after parametric quan- monitor DNA replication, chromosome segregation and cell tification based on experimental radiobiological data. division” (Harris and Levine 2005; Vogelstein et al. 2000). The model is divided into two parts: in the first part the Therefore, the adaptive response mechanism may increase calculations focus on the reduction of DNA damage (lesions) DNA repair up-regulation, leading to the reduction of pos- in quite a short time after the irradiation; in the second part tradiation mutation frequency (Dimova et al. 2008). late effects, namely stable mutations, are considered. The There are two main processes of triggering the radiation latter case is much easier to consider in experimental radio- adaptive response: via chronic irradiation or pulse single biological research. doses. In the first case, the effect can be observed e.g. within some people living in areas with a high background radiation (Scott et al. 2009; Dobrzyński et al. 2015a, 2015b). The sec- Adaptive response phenomenon ond case is equivalent to the priming dose effect, also called the Yonezawa phenomenon (originally for mice) (Yonezawa The proposed theoretical approach to the radiation adap- et al. 1990, 1996; Matsumoto et al. 2004; Tapio and Jacob tive response effect was originally presented a few years ago 2007), where the small priming dose (D ) received prior (Fornalski 2014) but its detailed biophysical explanation to the high challenging dose (D ) can reduce detrimental has been published very recently (Dobrzyński et al. 2019). effects of the latter (Shadley and Wolff 1987; Wang et al. Therefore, only a short and general mechanistic description 2013; Toossi et al. 2016; Hauptmann et al. 2016). The exem- will be presented here. plary scheme of this effect is presented in Fig. 1, where Generally, linear and non-linear effects can cause a the difference between the D + D scheme (with time Δt response described by the specific hunchbacked shape 1 2 between doses) and the single D one is denoted as delta (δ) (Feinendegen 2005, 2016) of time- and dose-related 1 3 Radiation and Environmental Biophysics (2022) 61:221–239 223 probability functions of the radioadaptation appearance, If some of the conditions presented above are not ful- respectively: filled, Eq. ( 3) shall be used instead. In particular, for two separate dose pulses, namely D and D , which are received 1 2 − D P(D) = D e 1 with the time interval Δt > 0, one can simply denote the sum- (1) − t marized probability distribution of the adaptive response as P(t) = t e p (D ,D ,t) = p (D ,t ) + p (D , t + Δt). When both AR 1 2 AR 1 0 AR 2 0 where D and t denotes radiation dose and time after irradia- doses are received in the same time (Δt = 0), one shall write tion, respectively, and {α}, {β}, ν, η are free parameters. p (D ,D ,t) = p (D + D ,t) because they can be simply AR 1 2 AR 1 2 Both functions can be merged into a single, time- and dose- treated as the single dose (D + D ). 1 2 dependent, probability distribution of the adaptive response appearance after the single dose pulse D received t time ago (Fornalski 2014; Dobrzyński et al. 2019): Kinetics of DNA lesions repair 2 2 − D− t 1 2 p = D t e , (2) AR 0 Let us assume that the dose pulse, D, generates some number of fast and direct damage, N, called DNA lesions. Regard- where all parameters are positive ones while ν and η param- ing the actual experimental findings (Rothkamm et al. 2007; eters were assumed to be equal to 2, which follows the result Manning et al. 2013), this dependence is assumed to be lin- of pooled simplified quantification by Feinendegen (2016). ear with additional background metabolic lesions (for a zero- Equation (2) represents the simplest version of the adaptive dose situation), therefore response per unit of time. Please note that the function p AR reaches its maximum value for the dose D = 2/α and for N = + D. max 1 (5) 0 1 the time t = 2/α after irradiation. In general, Eq. (2) can max 2 In other words, N from Eq. (5) represents the immediate be rewritten in the continuous form as (Fornalski 2014): number of physical damage (lesions) generated linearly by the dose D. All parameters {μ} are calibration parameters 2 2 −𝛼 D−𝛼 (𝜏 −t) ̇ 1 2 p = 𝛼 D 𝜏 − t e dt, ( ) (3) AR 0 of Eq. (5), of which values are given in the literature e.g. t=0 in direct initial lesions testing (Ward 1995) and their back- ground level. Especially, the parameter μ corresponds to the where Ḋ corresponds to the time-dependent dose-rate and frequency of spontaneous lesions (without radiation), and τ is the cell’s (or the organism’s) age. In practice, however, µ is a linear slope. One has to note, however, that the free Eq. (3) is usually used for chronic irradiation, where Ḋ is parameter µ is rather small (µ << µ ) and can be simply 0 0 1 given by some dose distribution, Ḋ(t) (Dobrzyński et al. neglected. Exemplary values of {μ} parameters for dam- 2019). But in the biophysical calculations it is easier to apply age in human lymphocytes after neutron irradiation equal Eq. (2) (or its combinations) (Dobrzyński et al. 2016; For- µ = 0.0005 and µ = 0.832 (IAEA 2011; Słonecka et al. 0 1 nalski et al. 2017; Fornalski 2019). When the dose distribu- 2018). tion is discrete, namely single dose pulses D in time steps The repair mechanisms start just after lesion (N) appear- k, the dedicated form of Eq. (3) ance, however, the probability of the adaptive response is strictly correlated with the number of repaired damage after 2 2 − D − (K−k) 1 k 2 p = D (K − k) e , (4) a period of time, which can be presented as (Foray et al. AR 0 k=0 2005) can be used e.g. in Monte Carlo simulations (Fornalski dN =−N p dt, (6) AR 2014; Loan et al. 2019). Equation (4) shall be understood as: each dose D received K–k steps ago (K is the age) generates where p is given by Eq. (2) for single dose pulse irradia- k AR a single signal given by Eq. (2) extended over time, which tion. It is assumed that this mechanism is the only method of is additive to the rest K − 1 signals described by an iden- repair after postradiation lesion appearance. Please note that tical mathematical form (Fornalski 2014). The summation Eq. (6) is the basis for the general function of the remaining presented in Eq. (4) allows one to freely add every single number of lesions in an actual moment in time, N(T). The adaptive response signal generated by each dose D under detailed calculations are presented in the Appendix 1. two conditions: Let us assume that N denotes the initial number of dam- 0,1 age (lesions) induced by dose D in moment zero (assumed the dose D is a short pulse with the duration of t ⟶ 0; that N = μ + μ D , see Eq. (5)). Moreover, dose D gen- k Dk 0,1 0 1 1 2 the time interval between two consecutive doses is large erates an additional number of damage, N = μ D , in 0,2 1 2 enough, Δt >> t . Dk 1 3 224 Radiation and Environmental Biophysics (2022) 61:221–239 (a) Challenging dose only, D (b) Priming and challenging doses, D +D 2 1 2 Fig. 2 The time-dependent relations of the probability distribution of to the challenging dose D (the latter received in the same moment the adaptive response, p , and the number of postradiation lesions, as in the plot a)); one can observe that the priming dose increased the AR N(t), given by Eq. (2) and Eqs. (21)–(22), respectively. Plot a presents probability of successful repair (enhanced value of p ) which causes AR a scenario where a single (reference) dose D is applied; one can a stronger decrease in the number of lesions, N(t). All input data were observe the hunchbacked shape of the p function and the decrease the same as in Fig. 1. The variable t is the global time where t = 0 cor- AR in the number of lesions, N(t). Plot b presents a scenario with a com- responds to the moment of D bination of doses D + D , where the priming dose D is given prior 1 2 1 moment Δt (where N > N ). Figure 2 presents the time 0,2 0,1 related probability of the adaptive response (p ) and the AR number of unrepaired damage in time N(T) for a single dose D and the combination of priming and challenging doses, D + D . Please note, that the N(T) relationship is a strongly 1 2 decreasing function, however, it never goes to zero due to the hunchbacked shape of the adaptive response probability function: within our model, the stronger the repair processes are, the closer to zero the N(T) function is at a large T. This is consistent with experimental findings, see Fig. 3 based on the paper by Müller et al. (2001). The Yonezawa (priming dose) experimental scheme, which is presented in Fig. 1, can be quantified as 1+2 = 1 − , Fig. 3 The residual DNA damage (lesions) represented by double- (7) 2 strand breaks (DSB) in human lymphocytes related to time, after X-ray irradiation of 2 Gy. Four different relationships represents four where N corresponds to the general number of lesions (or different sensitivities to radiation, from hyper-radiosensitivity (upper grey curve) to radioresistance (lower black curve)—this last case rep- mutations) in a single D scenario, while N is the general 2 1+2 resents the strongest adaptive response effect (Fornalski 2019). The figure was created based on the paper by Müller et al. (2001) and presentation of Feinendegen (2012) Please note, that N relationship intentionally omits μ parameter 0,2 0 to avoid double counting of background lesions/mutations in Eq. (7). 1 3 Radiation and Environmental Biophysics (2022) 61:221–239 225 Table 1 Detailed forms Function No. of equation(s) Solution of functions f presented in Eqs. (8)–(11) 0 2 − D − T 2 2 (8), (9) 1 1 2 f D , T D e T + 2 T + 2 1 3 2 1 2 (8), (9), (10) f D ,Δt 0 2 − D − Δt 2 2 1 1 2 D e Δt + 2 Δt + 2 3 2 1 2 (8), (9), (10), (11) f D ,0 0 2 − D 1 1 2 D e (11) 0 2 − D − Δt +Δt 2 f D ,Δt + Δt ( ) 1 1 2 1 2 1 1 2 D e Δt +Δt + 2 Δt + Δt + 2 3 1 2 2 1 2 1 2 0 2 − D − Δt 2 2 (11) 1 2 2 2 f D ,Δt D e Δt + 2 Δt + 2 2 2 3 2 2 2 2 2 0 2 − D − Δt 2 2 1 1 2 1 (11) f D ,Δt D e Δt + 2 Δt + 2 1 1 3 2 1 1 2 1 (11) f D ,0 2 0 2 − D 1 2 2 D e See Appendix 1 for details of calculations number of lesions (or mutations) in a D + D scenario. The than the number of mutations after the single D scenario, 1 2 2 delta parameter, δ, is the percentage difference between N but still lower than the sum of independent D and D ). The 2 1 2 and N (Fig. 1). latter case can be therefore treated as a special case of the 1+2 After some calculations, which are expressed in detail in Yonezawa effect, however, in most studies it is assumed the Appendix 1, Eq. (7) can be written as that δ = 0 (the case when the number of mutations after min D + D scenario is the same as the number of mutations 1 2 f D ,T −f D ,Δt f D ,T −f D ,0 ( ) ( ) ( ) ( ) 1 1 1 1 N 1 − e − N e after single dose D ), see Appendix 1 for more information. 0,2 0,1 2 (8) = , 0,2 Mutations in DNA where f are functions related to the adaptive response, see Appendix 1. Detailed forms of f functions are presented in Here one considers the phenomena of mutations, which are Table 1. permanent changes in DNA caused by mis-repaired lesions: For simplicity, Eq. (5) can be approximated by the simple all repair processes are finished now and all N (T) functions dependence of N ≈ µ D, where background lesions, µ , can 1 0 stabilized. That way one can consider Eq. (9), but for inn fi ite be also neglected (µ << µ ). In that situation, Eq. (8) can be 0 1 time, T ⟶ ∞. This assumption means that the repair time approximated by is long enough to repair all possible lesions and only muta- f D ,T −f D ,Δt 1 f D ,T −f D ,0 tions are left. Thus, the analogous form of Eq. (9) becomes ( ) ( ) ( ) ( ) 1 1 1 1 = 1 − e − e , (9) −f D ,Δt −f D ,0 ( ) ( 1 ) = 1 − e − e , (10) which is independent of parameters {µ} (Eq. 5). The approach presented in Eqs. (8–9) describes the which describes the Yonezawa effect for mutations, namely number of actual damage (lesions), still unrepaired, in the late effects. Appendix 1 contains details of the above calcu- moment of T. This gives no information about mutations, lations. Both f functions are presented in Table 1. which are created much later (>> T). One has to note, that Exemplary results of the parameter δ for the conditions Eq. (9) actually contains three free parameters (α , α , and 0 1 from Fig. 2 are: δ = 0.725 for lesions (Eq. (9)) and δ = 0.728 α ) plus two time variables (Δt and T, where 0 < Δt < T), for mutations (Eq. (10)), respectively. This latter case is pre- which are conditions of the dedicated experiment. In prac- sented in Fig. 1. One can generally see that the difference tice, however, the presence of the Yonezawa effect related to between lesions and mutations analyses is, within the scope fast damage (lesions) is more difficult from a radiobiological of the model, generally marginal for a large T (> Δt). point of view, because Eqs. (8–9) vary with time. Therefore, a more popular and representative situation is described by the second part (Chapter 4) of the proposed model, namely Two priming doses the Yonezawa effect on mutation frequency (late effects). The parameter δ can vary from 1 (ideal protection and The single priming dose, D , is just a special case. Another full reduction of all mutations) to some minimal value δ 1 min possible case can involve two similar priming doses: the first which can even be lower than zero (when the number of one D , the second one (D ) received Δt time after the first mutations after the D + D scenario is a little bit higher 1 2 1 1 2 1 3 226 Radiation and Environmental Biophysics (2022) 61:221–239 Fig. 4 The case from the Fig. 2 with additional priming dose D = D , received Δt = 3 UT 2 1 1 after D . The challenging dose D was received Δt = 2 UT 3 2 after D one, and the challenging dose (D ) received Δt time after Please also note, that Appendix 1 contains analogical cal- 3 2 D . One can note that D ≈ D < D . It follows from Eq. (9) culations for multiple priming dose scenarios. 2 1 2 3 and Fig. 2b that the adaptive response of the second dose (similar to the first one) results in an apparent decrease of the N(T) function as compared to the one resulting from the Practical application and results D dose alone. Thus, one should expect that the challenging dose D given at a later time starts its repair functions from Numerical methods smaller number of lesions than when applied at Δt after the first priming dose. In another words: two consecutive prim - Both approaches, namely Eq. (9) for lesions and Eq. (10) for ing doses, given in proper time and value, can boost the mutations, can be applied to experimental data to estimate repair process triggered by D . the values of α , α and α parameters. However, the differ - 3 0 1 2 This approach needs more laborious calculations, which ences in results obtained by those two equations are usually are presented in Appendix 1 as well. Finally, the parameter marginal for a large T, as discussed previously. δ can be now defined as 1 − N /N , so The application of the model, irrespective of the equa- 1+2+3 3 tion used, needs advanced numerical methods of non-lin- −f D ,Δt +Δt −f D ,Δt −f D ,Δt −f D ,0 ( ) ( ) ( ) ( ) 1 1 2 2 2 1 1 2 ear optimization to estimate all input parameters based on = 1 − e − e experimental output. Here two of them were used: Simpli- (11) −f D ,0 −f D ,0 fied Genetic Algorithm (SGA) as well as Bounded Limited- ( ) ( ) 1 2 − e , memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS-B) one. Both methods and their applications are described in which describes the Yonezawa effect when the challenging Appendix 2 in details. dose D is applied after two priming doses D and D . Addi- 3 1 2 tionally, f functions are presented in Table 1. Lesions Exemplary results of calculations, based on conditions from Fig. 2 with second priming dose, D = D , are pre- 2 1 Equation (9) was applied to real experimental data where sented in Fig. 4. human lymphocytes were irradiated in vitro (X-ray) in three consecutive radiobiological researches conducted by 1 3 Radiation and Environmental Biophysics (2022) 61:221–239 227 Fig. 6 The relationship between the delta (δ) parameter for DNA lesions and the time interval between priming and chal- Fig. 5 The relationship between the delta (δ) parameter for DNA lenging dose (Δt) for human lymphocytes (Shadley et al. 1987), lesions and the priming dose (D ) for human lymphocytes (Shad- where D = 10 mGy, D = 1.5 Gy, and T − Δt = 6 h. The dashed 1 2 ley and Wolff 1987), where D = 1.5 Gy, Δt = 16 h, and T = 22 h. line represents the assumed potential linear trend (the best fit by The Yonezawa effect disappears above approx. D = 200 mGy. The −1 2 δ = − 0.005 [h ] Δt + 0.487 with R ≈ 0.60) according to which the dashed line represents the potential purely empirical trend (best fit Yonezawa effect disappears above Δt ≈ 100 h. The straight line was with R ≈ 0.997) given here by unsymmetrical Gaussian function: selected as the simplest best fit according to the existing scatter of 2 (1/d) f(x) = exp(− (x − a) /b) + c + (x − a) , where a = − 0.036, b = 0.067, data points. Due to potential outliers, this fitting was also tested by c = − 1.12, and d = 3.668 the robust Bayesian regression method (Fornalski et al. 2010) but the −1 result was practically the same (δ = − 0.005 [h ] Δt + 0.483) Shadley and Collaborators (Shadley and Wolff 1987; Shad- ley et al. 1987; Shadley and Dai 1992). These studies were The collected data demonstrated how the response to low- dose radiation depends on whether the cells have been stimu- selected for analysis because of a large amount of data rep- resenting the Yonezawa effect. The first experiment (Shadley lated to divide. The results show that significantly fewer breaks were observed in cells pretreated with 0.01 Gy in G , S or G and Wolff 1987) studied the effect of 3-aminobenzamide 1 2 (3AB) on the repair functions present during the adaptive than in those pretreated in G . This suggests that the adapta- tion of cells to low doses requires mitogenic stimulation of response; additionally, the experiment studied the effect of the conditioning dose level on the priming dose effect. lymphocytes. Concerning the lifespan of the adaptive response, data shows that lymphocytes treated with 1500 mGy at 40, 48, The lymphocytes for testing were taken from the peripheral blood of healthy 30–40 y.o. females. The blood was exposed 66 h exhibited an adaptive response, and those treated later did not. Figure 6 shows the potential relationship between delta (δ) to low doses at 32–34 h after stimulation, the challenging dose was given at 48 h. For one group, 3AB was applied parameter and the time interval between priming and challeng- ing dose (Δt). One can deduce that for approx. Δt > 100 h the immediately after the high dose. The results showed that application of 3AB immediately after the higher dose did Yonezawa effect completely disappears in human lymphocytes. In the third analysed study (Shadley and Dai 1992), low reverse/eliminate the repairs of the adaptive response. Based on the results, low doses of 5, 10, 50, 100 mGy are capa- doses were applied 12 h after culture (PHA application), at 18 h (first G phase after PHA) after stimulation the higher ble of inducing an adaptive response when a higher dose is applied, while doses above 200 mGy do not have the ability dose was applied. In this case, however, DNA mutations (namely, the chromosomal aberrations) were analysed but the to adapt human lymphocytes to ionizing radiation. This is perfectly illustrated in Fig. 5, where the priming dose D experimental investigation was quite similar to the previous two studies (Shadley and Wolff 1987; Shadley et al. 1987) with from the approximated range 10–100 mGy gives the high- est value of δ. a short T. Here, both aberration and deletion frequencies were reduced when a priming dose was applied, although there was The second analysed study (Shadley et al. 1987) consid- ered the effect of the cell cycle phase during which certain a variability between the responses in samples from different donors. The results of this experiment show that G lympho- doses are applied as well as how long the effect of a prim - ing dose can last. Low doses were applied either before 2% cytes are capable of exhibiting a cytogenetic adaptive response. Table 2 shows the exact raw data published in the three phytohemagglutinin (PHA) addition (G ) or at times corre- sponding to G , S or G phase of the first cell cycle. Higher mentioned papers (Shadley and Wolff 1987; Shadley et al. 1 2 1987; Shadley and Dai 1992) to estimate the values of α , α doses are applied either in the same or the next cell cycle 0 1 (40, 48, 66, 90, or 114 h after PHA addition). and α parameters using both numerical methods. The two 1 3 228 Radiation and Environmental Biophysics (2022) 61:221–239 Table 2 The source data used a Source of data/number of table in D (Gy) D (Gy) Δt (h) T (h) N N δ 1 2 2 1+2 for estimation of α , α and α 0 1 2 original paper parameters, selected for DNA b b lesions and chromosomal Shadley and Wolff (1987) Table II 0.01 1.5 16 22 83/200 60/200 0.277 aberrations, based on three b b 0.05 1.5 16 22 83/200 58/200 0.301 studies by Shadley and b b 0.1 1.5 16 22 83/200 66/200 0.205 Collaborators (Shadley and b b Wolff 1987; Shadley et al. 1987; 0.2 1.5 16 22 83/200 83/200 0.000 b b d Sahdley and Dai 1992) for 0.3 1.5 16 22 83/200 99/200 − 0.193 human lymphocytes b b d 0.4 1.5 16 22 83/200 103/200 − 0.241 b b d 0.5 1.5 16 22 83/200 126/200 − 0.518 b b Shadley et al. (1987) Table I 0.01 1.5 44 50 101/300 78/300 0.228 b b 0.01 1.5 40 46 101/300 73/300 0.277 b b 0.01 1.5 36 42 101/300 68/300 0.327 b b 0.01 1.5 34 40 101/300 73/300 0.277 b b Table II 0.01 1.5 34 40 68/200 48/200 0.294 b b 0.01 1.5 32 38 68/200 51/200 0.250 b b 0.01 1.5 30 36 68/200 49/200 0.279 b b 0.01 1.5 28 34 68/200 47/200 0.309 b b 42/200 0.382 0.01 1.5 10 16 68/200 b b Table III 0.01 1.5 14 20 68/200 40/200 0.412 b b 0.01 1.5 20 26 76/200 43/200 0.434 b b 0.01 1.5 38 44 71/200 36/200 0.493 b b Table IV 0.01 1.5 18 24 84/200 55/200 0.345 b b 0.01 1.5 36 42 92/200 55/200 0.402 b b 0.01 1.5 60 66 88/200 83/200 0.057 b b 0.01 1.5 84 90 93/200 88/200 0.054 c c Shadley and Dai (1992) Table I 0.05 2 6 30 95/100 48/100 0.495 c c 0.05 4 6 30 188/100 143/100 0.239 c c 0.05 2 6 30 90/100 74/100 0.178 c c 0.05 4 6 30 240/100 166/100 0.308 c c 0.05 2 6 30 69/100 55/100 0.203 c c 0.05 4 6 30 118/100 65/100 0.449 c c 0.05 2 6 30 106/100 87/100 0.179 c c 0.05 4 6 30 218/100 176/100 0.193 c c 0.05 2 6 30 58/100 51/100 0.121 c c 0.05 4 6 30 192/100 147/100 0.234 Calculated according to Eq. (7) Number of chromatid and isochromatid breaks/number of cells examined Number of chromosome aberrations (dicentrics, rings and deletions)/number of cells examined Negative values do not represent the Yonezawa effect, but they are useful for model calibration first studies (Shadley and Wolff 1987; Shadley et al. 1987) The results obtained by both algorithms are qualitatively can be simply treated as fully consistent and analysed jointly, the same within the range of uncertainties, therefore both because both of them show DNA lesions in the Yonezawa numerical methods work well and they can be treated on scheme. Thus, the results are: equal footing. In the case of the SGA, the uncertainties were estimated by the upper-lower bound method. Simulations +0.5 −2 −3 +5.5 −1 • for the SGA: = 22.9 Gy h , = 79.4 Gy were run for each worst case scenario assuming that numbers 0 1 −4.0 −11.2 +0.0093 −1 of lesions or mutations measured in experiments follow a and = 0.0832 h . −0.0082 −2 −3 Poisson distribution. As explained in the Appendix 2, in • for the L-BFGS-B algorithm: α = 22.9 Gy h , −1 −1 the L-BFGS-B algorithm it was not possible to calculate α = 79.5 Gy and α = 0.0832 h . 1 2 uncertainties of the parameters. 1 3 Radiation and Environmental Biophysics (2022) 61:221–239 229 Table 3 The source data used a Source Tissue D (mGy) D (mGy) Δt (h) T (h) N fre- N δ 1 2 2 1+2 for estimation of α , α and α 0 1 2 quency frequency parameters, selected for DNA –3 –3 (·10 ) (·10 ) mutations, based on studies by Day et al. (2006, 2007) for Day et al. (2006) Prostate 0.001 1000 4 72 5.72 0.93 0.837 chromosomal inversions in 0.01 1000 4 72 5.72 1.70 0.703 mice’s spleen and prostate 1 1000 4 72 5.72 1.88 0.671 10 1000 4 72 5.72 0.98 0.829 Day et al. (2007) Prostate 0.01 1000 4 72 4.66 0.88 0.811 10 1000 4 72 4.66 1.13 0.758 Spleen 0.01 1000 4 72 3.15 0.77 0.756 10 1000 4 72 3.15 0.98 0.689 Calculated according to Eq. (7) In the next step, one can observe that the data and results and ultra-low priming doses caused an adaptive response of all three cited papers (Shadley and Wolff 1987; Shadley when a 1 Gy challenging dose was administered (Day et al. et al. 1987; Shadley and Dai 1992) are fully comparable, 2006), which is presented in Table 3. regardless of chromosomal aberrations analysis in the last Both sets of data, for spleen and prostate postra- one. Therefore, they are able to be treated as a meta-data diation mutation frequency in mice, show a strong for joint analysis, which is carried out in the “Discussion”. Yonezawa effect (δ > 0.6). The calculated param- −2 −3 −1 eters are: α = 11160 Gy h , α = 1400.9 Gy and 0 1 −1 Mutations α = 0.0116 h for spleen mutations (Day et al. 2007), and −2 −3 −1 −1 α = 15.92 Gy h , α = 1148.7 Gy and α = 0.00026 h 0 1 2 The analogical numerical research for mutations only can be for prostate mutations calculated jointly for both studies investigated using Eq. (10). For this case, the comprehensive (Day et al. 2006, 2007) (Table 3). experimental data collected by Day et al. (2006, 2007) were As mentioned earlier, the human lymphocytes chro- used (see Table 3) because they offer a large combination of mosomal aberration data from Table 2 (Shadley and Dai situations to be used in numerical analysis. In the study by 1992) are also related to mutations analysis. These data −2 −3 −1 Day et al. (2007), Atm knockout heterozygous pKZ1 mice results in: α = 89.23 Gy h , α = 174.86 Gy and 0 1 –6 −1 were treated by a whole-body X-irradiation with a priming α = 1.41·10 h , which differ from the results for lesions. dose (D ) and, after a 4 h interval, a challenging dose (D ). 1 2 3 days after exposure, the mice were sacrificed, their spleen and prostate removed and snap-frozen until needed for scor- Discussion ing of chromosome inversions. Both the 0.01 and 10 mGy priming doses caused a similar in magnitude adaptive The role of the time interval between two consecutive doses response to the challenging dose of 1 Gy, although a single of ionizing radiation has been pretty well known for years dose of 0.01 mGy seemed to induce a chromosomal inver- and is the basis of e.g. dose fractionation. But the first exper - sion frequency by itself. Based on the results by Day et al. imental finding that this time interval is related to repair (2007), being a Atm knockout heterozygote (Hishiya et al. processes was conducted in 1940s by Prof. John R. Raper 2005) does not affect the response to single low radiation during his works in Manhattan Project (Raper 1947). At doses (used here) or the induction of an adaptive response that time he irradiated groups of mice using beta radiation for inversions. and observed a “recovery from the radiation damage” when The second analysed study (Day et al. 2006), used the the conditioning sublethal dose was applied prior to the test very low priming doses of 0.001, 0.01, 1 or 10 mGy, which lethal dose. Raper’s experiments were repeated a few years were administered 4 h before a challenge dose of 1 Gy. Mice later and showed the same effect but no detailed explanation were sacrificed 3 days after treatment to detect pKZ1 chro- was proposed (Cronkite et al. 1950). Next, in the late 1950s, mosomal inversion assay in the prostate. The results show the historical Elkind-Sutton experiments were conducted that the 1–10 mGy priming doses reduced chromosomal (Elkind and Sutton 1959). Elkind and Sutton irradiated aberration (inversion) frequency to below the spontaneous mammalian cells with two doses, which they called condi- frequency level. The 0.001 mGy single priming dose had no tioning and test doses, analogically to Raper’s terminology. significant effect, while the 0.01 mGy single priming dose They found that the time delay between doses can change increased the inversion frequency. Despite this, all four low 1 3 230 Radiation and Environmental Biophysics (2022) 61:221–239 the shape of the survival curve of irradiated cells but both papers, which presented the effect (Fan et al. 1990; Liu et al. doses were large and no low conditioning dose was tested. 1992; Farooqi and Kesavan 1993; Cai et al. 1994), contain Important changes appeared in 1980s and 1990s, when too few results to be valuable for the model’s validation. many studies began reporting that the radiation adaptive In practice, the model is reduced to a relatively simple response effect showed strong results with a low priming equation, which connects the delta (δ) parameter (Fig. 1) dose and high challenging dose scheme (Olivieri et al. 1984; with priming and challenging doses, and the radiation adap- Shadley and Wolff 1987; Shadley et al. 1987; Shadley and tive response probability function. This equation, however, Dai 1992; Yonezawa et al. 1990; Fan et al. 1990; Liu et al. is presented in two versions: for DNA lesions, namely fast 1992; Farooqi and Kesavan 1993; Cai et al. 1994). In the damage (Eq. 9), and mutations, which are rather later effects 1990s, this special case of adaptive response was called the (Eq. 10). The quantitative results obtained by Eqs. (9) and priming dose effect or, a few years later, the Yonezawa effect (10) are very similar, therefore one can use one equation (Wang et al. 2013, 2018, 2021; Liu et al. 2019). Today, we (Eq. 9) for both types of data to improve and unify the mod- can also call it the Raper-Yonezawa effect to account for el’s parameters, as well as to reduce some of uncertainties. some historical aspects surrounding it. Anyway, from the We are aware that the number of lesions a long time after experimental point of view, the easiest way to test the adap- application of the challenging dose may only approximate tive response appearance is the priming dose scheme: the the number of mutations. Therefore our model apparently first small dose, called the priming or conditioning dose, simplifies the real situation in which the kinetics of late and some time later the higher one, called challenging dose. repairs should be better described. That situation results in smaller detrimental effects than We have to note that it is known (Berthel et al. 2019; when the same big dose is given as a single dose – of course Feinendegen et al. 2007; Mezentsev and Amudson 2011; only in situations when the radiation adaptive response was Long et al. 2007; Ding et al. 2005) that low and high doses activated. trigger separate groups of genes. As stated by Mezentsev In the XXI century there have been several biomathemati- and Amudson (2011), “Accumulating data suggest that the cal models which have tried to explain, among others, the biological responses to high and low doses of radiation are priming dose effect (Schöllnberger et al. 2001; Smirnova qualitatively different”, that may be linked to the activation and Yonezawa 2003, 2004; Esposito et al. 2010; Zhao and of “radiation-responsive genes after high- and low-dose Ricci 2010; Wodarz et al. 2014; Devic et al. 2018, 2020; exposures”. Therefore, our model in which the adaptive Bondarenko et al. 2021), however, they are usually based response at both doses is described by the same function on a set of differential equations, and in most cases they are of dose and time may not be adequate for a full explana- not deeply related to the adaptive response phenomenon. tion of the originally observed Yonezawa effect. This puts The presented paper intends to explain the biophysical ori- a natural limit on the value of the challenging dose. On the gin of the Yonezawa effect within the scope of a relatively other hand, the alpha parameters including their uncertain- simple model used so far in many of our papers (Fornalski ties, which are connected with the biology of the studied et al. 2017; Dobrzyński et al. 2016, 2019). The considera- object, endpoints, and proper timing of the whole process tions concern the effect of the use of a high challenging dose as well, may be so individual-dependent that the whimsical after a rather low priming dose given earlier. In the original behaviour of Yonezawa effect can be expected. papers by Yonezawa and coworkers (Yonezawa et al. 1990, The dose relationship presented by Eq. (9) has to be com- 1996; Matsubara and Yonezawa 2004; Matsumoto et al. mented on here as well. Certainly, the right-hand side of 2004; Smirnova and Yonezawa 2003, 2004), originally for Eq. (9) is not appropriate for very high doses D because mice, the challenging dose was so high that it caused the sys- it tends to zero for that case. This problem is much wider: tematic death of mice with time. However, when the priming firstly, a very high value of D would be lethal, thus no Yon- dose was added, this process was substantially hindered. ezawa effect may be observed because of the organism's The proposed new model describing the Yonezawa effect death. Secondly, as mentioned above, a very high value of contains only three free parameters. This model, however, is D activates different groups of genes responsible for DNA based on the radiation adaptive response probability function repair, which makes the radioadaptation more complicated which makes it more grounded in biophysics. The model was and this is not reflected in our relatively simple model. validated on a group of experimental data from papers by Therefore, the presented approach has a very important limi- Shadley et al. (Shadley and Wolff 1987; Shadley et al. 1987; tation: the model works well when the challenging dose, D , Shadley and Dai 1992) and Day et al. (2006; 2007). Those is not loo large. studies represent the most broad and comprehensive data in The alpha parameters calculated for all the data in Table 2 the Yonezawa scheme, which is necessary for proper param- are as follows: eter calculations to validate the model. Other exemplary 1 3 Radiation and Environmental Biophysics (2022) 61:221–239 231 proteins p21, CHK kinases, ataxia telangiectasia-mutated ATM or ataxia-telangiectasia and RAD3-related ATR) and their inner cell signalling (Pawlik and Keyomarsi 2004). After a cell enters cycle arrest two major DNA double- strand break repair pathways can occur: error prone NHEJ (Non-Homologous End Joining, dominant in G /S phase) and HR (Homologous Recombination, present in late S or G phase). The greater proportion of error-free repair in late S phase may explain its radioresistance (lower radiosensitiv- ity). If the DNA damage is fully repaired, the cell cycle con- tinues. The choice of which pathway becomes activated is determined by the conflict between maintenance and resec- tion of the DNA ends (Maier et al. 2016). The differences in radioadaptation are, however, relatively Fig. 7 The non-normalized probability functions of radiation adaptive small but nonetheless, the adaptive response in phase G –G response in human lymphocytes (Shadley et al. 1987) in phase G –G 0 1 0 1 (blue solid line), in phase G –S (orange dashed line) and after phase and after S (G ) is smaller than in G -S, which is clearly pre- 2 1 S (green dotted line), related to time (hours). The orange dashed line sented in Fig. 7. The mechanisms explaining such cell cycle corresponds to the lowest radiosensitivity of the cell and therefore the effects on the adaptive response to ionizing radiation are not strongest radioadaptation (color figure online) fully understood, but it has been suggested that the adap- tive response phenomenon may be due to cell cycle changes −2 −3 +8.3 +2.6 (Cramers et al. 2005) or arrest (Syljuåsen 2019) caused by the = 36.2 Gy h , = 120.2 0 1 −8.0 −1.5 low priming dose of radiation. Since sensitivity to radiation −1 +0.0085 −1 Gy and = 0.0845 h . −0.0060 varies with cell cycle stage, changes in cell cycle distribution may be responsible for the radiation adaptive responses (Hafer These results are related to human lymphocytes (Shad- et al. 2007). Recently, many studies have been focussing on ley and Wolff 1987; Shadley et al. 1987; Shadley and Dai pointing out which of the repair pathway components are tak- 1992) and their potential radioadaptation, which reaches its ing majority in radioadaptation induction (Hafer et al. 2007; maximum for the priming dose of D = 2/α ≈ 25.2 mGy and 1 1 Hendrikse et al. 2000; Boothman et al. 1998). Δt = 2/α ≈ 24 h after its exposure. It is worth mentioning The delta (δ) parameter, when the input data are too weak, here that in the original Yonezawa and Smirnova mice model can hardly be determined because less than 3 equations for 3 (Smirnova and Yonezawa 2003, 2004), their theoretical con- variables {α , α , α } would give an inn fi ite number of solu - 0 1 2 sideration of the biological mechanisms responsible for the tions. This is the reason why advanced numerical methods modifications of radiosensitivity is due to the change of radi- were necessary to calculate the model’s parameters. The delta osensitivity of the hematopoietic system. Consistently, the (δ) parameter, however, is very sensitive to even the smallest blood-forming system's cells were the objects of the studies changes of alphas, which display the Yonezawa effect in some (Smirnova and Yonezawa 2004). ranges only. This is clearly illustrated in Fig. 8, where the pre- The next finding, which can be observed in the detailed sented surface can easily reach maximal or minimal values of δ. data from Shadley et al. studies (Shadley et al. 1987), is that Indeed, the uncertainties of the alpha parameters inferred the level of radioadaptation is connected with the cell cycle from the data shown in Table 2 indicate the high selectivity phase and its radiosensitivity. The analysis of the data shows of the model used in the description of the Yonezawa effect. that the strongest radiation adaptive response generated by However, the calculated alpha parameters can be useful for the priming dose is connected with the lowest radiosensitiv- the prediction of delta factor for other dose schemes, which ity of the cell, which seems to be quite natural (Fig. 7). It has can be planned in future experiments. Moreover, this method long been known that cells in different cell cycle stages dis- can be useful for the radiation adaptive response probabil- play die ff rent sensitivity to radiation. Cells in the late S phase ity function assessment (like Fig. 7) for e.g. cell behaviour are usually most radioresistant and cells in the M phase are modelling during irradiation. most radiosensitive. A cellular response to DNA-damaging One has to note that the presented results were calculated for agents correlates not only with DNA replication and chro- experimental cases where the Yonezawa effect was pretty well mosome segregation stage. It also depends on the activation observed. But this is not always the case: this effect is observed of the repair pathway maintaining the genetic integrity and in approx. 50% of all expected cases (Tapio and Jacob 2007), activation of the cell cycle checkpoint. This complex mecha- which means that the radiation adaptive response appearance is nism is driven by the variety of key proteins concentrated in quite a selective process: the probability that this phenomenon the cell (such as tumour suppressor proteins p53, inhibitor 1 3 232 Radiation and Environmental Biophysics (2022) 61:221–239 −1 Fig. 8 a The shape of delta (δ) parameter function from 0.23 h , D = 10 mGy, D = 1.5 Gy, Δt = 34 h; plot b represents the 1 2 Eq. (9) for the exemplary sets of parameters and their ranges: same situation but two parameters’ ranges were narrowed: α from 50 −2 −3 −1 −1 −1 α = 36.21 Gy h , α from 20 to 1200 Gy , α from 0.02 to to 550 Gy , and α from 0.08 to 0.1 h 0 1 2 2 −1 +0.0093 appears equals approx. 0.5, but when it does appear, it can be = 0.0832 h are related to the human lymphocytes −0.0082 described by the p distribution with proper {α} parameters. AR DNA lesions reduction in Yonezawa scheme. The relatively The last item to discuss is terminology: it is often found narrow range of these parameters indicates that their values in scientific literature that the priming dose effect (which may be strongly dependent on the individual case. Indeed, can be called the Yonezawa—or Raper-Yonezawa—effect) it is a set of parameters for one type of cells and their indi- is practically equivalent to the radiation adaptive response vidual radiosensitivity. Therefore, the presented analysis phenomenon. However, this is not the case: the priming shows that the level of radiation adaptive response is strictly dose (Yonezawa) effect is just a special case of the adaptive connected with radiosensitivity (Fig. 7)—low radiosensitiv- response with a specific dose fractionation scheme. Another ity (so high radioresistance) is a result of a strong adaptive examples of the adaptive response is constant low-dose rate response of the cell, which is consistent with many recent irradiation (Dobrzyński et al. 2016) or so called “radiation findings presented in this paper. Finally, the proposed mech- training” by many small dose pulses (Socol et al. 2020), see anistic model was quantified based on experimental data— Appendix 1 (“Mutations in DNA”). Nevertheless, year by e.g. lesions in human lymphocytes and chromosomal inver- year we learn more about radioadaptation (UNSCEAR 2000; sions in mice—to be able to predict the Raper–Yonezawa Tapio and Jacob 2007; Guéguen et al. 2019) therefore it is effect for future experimental and theoretical investigations. high time for its wide biophysical and mathematical descrip- tion, at least of the most popular priming dose scenario. Appendix 1: Detailed calculations Conclusions DNA lesions The presented paper uses the radiation adaptive response theory to create a single equation (Eqs. (9) or (10)) for the Let us consider the relationship between repaired damage Raper-Yonezawa (priming dose) effect (Fig. 1) when D is (lesions) and the adaptive response, dN = − N p dt, see AR not too high, and the adaptive response probability func- Eqs. (6) and (2). The simplest solution of that equation, as tion after D can be described by same formula as the one an indefinite integral, is represented by after the priming dose D . The delivered equations were confronted with a group of experimental data on humans dN 2 2 − D− t 1 2 =− p dt =− D t e dt, (12) and mice to calculate the exemplary input parameters. For AR 0 +0.5 −2 −3 +5.5 −1 example = 22.9 Gy h , = 79.4 Gy and 0 1 −4.0 −11.2 1 3 Radiation and Environmental Biophysics (2022) 61:221–239 233 so one can calculate it as However, when another dose is received in the moment Δt, it is necessary to use Eq. (16) with time T shifted by Δt: 0 2 − D− t 1 2 − p dt = D e t + 2 t + 2 ≡ f D, t , ( ) AR 2 2 T−Δt − p dt = f (D, T −Δt) − f (D,0) AR (13) 0 (20) 1 2 and assume such a solution as a function f(D,t). Thus, the − (T−Δt) = e (T −Δt) + (T −Δt) + 1 − 1 . D 2 2 definite integral has a form: In the scenario of the typical Yonezawa effect, one can consider the three moments in time mentioned above: (1) − p dt = f (D, b) − f (D, a). (14) AR the first one, let us call it a moment zero, t = 0, when the first small priming dose (D ) was received, (2) the second one after a short period of time, Δt > 0, when the challeng- For example, after some calculations and integration for ing (much higher, D > D ) dose pulse was received, and (3) N from N to N(T) and for t from 0 to T, one can get 2 1 the third one, T, which corresponds to the actual moment, 0 < Δt < T, when the number of lesions is measured. − ∫ p dt AR (15) f (D,T)−f (D,0) N(T) = N e = N e . 0 0 Next, one can note that in the moment of Δt, the number of damage (lesions) from dose D which remain unrepaired This can be written in exact form as: equals to (see Eq. (15)): 2 f D ,Δt −f D ,0 0 ( ) ( ) 2 − D − T 1 1 1 2 N = N e , N(T) = N exp D e e T + 2 T + 2 − 2 (21) 0,1 0,1 0 2 2 (16) where N denotes the initial number of damage (lesions) 0,1 induced by dose D in moment zero (assumed that where N means the initial number of lesions (just after sin- N = μ + μ D , see Eq. (5)). gle D appearance, as described by Eq. (5)), N(T) is the actual 0,1 0 1 1 In time Δt, dose D generates an additional number of remaining lesions, and T corresponds to the actual time. It damage, N = μ D , where N > N (because D > D ). should be noted, that Eq. (16) has an inverted Gompertzian- 0,2 1 2 0,2 0,1 2 1 Therefore, in the moment of Δt, one needs to consider the like shape in the function of time (Fornalski et al. 2020). Let total N + N’ number of damage to repair. However, in us assume for further analyses, that the term 0,2 0,1 the same moment, additional repair mechanisms from the 2 − D 1 second dose appear, which by assumption are additive to the = 2 D e , (17) first ones (because repair mechanisms from the first dose are still working). That way, in the later time, T, the number of is constant for a dedicated dose pulse, D, which is also a unrepaired damage (lesions) equals: constant value in exact conditions. One can also note, that f(D,0) = ξ . Additionally, for further considerations, let us f D ,T −f D ,Δt +f D ,T−Δt −f D ,0 ( ) ( ) ( ) ( ) 1 1 2 2 N(T) = N + N e , 0,2 0,1 calculate the probability connected with the appearance of (22a) the adaptive response from dose D after some time Δt: f D ,Δt −f D ,0 f D ,T −f D ,Δt +f D ,T−Δt −f D ,0 ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 2 2 N T = N + N e e , ( ) Δt 0,2 0,1 (22b) − p dt = f (D, Δt) − f (D, 0) AR (18) which can be denoted as N for further considerations 1+2 2 (see Eq. (7)). Let us note that as D increases, the value of − Δt 2 = e Δt + Δt + 1 − 1 , D 2 2 2 also increases with D , which causes a rapid decrease D2 of N(T) (see analogical Fig. 3). To understand these rela- which is an analogical solution to Eq. (16). tionships, Fig. 2 presents the time related probability of After the next period of time, from Δt to T, the adaptive the adaptive response (p ) and the number of unrepaired AR response from the same dose D is calculated as: 1 2 1 2 − T − Δt 2 2 (19) − p dt = f (D, T) − f (D, Δt) = e T + T + 1 − e Δt + Δt + 1 . AR D 2 2 2 2 2 2 Δt 1 3 234 Radiation and Environmental Biophysics (2022) 61:221–239 damage (lesions) in time N(T) for a single dose D and the means that the repair time is long enough to repair all pos- combination of priming and challenging doses, D + D . sible lesions and only mutations are left. Of course Eq. (18) 1 2 Finally, at N > 0, the main quantification of the Yon- will remain the same, but Eq. (19) will change into 0,2 ezawa effect, the parameter δ, in the moment of T equals: f D ,Δt −f D ,0 f D ,T −f D ,Δt − p dt = f(D, ∞) − f(D, Δt) ( ) ( ) ( ) ( ) 1 1 1 1 AR N + N e e 0,2 0,1 1+2 = 1 − = 1 − , (28) Δt N N 2 0,2 1 2 − Δt (23) =− e Δt + Δt + 1 , D 2 2 where N = N exp[f(D ,T − Δt) − f(D ,0)]. One can note, 2 0,2 2 2 where f(D,∞) = 0. Analogically, Eq. (20) will change into that the denominator in the left-hand-side term of Eq. (23) represents the situation where the single dose D (without a ∞ priming dose) is given in the time t + Δt, as shown in Fig. 2. − p dt = f(D, ∞) − f(D,0) =− . (29) AR D After some calculations Eq. (23) can be rewritten as: f D ,T −f D ,Δt f D ,T − ( ) ( ) ( ) 1 1 1 D1 N 1 − e − N e 0,2 0,1 Thus, the analogous form of Eq. (25) becomes (24) = , 0,2 D −f D ,Δt 1 − ( ) 1 D1 = 1 − e − e , (30) Assuming that N ≈ µ D, because background lesions (µ ) 1 0 can be neglected (µ << µ ), one can rewrite Eq. (24) as 0 1 which describes the Yonezawa effect for mutations, see equivalent Eq. (10). f(D ,T)−f(D ,Δt) f(D ,T)− 1 1 1 D1 = 1 − e − e , (25) The limitation for δ , as the minimal possible value of min the delta parameter, is the same as in the previous chapter, see Eq. (27), because both Δt ⟶ ∞ and T ⟶ ∞. How- which is independent of parameters {µ} (Eq. 5). It should ever, to keep δ ≥ 0 for mutations, one needs to fulfill the be noted that Eqs. (24) and (25) are the same as Eqs. (8) and condition of (9), respectively. The parameter δ can vary from 1 to some minimal value D1 δ which can even be lower than zero (but in most stud- f D ,Δt ≥ − ln 1 − e , min (31) ies it is assumed that δ = 0). Generally, to keep the delta min parameter above δ ≥ 0, one needs to fulfill the condition of which is similar to Eq. (26). −f D ,T − ( ) 1 D1 f D , Δt ≥ − ln e − e , (26) Two priming doses which guarantees that the adaptive response signal from The two priming doses case is presented in Fig. 4. After the dose D will be still working when challenging D appears. 1 2 first time interval (i.e. just before the moment of Δt so just The Yonezawa effect completely disappears when the 1, before D appears) one can use reasoning analogous to the time distance (Δt) between D and D is too large for the 1 2 Δt previous case: Eq. (18) can be rewritten as − ∫ p dt , activation of repair mechanisms of the priming dose. Param- AR which can be used analogically to the Eq. (22). After the eter δ attains its minimal (negative) value of second time interval (i.e. just before the moment of −D −𝜉 Δt + Δt. so just before D appears), Eq. (19) can be sub- D1 𝛿 = lim 𝛿 = e < 0, 1 2, 3 (27) min Δt +Δt Δt Δt→∞ 1 2 2 stituted by − ∫ p dt , and Eq. (20) as − ∫ p dt ; AR AR Δt 0 this can be used analogically to Eq. (22). However, after the which is a clear information that in such a case the Yon- appearance of D (we are considering mutations, therefore ezawa effect disappears. T = ∞), one shall calculate the adaptive response from the ∞ ∞ first dose ( − ∫ p dt ), second dose (− ∫ p dt ) and AR AR Δt +Δt Δt 1 2 2 Mutations the third one (− ∫ p dt , which equals − , see Eq. (29)). AR D The process of mutations repair can be written as: Mutations are stable and unrepaired (or not properly 1+2+3 repaired) lesions. Therefore, one can consider Eqs. (18), (19) f D ,∞ −f D ,Δt +Δt f D ,∞ −f D ,Δt f D ,∞ −f D ,0 ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 2 2 2 3 3 = N + N e e e , 0,3 1+2 and (20), but for infinite time, T ⟶ ∞. This assumption (32a) 1 3 Radiation and Environmental Biophysics (2022) 61:221–239 235 To conclude, the radiation adaptive response effect f(D ,Δt )− 1 1 D1 N = N + N + N e 1+2+3 0,3 0,2 0,1 (namely, radioadaptation) is a wide phenomenon. Special f D ,Δt +Δt −f D ,Δt +f D ,Δt − ( ) ( ) ( ) (32b) case of radioadaptation is the priming dose effect (called 1 1 2 1 1 2 2 D2 the Raper-Yonezawa effect). Another example of the adap- −f D ,Δt +Δt −f D ,Δt − ( ) ( ) 1 1 2 2 2 D3 e . tive response is e.g. constant low-dose rate irradiation (see Eq. (35)), like in high background radiation areas The parameter δ can be now defined as 1 – N /N , so: 1+2+3 3 (Dobrzyński et al. 2015a, 2015b). D D −f D ,Δt +Δt −f D ,Δt 2 −f D ,Δt − 1 − − ( ) ( ) ( ) 1 1 2 2 2 1 1 D2 D1 D2 = 1 − e − e − e , D D 3 3 (33) Appendix 2: Numerical methods which describes the Yonezawa effect when the challenging dose D is applied after two priming doses D and D . Equa- 3 1 2 tion (33) is equivalent to the Eq. (11). Simplified Genetic Algorithm (SGA) Multiple priming doses The proposed methodology starts with the well-known least squares method, where Eqs. (9) (or (10)) has defined the To present more general situation, let us consider n identical function to be fitted. This function is complicated, thus an priming doses D separated by the same time shift Δt. The iterative method of finding the minimum of sum of squared situation where several single dose pulses are applied to an residuals had to be used. Thus, the most suitable method organism is called a “radiation training” (Socol et al. 2020). of parameters {α} assessment is the genetic algorithm. The challenging dose D* (where D* > D) is applied Δt after However, the use of the classical genetic algorithm (Ban- the last of n doses D were applied. zhaf et al. 1998; Whitley 1994) is not suitable for finding a Using the same reasoning as above, one can present the global minimum, for several reasons. One of them is limited general equation for the parameter δ as: ∗ f (D,(n−2)Δt)− f (D,(n−1)Δt)− f (D,nΔt)− −f (D,Δt)−f (D,2Δt)−⋯−f (D,nΔt) D D D D + D 1 + 1 + (1 +⋯)e e e e (34) = 1 − One can consider the situation where the number of dose possibility of coding a broad range of real numbers (such pulses is infinite (n ⟶ ∞) and the time distance between as parameters {α}) into artificial “genes” with keeping suf- them tends to zero (Δt ⟶ 0). This situation is equivalent ficient resolution. Another is that two steps of the classical to chronic irradiation by a constant dose rate (Ḋ = const), genetic algorithm (namely crossover and mutation) have a which can be easily found in areas with high natural back- tendency to change these parameters rapidly, resulting in an ground radiation (Dobrzyński et al. 2015a, 2015b), for escape from the best solution neighbourhood. Significant example. However, Eq. (34) would give an infinite level of changes to the algorithm were introduced including removal adaptive protection ( lim = 1 ), which is not possible. of the crossover step and applying small random changes Δt→0;n→∞ directly to the parameters instead of randomly mutating To avoid that problem, one should note that in this particular artificial “genes”. A much simpler yet more effective algo- situation the probability function of the radiation adaptive rithm mixing the features of genetic algorithms with particle response (Eq. (3)) saturates at some constant value: 2,3 4 swarm optimization and simulated annealing was then 2𝛼 0 ̇ obtained (Fig. 9). 2 −𝛼 D ̇ ̇ 1 P = lim p D = D e = 𝜉 , c AR D (35) T→∞ Simulation starts with creating a set (“a population”) of 99 individuals. Each individual represents all three-param- which was originally calculated in (Dobrzyński et al. eter values: α , α and α , chosen randomly as real posi- 0 1 2 2016), where Ḋ = const. This result will modify Eq. (16) tive numbers at the beginning of simulation. These values to N(T) = N exp(− P ) = const which allows one to make are used to calculate the fitness function of each individual, 0 C analogical calculations as in previous subchapters. This is, however, not the subject of this paper, because chronic http:// www. schol arped ia. org/ artic le/ Parti cle_ swarm_ optim izati on. irradiation with constant dose-rate is not considered in the https:// www. scien cedir ect. com/ topics/ engin eering/ parti cle- swarm- Yonezawa effect. Still, one can pose a question whether the optim izati on. natural radiation can function as a factor reducing possible 4 https:// www. scien cedir ect. com/ topics/ engin eering/ simul ated- annea effects of eventual higher doses (Mortazavi et al. 2012). ling- algor ithm. 1 3 236 Radiation and Environmental Biophysics (2022) 61:221–239 Bounded Limited‑memory Broyden– Fletcher–Goldfarb–Shanno (L‑BFGS‑B) algorithm The second approach also uses χ function to find the best estimation of alpha parameters. In this case, however, the optimization procedure uses a quasi-Newtonian algorithm called BFGS (Broyden–Fletcher–Goldfarb–Shanno) using a limited amount of computer memory with additional bound constraints of variables (Byrd et al. 1995; Zhu et al. 1997). L-BFGS-B method uses an estimate of the inverse Hessian matrix to drive its search in the variable space. The advan- tages of this algorithm are fast convergence and relatively low computational complexity. However, one should also be aware of its shortcomings: the L-BFGS-B algorithm may turn out to be divergent when the starting point is far from the solution sought and it does not guarantee finding the global minimum—it may happen that the parameters found only correspond to the local minimum. To minimize the risk that Fig. 9 The flow chart of the Simplified Genetic Algorithm (SGA) the found parameters would correspond to a local minimum used to evaluate parameters α , α and α 0 1 2 instead of a global minimum, the algorithm was started from multiple points in the parameter space within a reasonable which in our case is reciprocal of the sum of squared residu- range. The L-BFGS-B algorithm should not be used for very als (each residual is the difference between δ measured in the flat functions. The limitations of numerical precision make it experiment and δ calculated with given values of α , α and impossible to compute the gradient of such a function. This 0 1 α ). Next, a new set of individuals is selected from the old creates the last disadvantage: the calculated uncertainties one with the probability of being chosen proportional to the of the fitted function’s parameters are inconsistent, thus the fitness function of a given individual. This way individuals uncertainties were calculated by SGA method only. with the best fitness can be duplicated and these with the Acknowledgements Authors wish to thank Dr. Sylwester Sommer worst can be eliminated. Next, small random changes are from the Institute of Nuclear Chemistry and Technology (IChTJ, War- applied to parameters in the new population, which becomes saw, Poland) for the consultation in radiobiology. the current one. These steps make up one loop (“a genera- tion”) of the algorithm, which is repeated until an end con- Declarations dition is fulfilled. The condition can be defined as being unable to find better parameters as the simulation progresses Conflict of interest The authors declare that they have no conflict of or simply reaching the maximum generation number, which interest. in our case was 6,000,000. Open Access This article is licensed under a Creative Commons Attri- Since there is a risk of not finding a global solution in a bution 4.0 International License, which permits use, sharing, adapta- single simulation, simulations conducted by the described tion, distribution and reproduction in any medium or format, as long algorithm were run at least 50 times and manually checked as you give appropriate credit to the original author(s) and the source, for obvious mistakes (i.e. when the algorithm got stuck in a provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are local solution or wasn't able to even get on the right path to included in the article's Creative Commons licence, unless indicated global solution). These mistakes can be spotted by compar- otherwise in a credit line to the material. If material is not included in ing the fitness functions of many simulations. In our case, it the article's Creative Commons licence and your intended use is not was safe to assume that if they differ more than a 1% from permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a the best fitness function obtained in any simulation, the copy of this licence, visit http://cr eativ ecommons. or g/licen ses/ b y/4.0/ . result should be ignored. Luckily, there were not many mis- takes (less than 10%). 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Radiation and Environmental Biophysics – Springer Journals
Published: May 1, 2022
Keywords: Adaptive response; Radiation; Yonezawa effect; Priming dose; Challenging dose; Radioadaptation; Radiosensitivity; Radiation biophysics; Cancer physics; Lymphocytes
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