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A multiscale radiation biophysical stochastic model describing the cell survival response at ultra-high dose rate

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A multiscale stochastic model predicts the FLASH cell-sparing effect from peroxyl radical recombination.

desk verdict Useful multi-scale FLASH model, but tuned chemical constants and a likely double-count of oxygen effects mean the validation claims outrun the evidence. read the letter →

arxiv 2412.16322 v1 pith:EADN7CZJ submitted 2024-12-20 physics.bio-ph physics.med-ph

Marco Battestini , Marta Missiaggia , Sara Bolzoni , Francesco G. Cordoni , Emanuele Scifoni This is my paper · ORCID
classification physics.bio-phphysics.med-ph
keywords FLASHradiotherapyultra-highdoserateperoxylradicalrecombinationstochasticmicrodosimetricmodelcellsurvivaloxygenenhancementratioDNAdamagemultiscalemodeling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the cell-sparing effect seen in ultra-high-dose-rate (FLASH) radiotherapy has a mechanistic cause that can be computed: at high dose rates, peroxyl radicals recombine before they can damage DNA, so fewer lethal lesions form. To make this precise, the authors build the MultiScale Generalized Stochastic Microdosimetric Model (MS-GSM2), which couples the random arrival of radiation energy to a five-equation chemical network for oxygen, radicals, and scavengers, and then to a Markov model of DNA damage formation and repair. They show that, with biological rates fitted only to conventional-irradiation normoxic survival data, the model predicts the measured survival of three cell lines irradiated with electrons, helium ions, and carbon ions at dose rates from 0.12 Gy/s to 600 Gy/s and oxygen levels from 0.5% to 21%. If the model is right, the FLASH effect is not a mystery requiring oxygen depletion or any single exotic mechanism; it is a quantitative consequence of ordinary radical chemistry, and the differing redox environments of tumors and healthy tissue explain where the sparing appears in vivo.

What carries the argument

The carrying object is a chemical reaction network of nine reactions written as five ordinary differential equations (Eq. 2), one per chemical species per spatial domain, describing O2, H2O2, OH•, carbon-centered radicals R•, and peroxyl radicals ROO•, with faster water-radiolysis chemistry folded into instantaneous G-value source terms (the number of species produced per 100 eV of absorbed energy). Radiation enters as random jumps in these equations, and the time-integrated concentration of ROO• modulates the Poisson yield of indirect DNA damage, so faster dose delivery leaves the radicals less time and produces fewer lesions. This chemical layer is coupled to the Generalized Stochastic Microdosimetric Model (GSM2), a Markov model in which sub-lethal lesions can be repaired, converted to lethal lesions, or interact pairwise; cell survival is the probability that no lethal lesion remains.

What would settle it

Measure the yield of peroxyl-radical-mediated indirect DNA damage (for example via plasmid DNA damage assays) after a single few-µs pulse and after conventional irradiation at matched dose and oxygen level; if the ratio of indirect damage does not fall with dose rate as the five-ODE network predicts, or if the sparing effect persists when peroxyl radical recombination is chemically suppressed, the central claim is wrong.

Watch

Extended reading notes

Core claim

The central claim is that the FLASH sparing effect emerges from a reduction in indirect DNA damage caused by organic peroxyl radicals (ROO•) when irradiation is delivered faster than the radicals can react. The authors state that the model "grasps all the physics behind the potential mechanism underlying the FLASH effect" and supports peroxyl radical recombination as the main determinant, with oxygen and antioxidant concentrations setting the magnitude. The evidence is predictive: after calibrating the three biological repair rates on conventional, normoxic cell survival, the MS-GSM2 reproduces the experimental survival fractions for DU145 cells irradiated with 10 MeV electrons, A549 cells with 4.5 keV/µm helium ions, and CHO-K1 cells with 13 keV/µm carbon ions, at oxygenations from 0.5% to 21%, including the rise in survival at UHDR. The model also generates an in-silico map of the differential effect as a function of oxygen and antioxidant levels, used to argue that the in-vivo normal-tissue versus tumor difference reflects different redox environments rather than a cell-intrinsic difference.

Load-bearing premise

The load-bearing premise is that the simplified five-equation chemical network, with reaction constants adjusted within published ranges, faithfully captures how peroxyl radicals and oxygen behave after irradiation, so that the predicted drop in indirect DNA damage at high dose rate is real.

Editorial extensions

If this is right

  • The sparing effect should appear in any in-vitro system at sufficiently high dose rate even at 21% oxygen, because the mechanism is radical recombination rather than oxygen depletion.
  • The dose-rate threshold for sparing should depend on the redox environment: cells with fewer antioxidants should show sparing at lower doses, matching the in-vitro effects near 8 Gy versus the in-vivo effects above 10 Gy.
  • For ions delivered as a single macro-pulse, the model predicts the same sparing as for pulsed electrons once the actual time structure is simulated, reproducing the helium- and carbon-ion data without invoking a separate mechanism.
  • Because the model accepts an arbitrary irradiation time structure and outputs survival probability, it can in principle support biologically driven FLASH treatment planning.
  • The normal-tissue versus tumor difference in vivo is attributed to differing antioxidant (especially thiol/GSH) environments, so tissue redox status should predict where FLASH sparing appears.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct testable extension: hold oxygen and dose fixed and vary antioxidant (thiol) concentration in vitro; the model's Figure 3 predicts that the FLASH sparing magnitude should fall as antioxidant levels rise, which the paper does not itself measure.
  • Implicit in the model is that the relevant dose-rate scale is set by radical reaction times rather than average dose rate; two schedules with equal mean dose rate but different pulse spacing should show different sparing, a prediction the paper does not test.
  • The same machinery could be applied to DNA double-strand break or plasmid damage endpoints at UHDR rather than only survival; the paper cites plasmid data as supporting evidence but does not model that endpoint quantitatively.
  • The model folds sub-microsecond water radiolysis into instantaneous source terms, so it is silent on whether early inter-track chemistry contributes; extending the chemical stage could test that assumption without changing the biological layer.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents MS-GSM2, a multiscale extension of the Generalized Stochastic Microdosimetric Model (GSM2) intended to describe cell survival after irradiation at conventional and ultra-high dose rates. The model couples (i) a physical stage using amorphous track structure and stochastic energy deposition events, (ii) a chemical stage described by five coupled ODEs for species including O2, R•, and ROO•, and (iii) a biological stage in which DNA damage and repair are modeled by a Markov jump process. Indirect damage is assumed proportional to the time integral of peroxyl radical concentration, normalized at 21% O2 and conventional dose rate, while the overall damage yield is scaled by DSB(LET)/OER(LET,[O2]). The authors compare model predictions with three published in-vitro UHDR survival datasets (DU145 with 10 MeV electrons, A549 with helium ions, CHO-K1 with carbon ions) at multiple oxygenation levels, and use the model to propose a redox-based explanation for differential in-vivo effects. The central claim is that the model predicts the UHDR sparing effect across radiation types, energies, doses, and oxygen concentrations.

Significance. If the central claim were established, this would be an important contribution to FLASH radiobiology: the model is unusually comprehensive in integrating stochastic energy deposition, explicit pulse structure, a chemical reaction network, and DNA damage repair, and it reproduces qualitative trends in three independent in-vitro datasets. The in-silico prediction summarized in Figure 3 is a falsifiable statement about the dependence of the sparing effect on oxygen and antioxidant levels. However, the current manuscript does not fully support the strong claim that the model 'grasps all the physics' behind the FLASH effect, because the chemical rate constants are tuned within literature ranges, the text and Table 1 disagree on key constants, and the oxygen dependence of the damage yield appears to be applied twice. These issues are fixable, but they substantially weaken the reported predictive validation, especially for hypoxic conditions.

major comments (3)
  1. [Bio-chemical stage, Eq. (5)] The text states that direct damage is independent of the chemical environment, but the Poisson mean for direct lesions is (1−q)κ(LET,[O2])z, with κ(LET,[O2]) = DSB(LET)/OER(LET,[O2]) from Eq. (3). Since OER depends on [O2], the direct damage as written is oxygen-scaled, which contradicts the stated independence and is the conceptual root of the double-counting problem. The direct component should use an oxygen-independent yield such as DSB(LET), and the OER scaling should enter only through the indirect component.
  2. [Bio-chemical stage and SI, Computational information] The oxygen dependence enters twice at conventional dose rates. Because ϱ is fixed by normalization at 21% O2 and conventional irradiation, and because Eq. (2) produces less ROO• at lower [O2], the conventional-dose-rate hypoxic damage yield becomes κ_h[(1−q)+q·∫ROO_h/∫ROO_std]z, which is strictly smaller than the OER-scaled yield κ_h·z. The OER in Eq. (3) already reduces the hypoxic yield, and the peroxyl integral reduces it again. Consequently, the conventional hypoxic survival points in Figure 2 are not clean tests of the peroxyl-recombination mechanism, and the reported FLASH differential at low oxygenation may be inflated. The model should be renormalized so that conventional-dose-rate hypoxic yields match Eq. (3) by construction, or the OER scaling and chemical integral should be combined into a single non-redundant oxygen dependence.
  3. [Chemical stage and Table 1] The main text reports k1 = 10^8 (mol/l)^−1·s^−1 and k2 = 10^5 (mol/l)^−1·s^−1, while Table 1 lists k1 = 5·10^7 and k2 = 10^4. Because the simulated chemical network is therefore ambiguous, and because the sparing prediction depends on the peroxyl recombination dynamics, the exact constants used must be specified. In addition, k1, k2, and k9 are explicitly updated 'to describe the experiments better' within literature ranges; without a sensitivity analysis over those ranges, the agreement in Figure 2 cannot be presented as strong evidence of predictive power. Please report the actual values used in the simulations and quantify the sensitivity of the survival predictions to the choice of these rate constants.
minor comments (5)
  1. [Appendix, Model parameters] The text refers twice to 'Table 1' when describing fitted biological rates; the biological rates are actually listed in Table 2. Please correct the cross-reference.
  2. [Appendix, Algorithm 1] The condition 'else if i = 4Nd' in line 24 appears inconsistent with the 1+4Nd hazard functions defined at the top of the algorithm; please check the indexing for the track-arrival event.
  3. [Eq. (7)] The last equation in Eq. (7) uses fξd and ẑd without defining them in the main text; the relationship between ẑd and the amorphous track deposition zd should be stated explicitly.
  4. [References and SI] Weber et al. 2022a and 2022b are identical references, and the SI contains a dangling citation '[ ?]' in the description of the normalized mean square error. Please clean these up.
  5. [Discussion] The sentence claiming 'compelling evidence supporting that the MS-GSM2 grasps all the physics behind the potential mechanism underlying the FLASH effect' overstates what can be concluded from a model with tuned chemical constants and a normalized indirect-damage term; please temper the wording to match the validation actually provided.

Circularity Check

2 steps flagged · score 6.0 of 10

Oxygen dependence at conventional dose rate reduces to the fitted OER input, and the UHDR sparing effect relies on reaction constants tuned to the same in-vitro experiments; partial circularity.

  1. fitted input called prediction [Results, The chemical stage, Eq. (2) and the paragraph following it]
    "we update the values of some reaction constants within the ranges we found in the literature to describe the experiments better, as shown in the next section. In particular, in this work, we consider k1 = 108(mol/l)−1 · s−1 ... k2 = 105(mol/l)−1 · s−1 ... and k9 = 102(mol/l)−1 · s−1 [Spitz et al., 2019, Wardman, 2020]."

    The dose-rate-dependent sparing effect is governed by k1 (formation of peroxyl radicals) and k2 (ROO• recombination), the very reactions that reduce indirect damage at UHDR. The paper says these constants were updated to describe the experiments better, i.e. the UHDR in-vitro data in Figure 2 that are later called 'predicted'. The FLASH sparing effect is therefore partly a fit to the target data, not a parameter-free prediction. The ambiguity is compounded because the text gives k1=10^8 and k2=10^5 while Table 1 lists k1=5×10^7 and k2=10^4, so it is unclear which tuned network produced the reported predictions.

  2. renaming known result [Results, The bio-chemical stage, Eqs. (3)-(4); Validation, Figure 2 and the paragraph after it]
    "κ (LET, [O2]) z = DSB(LET) / OER (LET, [O2]) z ... The OER is described in [Scifoni et al., 2013, Epp et al., 1972] ... where γ is an additional fitted parameter ... The modulation term ϱ∫[ROO•](s) ds ... is normalized to 1 ... to be consistent with the DNA damage yield at standard conditions ... For each of these experiments, we calibrated the biological parameters ... on the experimental points at standard conditions ... The MS-GSM2 has completely predicted all the other results."

    The oxygen trend in conventional irradiation is inserted as the empirical OER formula (4) with a fitted γ from prior work by the same group (Scifoni et al., 2013), and the biological rates a, b, r are calibrated at normoxic conventional conditions. The conventional hypoxic points in Figure 2 are therefore not emergent tests of the peroxyl-recombination mechanism; they are the OER input expressed through the fitted survival model. The peroxyl integral is normalized to unity only at 21% O2 and also decreases with [O2], so the model applies an additional oxygen correction on top of the OER-scaled yield κ; the claimed agreement with OER for conventional irradiation is not satisfied by the equations, and the hypoxic conventional 'predictions' are overdetermined by construction.

full rationale

The model has a genuine emergent component: the dose-rate dependence of indirect damage follows from the ODE network and the stochastic delivery structure, so the FLASH-sparing direction is not simply restating the inputs. However, two load-bearing pieces of the validation are partially circular. The oxygen dependence of survival at conventional dose rate is imported as the fitted OER formula (Eq. 4) from prior work including the same senior author, and the biological parameters are calibrated at normoxic conventional conditions; the hypoxic conventional points in Fig. 2 are therefore not independent predictions. Second, the central UHDR effect is controlled by k1 and k2, which the text states were updated 'to describe the experiments better' — i.e. tuned toward the very in-vitro data used as validation, with an unresolved discrepancy between text and Table 1 for these constants. These issues make the 'predictions across various oxygenation levels' partly a fitted/renamed input, but they do not make the whole derivation equivalent to its inputs: the peroxyl recombination mechanism and its dose-rate dependence are additional structure. Score 6, not 8 or 10.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The model's predictive content is concentrated in the chemical ODE dynamics, but several parameters are calibrated to prior fits (OER gamma, Kundrat normalization) or to the target data (chemical constants k1/k2/k9, biological rates a/b/r). The oxygen-axis predictions largely inherit the fitted OER formula, while the dose-rate dependence is more emergent from the ODE network.

free parameters (4)
  • biological rates a, b, r = per cell line: DU145 (7.82e-3, 1.83e-2, 3.23) 1/h; A549 (4.70e-3, 1.34e-2, 4.51) 1/h; CHO-K1 (4.21e-3, 2.43e-2, 3.68)…
    Fitted to the normoxic, conventional-irradiation survival point for each cell line using cross-entropy minimization (SI and Table 2).
  • chemical rate constants k1, k2, k9 = main text: k1=1e8, k2=1e5, k9=1e2 (M s)^-1; Table 1: k1=5e7, k2=1e4, k9=1e2 (M s)^-1
    Adjusted within literature ranges "to describe the experiments better" (Chemical stage section); values conflict between main text and Table 1.
  • OER parameters gamma, KO2, KLET, M = gamma: fitted in Scifoni et al. 2013; KO2, KLET, M from Epp et al. 1972 or Scifoni et al. 2013
    Equation (4) is an empirical formula containing a fitted parameter gamma; the model inherits this oxygen dependence rather than deriving it.
  • indirect damage normalization rho = not given numerically; normalized so mean indirect damage equals Kundrat yield at 21% O2, conventional
    Chosen to match track-structure DSB yields at standard conditions, so standard-condition damage is imposed by construction.
assumptions (6)
  • domain assumption Amorphous track model (Eq. 1) describes energy deposition with cylindrical symmetry and r^-2 falloff up to R80.
    Used in physical stage to distribute energy deposition over 52 nuclear domains; neglects stochastic fluctuations of track structure beyond the radial average.
  • domain assumption Water radiolysis products reach their 1 microsecond G-values instantaneously; the early heterogeneous chemical stage is decoupled.
    Chemical stage section: the early heterogeneous stage is incorporated through production terms described by G-values, based on TRAX-CHEM.
  • ad hoc to paper The five-species, nine-reaction ODE system (Eq. 2) captures the chemical environment relevant to DNA damage.
    Reduction from Labarbe 2020 by cutting away negligible reactions and adjusting constants; completeness is not independently verified.
  • ad hoc to paper Indirect DNA damage yield is proportional to the time integral of peroxyl radical concentration [ROO•].
    Biochemical stage states that [ROO•] is directly linked to the number of indirect damages created, with a normalization constant rho matched to Kundrat yields.
  • domain assumption Empirical OER formula (Eq. 4) correctly captures oxygen dependence across LET and dose rates.
    Inherited from Scifoni et al. 2013 and Epp et al. 1972; contains a fitted parameter gamma, and is used to set the damage yield kappa.
  • domain assumption The GSM2 Markov model with rates a, b, r correctly describes DNA damage repair and cell survival.
    Biological stage uses the prior GSM2 framework with sub-lethal lesion repair, pairwise interaction, and conversion to lethal lesions.

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Pith. "Pith review of A multiscale radiation biophysical stochastic model describing the cell survival response at ultra-high dose rate." pith.science (2026). https://pith.science/paper/EADN7CZJ

@misc{pith2026241216322,
  author       = {Pith},
  title        = {Pith review of: A multiscale radiation biophysical stochastic model describing the cell survival response at ultra-high dose rate},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EADN7CZJ}},
  note         = {Machine review of arXiv:2412.16322}
}
read the original abstract

Ultra-high dose-rate (UHDR) radiotherapy, characterized by an extremely high radiation delivery rate, represents one of the most recent and promising frontier in radiotherapy. UHDR radiotherapy, addressed in the field as FLASH radiotherapy, is a disruptive treatment modality with several benefits, including significantly shorter treatment times, unchanged effectiveness in treating tumors, and clear reductions in side effects on normal tissues. While the benefits of UHDR irradiation have been well highlighted experimentally, the biological mechanism underlying the FLASH effect is still unclear and highly debated. Nonetheless, to effectively use UHDR radiotherapy in clinics, understanding the driving biological mechanism is paramount. Since the concurrent involvement of multiple scales of radiation damage has been suggested, we developed the MultiScale Generalized Stochastic Microdosimetric Model (MS-GSM2), a multi-stage extension of the GSM2, which is a probabilistic model describing the time evolution of the DNA damage in an irradiated cell nucleus. The MS-GSM2 can investigate several chemical species combined effects, DNA damage formation, and time evolution. We demonstrate that the MS-GSM2 can predict various in-vitro UHDR experimental results across various oxygenation levels, radiation types, and energies. The MS-GSM2 can accurately describe the empirical trend of dose and dose rate-dependent cell sensitivity over a wide range, consistently describing multiple aspects of the FLASH effect and reproducing the main evidence from the in-vitro experimental data. Our model also proposes a consistent explanation for the differential outcomes observed in normal tissues and tumors, in-vivo and in-vitro.

Figures

Figures reproduced from arXiv: 2412.16322 by the authors.

Figure 1
Figure 1. Schematic representation of the MS-GSM2 workflow. able to explain it fully. However, what has emerged from all these efforts is that multiple scales of radiation damage action should be— involved [Weber et al., 2022a]. The crucial role of chemical environment and of the redox system has been underlined [Labarbe et al., 2020, Weber et al., 2022a, Wardman, 2020, Favaudon et al., 2022]. Over the last years, a plethora … view at source ↗
Figure 2
Figure 2. Comparison of the MS-GSM2 results with experimental data. impinging the cells at a LET of 4.5 keV/µm. In particular, the dose was delivered with a mean dose rate of 0.12 Gy/s for Conv, with a mean dose rate of 205.13 Gy/s. Figure 2c shows the comparison of survival fractions of Chinese hamster ovary cells CHO-K1, between the MS-GSM2 prediction (in black) and the experiment [Tinganelli et al., 2022a] (in blue), at di… view at source ↗
Figure 3
Figure 3. Perspective analysis of the impact of the chemical environment on the emergence of the FLASH [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

  1. [1]

    Introduction A current frontier in the treatment of cancer is undoubtely the so-called FLASH radiotherapy, a revolutionary radiation therapy technique based on the ultra-high dose rate (UHDR) irradiation, i.e., a faster beam delivery (total irradiation time < 100 ms and a mean dose rate of the total irradiation > 40 Gy/s for a single high dose, usually hi...

    work page Pith review arXiv 2022

  2. [2]

    The physical stage We describe a cell nucleus using a cylinder of radiusRN = 8 µm (xy-plane) and length 2RN (z-direction), divided into Nd = 52 cubic domains. The absorbed dose of every energy deposition event z is distributed on the Nd domains of the xy plane of the cell nucleus, according to the Amorphous Track (AT) model described in [Scholz and Kraft,...

    work page 1996

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    Additional information about this can be found in the SI Appendix

    At the same time, it decreases as the oxygenation level decreases and the dose rate increases. Additional information about this can be found in the SI Appendix. Overall, the damage yield creation can be summarized by the following network X ˙d − → ( X + ZX;dir with probability 1 − q ∈ [0, 1] , X + ZX;indir with probability q ∈ [0, 1] , . (5) where ZX;dir...

    work page 2022

  4. [4]

    as indicated in the original publication, for electron at 10 MeV in Conv (mean dose rate of 14 Gy/min) and UHDR (mean dose rate of 600 Gy/s) irradiations of 18 Gy. In particular, the dose was delivered with an integer number of pulses, a pulse time of 3.5 µs, and a pulse repetition frequency of 200 Hz in the case of Conv irradiation, while with an integer...

    work page 2021

  5. [5]

    to normoxia (21% O 2). This is achievable due to the remarkable flexibility and comprehensiveness of the model, which simultaneously encompasses various physical, chemical, and biological aspects across different spatial and temporal scales. In particular, temporally, the MS-GSM 2 can span from 10 −6 to 100 s with the chemical stage and up to 105 s for th...

    work page 2023

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    A key advantage of the MS-GSM2 is its ability to account explicitly for the actual temporal structure 9 of dose delivery

    Currently, there are no other mechanistic models capable of simultaneously incorporating all these effects and, consequently, reproducing all the main in-vitro experimental data at UHDR regime, i.e., [Adrian et al., 2020, Tessonnier et al., 2021, Tinganelli et al., 2022a]. A key advantage of the MS-GSM2 is its ability to account explicitly for the actual ...

    work page 2023

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    [Adrian et al., 2020] Adrian, G., Konradsson, E., Lempart, M., B¨ ack, S., Ceberg, C., and Petersson, K. (2020). The flash effect depends on oxygen concentration. The British Journal of Radiology , 93(1106):20190702. PMID: 31825653. [Alanazi et al., 2020] Alanazi, A., Meesungnoen, J., and Jay-Gerin, J.-P. (2020). A Computer Modeling Study of Water Radioly...

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    [Battestini et al., 2022] Battestini, M., Schwarz, M., Kr¨ amer, M., and Scifoni, E. (2022). Including volume effects in biological treatment plan optimization for carbon ion therapy: Generalized equivalent uniform dose-based objective in trip98. Frontiers in Oncology, 12:555. [Boscolo et al., 2018] Boscolo, D., Kr¨ amer, M., Durante, M., Fuss, M., and Sc...

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    and set Ui := log 1 Ri ; 7: compute τi solving Z t+τi t hi(X(s), ξs)ds = Si − Ti . We use the convention that if hi(X(t), ξs) = 0, then τi = ∞; 8: end for 9: select τ := min(i=1,...,1+4Nd) τi; 10: select the corresponding rate hi at which the minimum is attained; 11: solve the...

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    At the same time, it naturally decreases as the oxygenation level decreases and the dose rate increases, in such a way that the average damage yield κ (LET, [O2], [ROO•]t) is in agreement with the Oxygen Enhancement Ratio (OER) formulation [Scifoni et al., 2013, Epp et al., 19...

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    We consider the new intervals, i.e

    We calculate the biological rates lj,min (min (NMSE50)) and uj,max = j (max (NMSE50)), with j ∈ a, b, r. We consider the new intervals, i.e. [log lj,min, log uj,max], and we repeat N-times the previous procedure, selecting, in the end, the best combination of a, b, and r. As d...

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