# Computational Biophysical Modeling of the Radiation Bystander Effect in Irradiated Cells

^{1}

^{2}

^{*}

## Abstract

**:**

## Simple Summary

## Abstract

## 1. Introduction

- by the gap junctional pathway, where the signal is transferred by small gaps in the cell directly to the neighboring cell;
- by signal emission into the surrounding environment (e.g., water) and through activating the MAP kinase in bystander cells through the cell’s wall [4], which plays a role in regulating the response to external signals entering the cell in the form of NF-kB proteins [5], which act as transcription factors (which bind DNA in the appropriate region).

## 2. Review of Existing Models

#### 2.1. Chinese Model Xia–Liu–Xue–Wang–Wu

- the signal hitting a cell (that is already damaged) will disappear from the simulation due to being absorbed by the cell;
- as soon as a signal hits a cell that is not damaged, it will continue to move in any direction.

#### 2.2. BSDM Model

- that the lifetime of the signals is greater than the expected diffusion time;
- that the reaction between the signal and the cell occurs when the distance between them is less than half the size of the cell (R);
- that a cell can emit two kinds of bystander effect signals: S-type signals—causing cell death; T signals—transforming into a cancer cell;
- that bystander effect signals do not react with each other;
- that diffuse cell movements are ignored;
- that neighborhood signals do not react with the cells that emit them.

_{d}and B

_{S}. By definition, Formula (2) can be written as follows:

_{1}and c

_{2}as so-called expansion coefficients. Assuming that ${S}_{d}\equiv x=\frac{\left({N}_{0}-{N}_{d}\right)}{{N}_{0}}$ and using the appropriate substitution, the following equations can be obtained, enabling the calculation of individual fractions:

_{1}and c

_{2}.

#### 2.3. BaD Model

- in the case of neoplastic transformation, the “all or nothing” principle prevails, according to which the radiation dose increase does not cause the increase in the observed neoplastic effects, but only the transformation process itself;
- in the case of a cell hit directly by radiation, the bystander effect is negligible.

- in the first one, a whole group of cells was exposed to ionizing radiation;
- in the second, only 10% of the total cell population was irradiated;
- in the last one, the cells were irradiated, where the dose is described by the Poisson distribution from the beam axis (irradiated cell).

#### 2.4. The Ebert–Suchowerska–Jackson–McKenzie Model

_{local}), as follows:

#### 2.5. Japanese Hattori–Yokoya–Watanabe Model

- cells;
- medium (e.g., empty space, water);
- wall (walls of a vessel containing a group of cells or other tissue, e.g., bone).

- cell irradiation;
- generating bystander effect signals;
- DNA damage induced by direct irradiation and by bystander signals;
- cell response to induced DNA damage.

- through the medium—MDP (medium-mediated pathway);
- by gap junctions—GJP (gap junctional pathway).

- ${\mathsf{\Phi}}_{i,j}\left(t\right)$ is a bystander signal concentration in the position (i,j) in the time, t;
- $\Delta t$ is a time interval (time step);
- ${d}^{2}$ is a cell’s two dimensional surface;
- ${\mathsf{\Phi}}_{W}$ is a diffusion constant;
- ${k}_{1,}{l}_{1}$ are coordinates of cells which are located linearly close to the irradiated cell (i,j) in positions: (i + 1, j), (i − 1, j), (i, j + 1) and (i, j − 1);
- ${k}_{2,}{l}_{2}$ are analogical coordinates but diagonal: (i + 1, j + 1), (i − 1, j − 1), (i − 1, j + 1) and (i + 1, j − 1);

- ${}_{}{}^{M}W{{}^{\prime}}_{k,l}={M}_{W}$, when the position (k,l) is a cell;
- ${}_{}{}^{M}W{{}^{\prime}}_{k,l}={M}_{W}$, when the position (k,l) is a medium;
- ${}_{}{}^{M}W{{}^{\prime}}_{k,l}=0$, when the position (k,l) is a wall;
- ${}_{}{}^{G}W{{}^{\prime}}_{k,l}={G}_{W}$, when the position (k,l) is a cell;
- ${}_{}{}^{G}W{{}^{\prime}}_{k,l}=0$, when the position (k,l) is a medium;
- ${}_{}{}^{G}W{{}^{\prime}}_{k,l}=0$, when the position (k,l) is a wall.

- ${}_{}{}^{M}\alpha $ and ${}_{}{}^{G}\alpha $ are constants related to creation of signals;
- ${}_{}{}^{M}\beta $ and ${}_{}{}^{G}\beta $ are constants related to reduction (decay) of signals;
- ${D}_{i,j}\left(t\right)$ is a radiation dose to cell.

- ${}_{}{}^{R}Z{}_{i,j}$ are damages created by direct irraditation;
- ${}_{}{}^{M}Z{}_{i,j}$ are damages created by MDP signals;
- ${}_{}{}^{G}Z{}_{i,j}$ are damages created by GJP signals;
- ${}_{}{}^{B}Z{}_{i,j}$ are damages created by some other external events;
- ${}_{}{}^{r}Z{}_{i,j}$ are damages already repaired.

#### 2.6. Monte Carlo Model

_{b}is met), it sends signals to the bystander cells that can damage them with a probability expressed as a distribution of [24] the following:

#### 2.7. Models Comparison

## 3. Methodology

#### 3.1. Description of the Algorithm

- diffusion signal propagation;
- saturation of its probability distribution for a certain dose;
- Poisson shape of the dependence of the number of damages on the distance from the hit cell.

#### 3.2. Exemplary Results of Simulation

- Dose per step—0.001 Gy;
- damage resulting both from direct irradiation and interaction with bystander signals—31 ± 1;
- damage caused by bystander signals only—2 ± 0.8.

- Dose per step—0.005 Gy;
- damage resulting both from direct irradiation and interaction with bystander signals—159 ± 6;
- damage caused by bystander signals only—10 ± 5.

- Dose per step—0.01 Gy;
- damage resulting both from direct irradiation and interaction with bystander—312 ± 9;
- damage caused by bystander signals only—20 ± 7.

^{−6}. After the 2nd time step, damage occurred in 7 cells, 1 of them being that which was directly irradiated. The remaining 6 cells were located in the nearest neighborhood to the directly irradiated cells: most damages occurred in a cell with coordinates (x, y, z) = (5,5,5) with the frequency 1.94 × 10

^{−5}, and in the other cells 1.78 × 10

^{−8}. After the third time step, damage appeared in 25 cells. The number of damages obtained as a result of the simulation was calculated using Equation (37), but in a more general approach, these values after scaling will be the probability of the appearance of the bystander effect in a particular cell at a given time step.

^{d}) (which was originally derived from the theory of nucleation and crystal growth [27]); however, this does not mean that after saturation the frequency of damage in all cells is the same—it follows a distribution similar to the Poisson one.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Nagasawa, H.; Little, J.B. Induction of sister chromatid exchanges by extremely low doses of alpha-particles. Cancer Res.
**1992**, 52, 6394–6396. [Google Scholar] [PubMed] - Wideł, M.; Przybyszewski, W.; Rzeszowska-Wolny, J. Popromienny efekt sąsiedztwa, ważny element odpowiedzi na promieniowanie jonizujące–potencjalne implikacje kliniczne. Post. Hig. Med. Dośw.
**2009**, 63, 377–388. [Google Scholar] - Hattori, Y.; Yokoya, A.; Watanabe, R. Cellular automaton-based model for radiation-induced bystander effects. BMC Syst. Biol.
**2015**, 9, 90. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bryk, D.; Oljarz, W.; Zapolska-Downar, D. Kinazy aktywowane mitogenami i ich znaczenie w patogenezie miażdżycy. Postępy Hig. Med. Dosw.
**2014**, 68, 10–22. [Google Scholar] [CrossRef] [PubMed] - Ghosh, S. Handbook of Transcription Factor NF-kB; CRC: Boca Raton, FL, USA, 2006. [Google Scholar]
- Prise, K.M.; Folkard, M.; Michael, B.D. A review of the bystander effect and its implications for low-dose exposure. Radiat. Prot. Dosim.
**2003**, 104, 347–355. [Google Scholar] [CrossRef] [PubMed] - Lenarczyk, M.; Słoikowska, M.G.; Matjle, T. Indukcja popromiennej odpowiedzi adaptacyjnej w retikulocytach krwi obwodowej myszy. Rocz. Państwowego Zakładu Hig.
**1997**, 3, 239–344. [Google Scholar] - Iyer, R.; Lehnert, B.E. Low dose, low-LET ionizing radiation-induced radioadaptation and associated early responses in unirradiated cells. Mutat. Res.
**2002**, 503, 1–9. [Google Scholar] [CrossRef] - Ballarini, F.; Ottolenghi, A. Low-dose radiation action: Possible implications of bystander effects and adaptive response. J. Radiol. Prot.
**2002**, 22, A39. [Google Scholar] [CrossRef] [PubMed] - Mothersill, C.; Smith, R.; Wang, J.; Rusin, A.; Fernandez-Palomo, C.; Fazzari, J.; Seymour, C. Biological Entanglement—Like Effect After Communication of Fish Prior to X-ray Exposure. Dose-Response
**2018**, 16, 1559325817750067. [Google Scholar] [CrossRef] - Xia, J.; Liu, L.; Xue, J.; Wang, Y.; Wu, L. Modeling of radiation-induced bystander effect using Monte Carlo methods. Nucl. Instrum. Methods Phys. Res. B
**2009**, 267, 1015–1018. [Google Scholar] [CrossRef] [Green Version] - Kopeć-Szlęzak, J. Cytokiny w procesach, odpornościowych. Onkol. Pol.
**2005**, 8, 217–222. [Google Scholar] - ‘Atomic Weight’: The Name, Its History, Definition, and Units. Available online: http://publications.iupac.org/pac/1992/pdf/6410x1535.pdf (accessed on 1 December 2021).
- Khvostunov, I.K.; Nikjoo, H. Computer modeling of radiation-induced bystander effect. J. Radiol. Prot.
**2002**, 22, A33–A37. [Google Scholar] [CrossRef] - Nikjoo, H.; Khvostunov, I.K. A theoretical approach to the role and critical issues associated with bystander effect in risk estimation. Hum. Exp. Toxicol.
**2004**, 23, 81–86. [Google Scholar] [CrossRef] - Nikjoo, H.; Khvostunov, I.K. Modeling of radiation-induced bystander effect at low dose and low LET. Int. J. Low Radiat.
**2006**, 3, 143–158. [Google Scholar] [CrossRef] - Nikjoo, H.; Khvostunov, I.K. Biophysical model of the radiation-induced bystander effect. Int. J. Radiat. Biol.
**2003**, 79, 43–52. [Google Scholar] [CrossRef] - Gaillard, S.; Pusset, D.; de Toledo, S.M.; Fromm, M.; Azzam, E.I. Propagation Distance of the α-Particle-Induced Bystander Effect: The Role of Nuclear Traversal and Gap Junction Communication. Radiat. Res.
**2009**, 171, 513–520. [Google Scholar] [CrossRef] [Green Version] - Sasaki, K.; Waku, K.; Tsutsumi, K.; Itoh, A.; Date, H. A Simulation Study of the Radiation-Induced Bystander Effect: Modeling with Stochastically Defined Signal Reemission. Comput. Math. Methods Med.
**2012**, 2012, 389095. [Google Scholar] [CrossRef] - Brenner, D.J.; Little, J.B.; Sachs, R.K. The bystander effect in radiation oncogenesis: II. A Quant. Model. Radiat. Res.
**2001**, 155, 402–408. [Google Scholar] [CrossRef] [Green Version] - Ebert, M.A.; Suchowerska, N.; Jackson, M.A.; McKenzie, D.R. A mathematical framework for separating the direct and bystander components of cellular radiation response. Acta Oncol.
**2010**, 49, 1334–1343. [Google Scholar] [CrossRef] [PubMed] - Fornalski, K.W. Simple empirical correction functions to cross sections of the photoelectric effect, Compton scattering, pair and triplet production for carbon radiation shields for intermediate and high photon energies. J. Phys. Commun.
**2018**, 2, 035038. [Google Scholar] [CrossRef] - Taylor, J.R. Wstęp Do Analizy Błędu Pomiarowego; Wyd. Naukowe PWN: Warszawa, Poland, 2016. [Google Scholar]
- Fornalski, K.W. Mechanistic model of the cells irradiation using the stochastic biophysical input. Int. J. Low Radiat.
**2014**, 9, 370–395. [Google Scholar] [CrossRef] - Fornalski, K.W.; Dobrzyński, L.; Janiak, M.K. Stochastyczny model odpowiedzi komórek na promieniowanie. Postępy Tech. Jądrowej
**2011**, 54, 23–37. [Google Scholar] - Wysocki, P. Modelowanie Monte Carlo Efektu Sąsiedztwa Oraz Dynamiki Procesu Nowotworzenia Dla Grupy Komórek Narażonych na Promieniowanie Jonizujące. Master’s Thesis, Faculty of Physics, Warsaw University of Technology, Warszawa, Poland, 2019. [Google Scholar]
- Dobrzyński, L.; Fornalski, K.W.; Reszczyńska, J.; Janiak, M.K. Modeling Cell Reactions to Ionizing Radiation: From a Lesion to a Cancer. Dose-Response
**2019**, 17, 1559325819838434. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shuryak, I.; Sachs, R.K.; Brenner, D.J. Biophysical Models of Radiation Bystander Effects: 1. Spatial Effects in Three-Dimensional Tissues. Radiat. Res.
**2007**, 168, 741–749. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Little, M.P.; Filipe, J.A.N.; Prise, K.M.; Folkard, M.; Belyakov, O.V. A model for radiation-induced bystander effects, with allowance for spatial position and the effects of cell turnover. J. Theor. Biol.
**2005**, 232, 329–338. [Google Scholar] [CrossRef] - Ballarini, F.; Alloni, D.; Facoetti, A.; Mairani, A.; Nano, R.; Ottolenghi, A. Modelling radiation-induced bystander effect and cellular communication. Radiat. Prot. Dosim.
**2006**, 122, 244–251. [Google Scholar] [CrossRef]

**Figure 1.**Block diagram of the bystander effect algorithm used in a program simulating the response of a group of cells to ionizing radiation using the Monte Carlo method [26]. The presented probability functions are given by Equations (35) and (36).

**Figure 2.**The number of cells with bystander damages in the function of time (black dots). The Avrami curve with the Equation 512−511.55·exp (−(0.13·x)

^{3,4}) [26] was fitted.

**Figure 3.**The number (as fraction) of signals of the bystander effect in the function of the distance from the irradiated cell (black points), to which the Poisson distribution was fitted (red curve, R

^{2}= 0.67) [26].

Number of Signals | ${\mathit{P}}_{\mathit{d}\mathit{a}\mathit{m}}$ | ${\mathit{P}}_{\mathit{e}\mathit{m}}$ | Lifetime |
---|---|---|---|

10 | 0.0040 | 0.9 | 1.5 h |

20 | 0.0018 | 0.95 | 1.5 h |

30 | 0.0020 | 0.15 | 1.5 h |

**Table 2.**The calculated numbers of cells received the bystander effect signals, B(x) and x, for the BSDM model. Source of the experimental data: [16].

Number of Cells Which Survived Direct Irradiation | Number of Bystander Effect Signals | Average Number of Cells Which Received Bystander Effect Signal(s) | B(x) | x |
---|---|---|---|---|

0 | 1000 | 0 | 0 | 0 |

50 | 1000 | 16.5 | 0.0416 | 0.125 |

100 | 1000 | 33 | 0.0825 | 0.25 |

200 | 1000 | 64.9 | 0.1626 | 0.5 |

300 | 1000 | 95.8 | 0.2395 | 0.75 |

400 | 1000 | 126 | 0.315 | 1 |

${\mathit{\alpha}}_{\mathit{R}}\left[\mathit{G}{\mathit{y}}^{-1}\right]$ | ${\mathit{\beta}}_{\mathit{R}}\left[\mathit{G}{\mathit{y}}^{-2}\right]$ | $\mathit{\mu}.\mathit{W}$ | $\mathit{\lambda}\left(\mathit{G}{\mathit{y}}^{-1}\right)$ | ${\mathit{\chi}}^{2}$ | |
---|---|---|---|---|---|

Method 1 | 0.053 | 0.061 | 1.28 | N/A | 0.00062 |

Method 2 | 0.00 | 0.083 | 6.10 | 0.033 | 0.0054 |

Method 3 | 0.00 | 0.13 | 20.2 | 0.013 | 0.015 |

${\mathit{\alpha}}_{\mathit{R}}\left(\mathit{G}{\mathit{y}}^{-1}\right)$ | ${\mathit{\beta}}_{\mathit{R}}\left(\mathit{G}{\mathit{y}}^{-2}\right)$ | $\mathit{\mu}.\mathit{W}$ | $\mathit{\lambda}\left(\mathit{G}{\mathit{y}}^{-1}\right)$ | ${\mathit{\chi}}^{2}$ | |
---|---|---|---|---|---|

Method 1 | 0.014 | 0.0024 | 4.71 | N/A | 0.0041 |

Method 2 | 0.00 | 0.0070 | 70.12 | 0.0015 | 0.029 |

Method 3 | 0.00 | 0.010 | 55.00 | 00022 | 0.052 |

**Table 5.**The comparison of bystander effect computational models described in Section 2.

Model and Its Reference Source | Short Description: | Main Technique Used |
---|---|---|

Chinese Model Xia–Liu–Xue–Wang–Wu [11] | - distribution of bystander signals is strictly defined and described using the simplified Einstein diffusion equation;
- a signal hitting a cell that is already damaged will disappear from the simulation due to the absorption; a signal hitting a cell that is not damaged will continue to move in any direction;
- cell that comes into contact with the bystander effect signals can send another signal;
| Stochastic model and Monte Carlo technique |

BSDM Model [14,15,16,17] | - distribution of bystander signals is strictly defined and described using the simplified Einstein diffusion equation;
- the reaction between the signal and the cell occurs when the distance between them is less than half the size of the cell;
- a cell can emit two kinds of bystander effect signals—causing cell death or cancer transformation;
- diffuse cell movements are ignored;
- presupposes that the exact number of cells has been irradiated;
| Deterministic model with empirical approximations to the data |

BaD Model [20] | - describes bystander and direct irradiation changes in the cell;
- in the case of neoplastic transformation, the “all or nothing” principle prevails, according to which the radiation dose increase does not cause the increase in the observed neoplastic effects, but only the transformation process itself;
- in the case of a cell hit directly by radiation, the bystander effect is negligible;
- signal propagation mechanism is not described in detail, so it can be used both for cells which are in contact or in distant from each other;
- presupposes that the exact number of cells has been irradiated;
| Stochastic binary model with empirical approximations to the data |

The Ebert–Suchowerska–Jackson–McKenzie Model [21] | - responses of a group of cells to direct irradiation and to bystander effect signals (resulting from earlier irradiation) are independent of each other;
- bystander effect signals propagate by diffusion and end up in bystander cells that have not been directly irradiated;
- model assumes the solo type of signal that can cause cell death, cell damages (that can be repaired), and might not cause any changes;
| Wide intercellular deterministic model with empirical approximations to the data |

Japanese Hattori–Yokoya–Watanabe Model [3] | - simulation is based on a two-dimensional cellular automaton in which three main components (states) can be distinguished: cells, medium (e.g., empty space, water), or wall (walls of a vessel containing a group of cells or other tissue, e.g., bone);
- an irradiated cell can send bystander signal to nearby cells by two ways: through the medium or by gap junctions;
- model assumes that the signals cannot interact with each other and their number is proportional to the dose;
| Iterative cellular automata model |

Monte Carlo Model [24,25] | - model is completely stochastic and describes comprehensively, but on a general level, possibly all biophysical phenomena concerning the irradiated cell, taking into account, e.g., cell death, transformation into a cancer cell, spontaneous formation of damage, repair of damage, etc., which, as mentioned, uses the Monte Carlo processes and the probability tree in its algorithm;
- the probability of the bystander effect appearance is given by the saturated probability function ${P}_{b}={\beta}_{1}\left[1-\mathrm{exp}\left(-{\beta}_{2}D\right)\right]$.
| Monte Carlo technique with probability tree |

Parameter | Exemplary Value |
---|---|

a | 10^{−5} |

M_{w} | 10^{−10} |

M_{α} | 1 |

M_{β} | 4.6 × 10^{−6} |

G_{w} | 5 × 10^{−11} |

G_{α} | 1 |

G_{β} | 1.18 × 10^{−3} |

λ_{D} | 60 |

λ_{M} | 0.006 |

λ_{G} | 0.06 |

β_{1} | 0.001 |

β_{2} | 300 |

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**MDPI and ACS Style**

Wysocki, P.; Fornalski, K.W.
Computational Biophysical Modeling of the Radiation Bystander Effect in Irradiated Cells. *Radiation* **2022**, *2*, 33-51.
https://doi.org/10.3390/radiation2010003

**AMA Style**

Wysocki P, Fornalski KW.
Computational Biophysical Modeling of the Radiation Bystander Effect in Irradiated Cells. *Radiation*. 2022; 2(1):33-51.
https://doi.org/10.3390/radiation2010003

**Chicago/Turabian Style**

Wysocki, Paweł, and Krzysztof W. Fornalski.
2022. "Computational Biophysical Modeling of the Radiation Bystander Effect in Irradiated Cells" *Radiation* 2, no. 1: 33-51.
https://doi.org/10.3390/radiation2010003