# Elucidating the Effects of Ionizing Radiation on Immune Cell Populations: A Mathematical Modeling Approach with Special Emphasis on Fractional Derivatives

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling of Immune System

#### 2.1. Merits of Modeling Biological Processes with Fractional Derivatives

#### 2.2. Fractional Integrals and Fractional Derivatives

## 3. Mathematical Formulation

#### Distribution of Initial Number of DSBs

**Definition**

**1.**

## 4. Fractional Differential Equations Model Formulation

**Case 1.**When ${k}_{max}$, $k$, and $m$ are taken to be equal to 1, the population can be structured into three groups of immune cells, which are ${N}_{0,0},{N}_{1,0},{N}_{0,1}$. Then, Equation (19) is reduced to a system of FODEs, which is given as follows:

**Case 2.**When ${k}_{max}=2$, then the possible $k$ and $m$ are taken as $k=0,1,2$ and $m=0,1,2$. The population is structured into groups of immune cells, ${N}_{0,0},{N}_{1,0},{N}_{0,1},{N}_{1,1},{N}_{2,0},{N}_{0,2}$. Then, Equation (21) is reduced to a system of FODEs, which is given as follows:

**Case 3.**When ${k}_{max}=3$ and $k$ and $m$ are taken as $k=0,1,2,3$ and $m=0,1,2,3,$ then the population is structured into groups of immune cells, which are ${N}_{0,0},{N}_{1,0},{N}_{0,1},{N}_{1,1},{N}_{2,0},{N}_{0,2},{N}_{2,1},{N}_{1,2},{N}_{3,0},{N}_{0,3}$. In this case, Equation (21) is reduced to a system of FODEs, which is given as follows:

**N**is a matrix that represents groups of immune cells having $k$ DSBs and $m$ misrepair DSBs and

**A**is a square matrix.

**N**and the size of matrix

**A**are derived based on the value of ${k}_{max}$, which is the maximum number of DSBs in a population of cells. The number of FODEs that exists in the system is $M$ with the dimension of matrix

**N**is $M\times M$ matrix given by:

#### 4.1. Solution of the FODE Model

#### 4.2. Model Simulation Algorithm

- Generate random initial conditions, $\mathit{N}\left(0\right)$. The dose $D$ is incorporated into the system via initial conditions. This is achieved by fixing dose $D$, to compute the mean $\lambda $ to obtain the initial conditions.
- Solve the system up to time $t=T$ for $T=24$ h. We chose the number to ensure that the repair process is completed [70].
- Compute the fraction of surviving immune cells $S=\frac{{{\displaystyle \sum}}_{k,m}{N}_{k,m}\left(T\right)}{{{\displaystyle \sum}}_{k,m}{N}_{k,m}\left(0\right)}.$
- Plot lnS versus the dose D. Due to the randomness of the initial conditions, we repeat steps (1)–(3) for twenty runs in order to obtain the averaged value of the surviving fraction in logarithmic scale,$lnS$. This step shows the completion of the loop for a single dose $D$, corresponding to a single data point. To obtain many data points on the survival curve, steps (1)–(3) and the averaged value of surviving fraction in a logarithmic scale, $lnS$ need to be repeated for each dose D.
- Fit the obtained data to the LQ (see Equation (1)) relation using fmin-search in MATLAB. This method employs the search method of Lagarias et al. [71], which determines the minimum of our unconstrained multivariable function using derivative-free. A flowchart of the model simulation algorithm is provided in Figure 1.

#### 4.3. Simulation of the Model

## 5. Comparison between Current Model and Previous Model Proposed by Siam et al. [34]

## 6. Results Discussion and Summary

^{−9}, i.e., |a

_{n}−a

_{n−1}| < 10

^{−9}. Moreover, in order to describe the evolution of cell population dynamics following ionizing radiation, the mechanistic model presented herein is developed within the framework of the fractional differential equations by taking into account the Mittag-Leffler Function as well as the Caputo derivative. The parameter estimation approach used herein qualifies the death rate coefficient of the model species, giving rise to a value of parameters that best fits the model simulation to the previous theoretical model [34] as well as the experimental data that will be collected for this purpose. Global optimization algorithms are used to obtain the minimum value of the sum of squares error (SSE), derived within the least square method, which supposes that the appropriate fit line of the experimental data is the one having the minimum value of the least square error.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Sowa, P.; Rutkowska-Talipska, J.; Sulkowska, U.; Rutkowski, K.; Rutkowskie, R. Ionizing and non-ionizing electromagnetic radiation in modern medicine. Pol. Ann. Med.
**2012**, 19, 134–138. [Google Scholar] [CrossRef] - Woodside, G. Environmental, Safety, and Health Engineering; John Wiley & Sons: Hoboken, NJ, USA, 1997; p. 476. [Google Scholar]
- Ryan, J. Ionizing Radiation: The Good, the Bad, and the Ugly. J. Investig. Dermatol.
**2012**, 132, 985–993. [Google Scholar] [CrossRef] [Green Version] - L’Annunziata, M.; Baradei, M. Handbook of Radioactivity Analysis; Academic Press: Cambridge, MA, USA, 2003; p. 58. ISBN 978-0-12-436603-9. [Google Scholar]
- Grupen, C.; Cowan, G.; Eidelman, S.; Stroh, T. Astroparticle Physics; Springer: Berlin/Heidelberg, Germany, 2005; p. 109. ISBN 978-3-540-25312-9. [Google Scholar]
- Siegel, R.L.; Miller, K.D.; Jemal, A. Cancer statistics. CA Cancer J. Clin.
**2018**, 68, 7–30. [Google Scholar] [CrossRef] - Belpomme, D.; Hardella, L.; Belyaevade, I.; Burgioa, E.O.; Carpenteragh, D. Thermal and non-thermal health effects of low intensity non-ionizing radiation: An international perspective. Environ. Pollut.
**2018**, 242, 643–658. [Google Scholar] [CrossRef] [PubMed] - Barrea, A.; Hernandez, M.E. Optimal control of a delayed breast cancer stem cells nonlinear model. Optim. Control Appl. Methods
**2016**, 37, 248–258. [Google Scholar] [CrossRef] - Özköse, F.; Yılmaz, S.; Yavuz, M.; Öztürk, İ.; Şenel, M.T.; Bağcı, B.Ş.; Doğan, M.; Önal, Ö. A Fractional Modeling of Tumor–Immune System Interaction Related to Lung Cancer with Real Data. Eur. Phys. J. Plus
**2022**, 137, 40. [Google Scholar] [CrossRef] - Mukhopadhyay, R.; Bhattacharyya, T. Spatio temporal variations in mathematical model of macrophage-tumorinteraction. Nonlinear Anal. Hybrid Syst.
**2008**, 2, 819–831. [Google Scholar] [CrossRef] - Khajanchi, J.; Nieto, J. Mathematical modeling of tumor-immune competitive system, considering the role of time delay. Appl. Math. Comput.
**2019**, 340, 180–205. [Google Scholar] [CrossRef] - Sarkar, R.R.; Sandip, B. Cancer self-remission and tumor stability—A stochastic approach. Math. Biosci.
**2005**, 196, 65–81. [Google Scholar] [CrossRef] - Hu, X.; Jang, S.R.-J. Dynamics of tumor-CD4T+-cytokines-host cells interactions with treatments. Appl. Math. Comput.
**2018**, 1, 700–720. [Google Scholar] [CrossRef] - Lei, G.; Zhang, Y.; Koppula, P.; Liu, X.; Zhang, J.; Lin, S.H.; Gan, B. The role of ferroptosis in ionizing radiation-induced cell death and tumor suppression. Cell Res.
**2020**, 30, 146–162. [Google Scholar] [CrossRef] [PubMed] - Prakasha, D.G.; Veeresha, P. Analysis of lakes pollution model with mittag-leffer kernel. J. Ocean. Eng. Sci.
**2020**, 5, 310–322. [Google Scholar] [CrossRef] - Veeresha, P.; Prakasha, D.G. A reliable analytical technique for fractional Caudrey-Dodd-gibbon equation with Mittag-Leer kernel. Nonlinear Eng.
**2020**, 9, 319–328. [Google Scholar] [CrossRef] - Tavassoli, M.H.; Tavassoli, A.; Rahimi, M.R.O. The geometric and physical interpretation of fractional order derivatives of polynomial functions. Differ. Geom.-Dyn. Syst.
**2013**, 15, 93–104. [Google Scholar] - Farayola, M.F.; Shae, S.S.; Siam, F.M.; Khan, I. Mathematical modeling of radiotherapy cancer treatment using Caputo fractional derivative. Comput. Methods Programs Biomed.
**2020**, 188, 105306. [Google Scholar] [CrossRef] - Pimentel, V.O.; Rekers, N.H.; Yaromina, A.; Lieuwes, N.G.; Biemans, R.; Zegers, C.M.L.; Germeraad, W.T.V.; Van Limbergen, E.J.; Neri, D.; Dubois, L.J.; et al. Radiotherapy causes long-lasting antitumor immunological memory when combined with immunotherapy. Radiother. Oncol.
**2018**, 127, S22. [Google Scholar] - Qureshi, S. Real life application of Caputo fractional derivative for measles epidemiological autonomous dynamical system. Chaos Solitons Fractals
**2020**, 134, 109744. [Google Scholar] [CrossRef] - Sopasakis, P.; Sarimveis, H.; Macheras, P.; Dokoumetzidis, A. Fractional calculus in pharmacokinetics. J. Pharmacokinet. Phar.
**2018**, 45, 107125. [Google Scholar] [CrossRef] - Ahmed, N.; Shah, N.A.; Taherifar, S.; Zaman, F.D. Memory effects and of the killing rate on the tumor cells concentration for a one-dimensional cancer model. Chaos Solitons Fractals
**2021**, 144, 110750. [Google Scholar] [CrossRef] - Machado, J.T.; Kiryakova, V. Recent history of the fractional calculus: Data and statistics. Handb. Fract. Calc. Appl.
**2019**, 1, 1–21. [Google Scholar] - Bisci, G.M.; Rădulescu, V.D.; Servadei, R. Variational Methods for Nonlocal Fractional Problems; Cambridge University Press: Cambridge, UK, 2016; Volume 162. [Google Scholar]
- Kumar, S.; Kuma, A.; Momani, S.; Aldhaifallah, M.; Nisar, K.S. Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two-dimensional systems. Adv. Differ. Equ.
**2019**, 2019, 413. [Google Scholar] [CrossRef] - Abdelaziz, M.A.; Ismail, A.I.; Abdullah, F.A.; Mohd, M.H. Bifurcations and chaos in a discrete SI epidemic model with fractional order. Adv. Differ. Equ.
**2018**, 1, 44. [Google Scholar] [CrossRef] [Green Version] - Veeresha, P.; Prakasha, D.G.; Baskonus, H.M. Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method. Math. Sci.
**2019**, 13, 115–128. [Google Scholar] [CrossRef] [Green Version] - Abdelaziz, M.A.; Ismail, A.I.; Abdullah, F.A.; Mohd, M.H. Codimension one and two bifurcations of a discrete-time fractional-order SEIR measles epidemic model with constant vaccination. Chaos Solitons Fractals
**2020**, 140, 110104. [Google Scholar] [CrossRef] - Al-Khedhairi, A.; Elsadany, A.A.; Elsonbaty, A. On the Dynamics of a Discrete Fractional-Order Cournot-Bertrand Competition Duopoly Game. Math. Probl. Eng.
**2022**, 2022, 8249215. [Google Scholar] [CrossRef] - Rashid, S.; Jarad, F.; Ahmad, A.G.; Abualnaja, K.M. New numerical dynamics of the heroin epidemic model using a fractional derivative with Mittag-Leffler kernel and consequences for control mechanisms. Results Phys.
**2022**, 35, 105304. [Google Scholar] [CrossRef] - Lu, Z.; Yu, Y.; Chen, Y.; Ren, G.; Xu, C.; Wang, S. Stability analysis of a nonlocal SIHRDP epidemic model with memory effects. Nonlinear Dyn.
**2022**, 109, 121–141. [Google Scholar] [CrossRef] - Bazan, J.G.; Le, Q.-T.; Zips, D. Radiobiology of Lung Cancer. In IASLC Thoracic Oncology, 2nd ed.; Elsevier: Amsterdam, The Netherlands, 2017; pp. 330–336. [Google Scholar]
- Loan, M.; Bhat, A. Effect of Over dispersion of Lethal Lesions on Cell Survival Curves. arXiv
**2021**. [Google Scholar] [CrossRef] - Siam, F.M.; Grinfeld, M.; Bahar, A.; Rahman, H.A.; Ahmad, H.; Johar, F. A mechanistic model of high dose irradiation damage. Math. Comput. Simul.
**2018**, 151, 156–168. [Google Scholar] [CrossRef] - Williams, J.P.; McBride, W.H. After the bomb drops: A new look at radiation-induced multiple organ dysfunction syndrome (MODS). Int. J. Irradiat. Biol.
**2011**, 87, 851–868. [Google Scholar] [CrossRef] [PubMed] - Baar, M.; Coquille, L.; Mayer, H.; Hölzel, M.; Rogava, M.; Tüting, T.; Bovier, A. A stochastic model for immunotherapy of cancer. Sci. Rep.
**2016**, 6, 24169. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Eftimie, R.; Bramson, J.; Earn, D. Interactions between the immune system and cancer: A brief review of non-spatial mathematical models. Bull. Math. Biol.
**2011**, 73, 232. [Google Scholar] [CrossRef] [PubMed] - Gupta, P.B.; Fillmore, C.M.; Jiang, G.; Shapira, S.D.; Tao, K.; Kuperwasser, C.; Lander, E.S. Stochastic state transitions give rise to phenotypic equilibrium in populations of cancer cells. Cell
**2011**, 146, 633644. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Durrett, R. Cancer modeling: A personal perspective. Not. AMS
**2013**, 60, 304309. [Google Scholar] [CrossRef] - Mohseni-Salehi, F.S.; Zare-Mirakabad, F.; Sadeghi, M.; Ghafouri-Fard, S. A Stochastic Model of DNA Double-Strand Breaks Repair Throughout the Cell Cycle. Bull. Math. Biol.
**2020**, 82, 11. [Google Scholar] [CrossRef] - Huether, S.E.; McCance, K.L. Understanding Pathophysiology-E-Book; Elsevier Health Sciences: Amsterdam, The Netherlands, 2019. [Google Scholar]
- Li, L.; Chen, X.C.; Li, X.J.; Li, Z.H.; Jian, Y.; Wu, Y.Z.; Zhang, J.P.; Ren, M.; Zhang, B. Analytical model for total ionizing dose-induced excess base current in PNP BJTs. Microelectron. Reliab.
**2020**, 113, 113939. [Google Scholar] [CrossRef] - Sazykina, T.G.; Kryshev, A.I. Simulation of population response to ionizing irradiation in an ecosystem with a limiting resource Model and analytical solutions. J. Environ. Radioact.
**2016**, 151, 50–57. [Google Scholar] [CrossRef] - Widdicombe, T.; Borrelli, R.A. MCNP modelling of irradiation effects of the Dragonfly missions RTG on Titan II: Atmospheric ionization effects. Acta Astronaut.
**2021**, 186, 517–522. [Google Scholar] [CrossRef] - Saeedian, M.; Khalighi, M.; Azimi-Tafreshi, N.; Jafari, G.R.; Ausloos, M. Memory effects on epidemic evolution: The susceptible-infected recovered epidemic model. Phys. Rev. E
**2017**, 95, 022409. [Google Scholar] [CrossRef] [Green Version] - Rashid, H.; Siam, F.M.; Maan, N.; Abd Rahman, W.N.W. Ionizing Irradiation Effects Modelling in Cells Population with Gold Nanoparticles. Malays. J. Fundam. Appl. Sci.
**2021**, 17, 659–669. [Google Scholar] [CrossRef] - Chakraverty, S.; Jena, R.M.; Jena, S.K. Time-fractional order biological systems with uncertain parameters. In Synthesis Lectures on Mathematics and Statistics; Springer: Berlin/Heidelberg, Germany, 2020; Volume 12, 160p. [Google Scholar]
- Anderson, D.R.; Camrud, E.; Ulness, D.J. On the nature of the conformable derivative and its applications to physics. arXiv
**2018**. [Google Scholar] [CrossRef] - Jena, R.M.; Chakraverty, S.; Rezazadeh, H.; Domiri Ganji, D. On the solution of time-fractional dynamical model of Brusselator reaction-diffusion system arising in chemical reactions. Math. Methods Appl. Sci.
**2020**, 43, 3903–3913. [Google Scholar] [CrossRef] - Al-khedhairi, A.; Elsadany, A.A.; Elsonbaty, A. Modelling immune systems based on Atangana-Baleanu fractional derivative. Chaos Solitons Fractals
**2019**, 129, 25–39. [Google Scholar] [CrossRef] - Azkase, F.; Aženel, M.T.; Habbireeh, R. Mathematical Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy. Model. Numer. Simul. Appl.
**2021**, 1, 67–83. [Google Scholar] - Ashraf, F.; Ahmad, A.; Saleem, M.U.; Farman, M.; Ahmad, M.O. Dynamical behavior of HIV immunology model with non-integer time fractional derivatives. Int. J. Adv. Appl. Sci.
**2018**, 5, 39–45. [Google Scholar] [CrossRef] - Petras, I. Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation; Springer Science and Business Media: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Kilbs, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Kumar, S.; Kumar, A.; Odibat, Z.M. A nonlinear fractional model to describe the population dynamics of two interacting species. Math. Methods Appl. Sci.
**2017**, 40, 4134–4148. [Google Scholar] [CrossRef] - Zou, C.; Zhang, L.; Hu, X.; Wang, Z.; Wik, T.; Pecht, M. A review of fractional-order techniques applied to lithium-ion batteries, lead-acid batteries, and supercapacitors. J. Power Sources
**2018**, 390, 286–296. [Google Scholar] [CrossRef] [Green Version] - Teodoro, G.S.; Machado, J.T.; De Oliveira, E.C. A review of definitions of fractional derivatives and other operators. J. Comput. Phys.
**2019**, 388, 195–208. [Google Scholar] [CrossRef] - Abdeljawad, T. On Riemann and Caputo fractional differences. Comput. Math. Appl.
**2011**, 62, 1602–1611. [Google Scholar] [CrossRef] [Green Version] - Mainardi, F. Why the Mittag-Leffler function can be considered the queen function of the fractional calculus? Entropy
**2020**, 22, 1359. [Google Scholar] [CrossRef] - Ahmadova, A.; Huseynov, I.T.; Fernandez, A.; Mahmudov, N.I. Trivariate Mittag-Leffler functions used to solve multi-order systems of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 97, 105735. [Google Scholar] [CrossRef] - Erdelyi, A. Higher Transcendental Functions; McGRAW-HILL Book Company: New York, NY, USA, 1953; p. 59. [Google Scholar]
- Nasir, M.H.; Siam, F.M. Simulation and sensitivity analysis on the parameter of non-targeted irirradiation effects model. J. Teknol.
**2019**, 81, 133–142. [Google Scholar] [CrossRef] [Green Version] - Singh, S.K.; Wang, M.; Staudt, C.; Iliakis, G. Post-irradiation chemical processing of DNA damage generates double-strand breaks in cells already engaged in repair. Nucleic Acids Res.
**2011**, 39, 8416–8429. [Google Scholar] [CrossRef] [PubMed] - Antonelli, F.; Campa, A.; Esposito, G.; Giardullo, P.; Belli, M.; Dini, V.; Tabocchini, M.A. Induction and repair of DNA DSB as revealed by H2AX phosphorylation foci in human fibroblasts exposed to low-and high-LET irradiation: Relationship with early and delayed reproductive cell death. Irradiat. Res.
**2015**, 183, 417–431. [Google Scholar] [CrossRef] [PubMed] - Herrera Ortiz, A.F.; Fernandez Beaujon, L.J.; Garcia Villamizar, S.Y.; Fonseca Lpez, F.F. Magnetic resonance versus computed tomography for the detection of retroperitoneal lymph node metastasis due to testicular cancer: A systematic literature review. Eur. J. Radiol. Open
**2021**, 8, 100372. [Google Scholar] [CrossRef] - Understanding Cancer, National Cancer Institute. Available online: https://www.cancer.gov/publications/dictionaries/geneticsdictionary/def/chromosome (accessed on 21 February 2023).
- Garrappa, R.; Giusti, A.; Mainardi, F. Variable-order fractional calculus: A change of perspective. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 102, 105904. [Google Scholar] [CrossRef] - Brandibur, O.; Garrappa, R.; Kaslik, E. Stability of Systems of Fractional-Order Differential Equations with Caputo Derivatives. Mathematics
**2021**, 9, 914. [Google Scholar] [CrossRef] - Haghani, F.K.; Soleymani, F. An improved Schulz-type iterative method for matrix inversion with application. Trans. Inst. Meas. Control
**2014**, 36, 983–991. [Google Scholar] [CrossRef] - Anelone, A.J.; Spurgeon, S.K. Modelling and Simulation of the Dynamics of the Antigen-Specific T Cell Response Using Variable Structure Control Theory. PLoS ONE
**2016**, 18, e0166163. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lagarias, J.C.; Reeds, J.A.; Wright, M.H.; Wright, P.E. Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions. SIAM J. Optim.
**1998**, 9, 112–147. [Google Scholar] [CrossRef] [Green Version] - Kusunoki, Y.; Hayashi, T. Long-lasting alterations of the immune system by ionizing radiation exposure: Implications for disease development among atomic bomb survivors. Int. J. Radiat. Biol.
**2008**, 84, 1–14. [Google Scholar] [CrossRef] [PubMed] - Manda, K.; Glasow, A.; Paape, D.; Hildebrandt, G. Effects of ionizing radiation on the immune system with special emphasis on the interaction of dendritic and T cells. Front. Oncol.
**2012**, 2, 102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mukherjee, D.; Coates, P.J.; Lorimore, S.A.; Wright, E.G. Responses to ionizing radiation mediated by inflammatory mechanisms. J. Pathol.
**2014**, 232, 289–299. [Google Scholar] [CrossRef] [PubMed] - Derer, A.; Frey, B.; Fietkau, R.; Gaipl, U.S. Immune-modulating properties of ionizing radiation: Rationale for the treatment of cancer by combination radiotherapy and immune checkpoint inhibitors. Cancer Immunol. Immunother.
**2016**, 65, 779–786. [Google Scholar] [CrossRef] [PubMed]

Parameters | Lower Boundary | Upper Boundary | Reference |
---|---|---|---|

$\delta $ | $2\text{}G{y}^{-1}cell{s}^{-1}$ | $40\text{}G{y}^{-1}cell{s}^{-1}$ | [34] |

${\alpha}_{1}$ | $0.0277{h}^{-1}$ | $20.97{h}^{-1}$ | [34] |

${\alpha}_{2}$ | $0{h}^{-1}$ | $0.005{h}^{-1}$ | [34] |

$\rho $ | 0 | 1 | [34] |

${V}_{max}$ | $0.1{h}^{-1}$ | $3{h}^{-1}$ | [34] |

${K}_{M}$ | $0\mu M$ | $5\mu M$ | [34] |

Item | Previous Model IDE | Current Model FDE |

System | $\frac{dN}{dt}=AN$ | $\frac{{d}^{\alpha}N}{d{t}^{\alpha}}=AN$ |

The solution | $N\left(t\right)=N\left(0\right)exp\left(At\right)$ | $N\left(t\right)=N\left(0\right){E}_{\alpha}\left(At\right)$ |

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**MDPI and ACS Style**

Alzahrani, D.Y.; Siam, F.M.; Abdullah, F.A.
Elucidating the Effects of Ionizing Radiation on Immune Cell Populations: A Mathematical Modeling Approach with Special Emphasis on Fractional Derivatives. *Mathematics* **2023**, *11*, 1738.
https://doi.org/10.3390/math11071738

**AMA Style**

Alzahrani DY, Siam FM, Abdullah FA.
Elucidating the Effects of Ionizing Radiation on Immune Cell Populations: A Mathematical Modeling Approach with Special Emphasis on Fractional Derivatives. *Mathematics*. 2023; 11(7):1738.
https://doi.org/10.3390/math11071738

**Chicago/Turabian Style**

Alzahrani, Dalal Yahya, Fuaada Mohd Siam, and Farah A. Abdullah.
2023. "Elucidating the Effects of Ionizing Radiation on Immune Cell Populations: A Mathematical Modeling Approach with Special Emphasis on Fractional Derivatives" *Mathematics* 11, no. 7: 1738.
https://doi.org/10.3390/math11071738