# An Agent-Based Model of Radiation-Induced Lung Fibrosis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Early and Late Fibrotic Response

^{3}, used as a surrogate of the more widely adopted Hounsfield Units. Figure 4 shows the early fibrotic response, that is, the absolute increase in the ECM concentration (with respect to the homeostatic values) at multiple doses 3 months after the irradiation. We separated the early component from the late one as was conducted in previous analyses [34] and fitted the experimental data using a logistic function as shown in [35,36,42] (we replaced the Hounsfield units (HU) with the ECM concentration):

_{50}is the dose at 50% of the maximum ΔECM concentration. As reported in multiple studies [34,35,43], we assumed negligible ECM changes at late time points for doses < 5 Gy and excluded those data from the experimental dataset. Given that the maximum dose (17.6 Gy) damaged 99% of the healthy AEC2, higher doses would not result in further increases of the ΔECM, and, therefore, we assume that the measured ΔECM

_{max}is close to its saturation value. Overall, we observed a good agreement between the experimental data, shown in Figure 5, and the fitting curve. Although both the early and late fibrotic responses could be fitted by a sigmoid, we observed the halving of the maximum ECM concentration in the late phase for the doses that induced RILF. Finally, we measured A = 0.0030 and A = 0.0040 for the 60% and 80% phagocytic fraction, respectively.

#### 2.2. Alveoli Survival

#### 2.3. RILF Severity Index

^{3}) and $\Delta {V}_{surv,FSU}\downarrow $ is the decrease in functioning distal lung volume given the total volume of the surviving FSUs (in cm

^{3}). The experimental data are shown in Figure 7 and were fitted using the equation provided in [37] and presented in Section 2.1 (Equation (2)). Again, we assumed negligible ECM changes at late time points for doses < 5 Gy and excluded those data from the experimental dataset.

#### 2.4. Effects of AEC2 Apoptotic to Senescent Ratio

## 3. Discussion

_{50}(while the fibrosis index differs by a multiplicative constant). Taken together, these outcomes support the RSI as an alternative to the FI for computational simulations where a direct measure of the decrease in the functioning volume might not be available.

## 4. Materials and Methods

#### 4.1. Software Platform and Modelling Environment

^{3}cube to fit an alveolar duct, and the 3D diffusion grid was split into cubes of side 500 μm to match the typical size of a CT voxel. The simulation time step was set to 1 s to ensure that the Courant–Friedrichs–Lewy stability condition was satisfied for the diffusion coefficient of each substance involved in the simulations. To reduce the total simulation time, the frequency of the default BDM standalone operations that regulate mechanical interactions among cells and cell behaviours was set to 10 simulation time steps. All the outputs derived from the simulations (as described in the following section) were printed on ROOT files and later on analysed using custom Python scripts (see, for example, [58]).

#### 4.2. Geometric Frame and Operations

- determine the number of surviving alveoli;
- simulate the secretion of IL13 from lymphocytes (which, for simplicity, are not included in the model) at the centre of each alveolus [61];
- simulate the inflow of fibroblasts in the interstitial space with a rate that depends on the number of AEC2 per alveolus. According to the literature [62,63], prostaglandin E2 secreted by AEC2 inhibits fibroblasts’ chemotaxis and proliferation. We modelled this phenomenon using a reverse Hill function with coefficient 1 (to avoid abrupt changes in the fibroblasts’ flow) multiplied by a constant rate. We assumed the Hill constant to be equal to the initial number of AEC2 per alveolus to reduce the number of unknown parameters and tuned the constant influx rate to keep the number of fibroblasts in homeostatic conditions constant;

#### 4.3. Cell Behaviours

#### 4.3.1. Secretion

- M2: PDGF (constant rate), MMP (constant rate), TIMP (constant rate), IL13 (constant rate), TGFβ
_{active}(rate is positively affected by IL13 concentration); - M1: TNFα (constant rate);
- F: TGFβ
_{inactive}(constant rate), ECM (rate is affected with inverse proportionality by ECM concentration; no secretion if the concentration exceeds a saturation threshold); - MF: ECM (rate is positively affected by TGFβ
_{active}concentration); - Senescent AEC2: TNFα (constant rate), MCP1 (constant rate), FGF2 (rate is positively affected by TGFβ
_{active}concentration).

#### 4.3.2. Damage Spreading and Clearance

_{active}in the process [67,68,69,70], we did not make any assumptions on the signalling molecules and instead modelled the intensity of the signal more generally using the intercellular distance. As in [66], our implementation relies on a threshold mechanism: when the number of senescent cells in the neighbourhood of a healthy one exceeds a threshold (which was tuned together with the senescent clearance mechanism described below to ensure complete recovery for low fractions of damaged cells), the “time above threshold” τ of the healthy cell is increased by one unit and decreased otherwise (if greater than 0). Since the neighbourhood of the healthy cell is defined as the group of cells within a two AEC2 diameter distance (to simulate short-range interaction), but not limited to the same alveolus, this mechanism allows for both intra- and inter-alveolus damage spreading. Finally, for a healthy cell with τ > 0, the probability of transitioning to the damaged type is given by:

#### 4.3.3. Migration

- The epithelial cell detects all its neighbours within an alveolar radius distance in the alveolus it belongs to (AEC1 detect other only AEC1, while AEC2 detect both AEC1 and AEC2);
- For each of the detected neighbours, the distance from the acting cell is measured;
- Using the distances from the step above, the cell determines the centre of mass of its neighbours, where each distance is weighted with the inverse of its squared normalized value;
- The cell migrates in the opposite direction from the location of the centre of mass, given a constant migration speed and the length of the simulation step.

#### 4.3.4. Proliferation

_{active}, IL13, and FGF2, the latter being present only in the presence of senescent AEC2 [78,79,80].

#### 4.3.5. Differentiation

_{active}[83].

#### 4.3.6. Apoptosis

#### 4.4. Initial Conditions

#### 4.4.1. Homeostasis

#### 4.4.2. Radiation-Induced Damage

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Visual output of the 3D ABM of a human alveolar duct. The structure is made up of 3 stacked layers, and each layer consists of 6 tangent alveoli. The centres of the alveoli are located on equidistant circles with a radius equal to the duct radius. Cells are represented as coloured spheres, with green = AEC2 cells, blue = AEC1 cells, brown = M1 macrophages, orange = M2 macrophages, light blue = fibroblasts, red = myofibroblasts.

**Figure 2.**Time evolution of the extracellular substances for multiple doses. Peaks in the MCP-1, TNFα, and FGF2 concentration, secreted by senescent AEC2, are observed immediately after the irradiation (see [38]). As shown in [39], a dose-dependent increase in the concentration of TGFβ (and the other macrophage-derived cytokines) is observed, followed by a restoration of the homeostatic levels in the late fibrotic phase as the senescent cells are fully cleared. A characteristic two-component pattern, where an initial dose-dependent peak is followed by a settlement, can be seen in the ECM concentration (as observed in [34]). The ECM, secreted by both fibroblasts and myofibroblasts, returns to homeostatic levels in the late phase only for doses ≤5 Gy.

**Figure 3.**Time evolution of the total number of cells for multiple doses. As shown in [5], the AEC2 population is fully restored at low doses, while above 5 Gy, the increase in the number of senescent AEC2 (which grows with the dose) leads to the complete depletion of the healthy ones in a fraction of the alveoli (see Section 2.2). Consequently, the total number of healthy AEC2 decreases and then settles, followed by a decay in the AEC1 population. M1 macrophages, triggered by the damaged epithelium, accumulate proportionally to the number of senescent AEC2 and later on differentiate into M2 macrophages. In the later stage of fibrosis, following the clearance of the senescent cells, a decrease in the macrophage population is observed [40,41]. Finally, the population of mesenchymal cells, whose proliferation is stimulated by macrophages and senescent AEC2 in the early phases, settles at higher levels (with respect to homeostasis) in the alveoli depleted of healthy AEC2.

**Figure 4.**Dose-response curve for the early ECM concentration increase after 3 months with phagocytic fraction = 100% (n = number of experiments).

**Figure 5.**Dose-response curve for the ECM concentration increase after 1000 days with phagocytic fraction = 100% (n = number of experiments).

**Figure 6.**Alveoli survival after 1000 days with phagocytic fraction = 100% (n = number of experiments).

**Figure 7.**RILI severity index after 1000 days with phagocytic fraction = 100% (n = number of experiments).

**Figure 8.**Alveoli survival after 1000 days with phagocytic fraction = 100% and different senescent to damaged AEC2 ratios. Senescent/Damaged = x% means that, after the irradiation, a fraction x of the healthy AEC2 shifted to the senescent state, while the remaining 1 − x fraction underwent apoptosis.

**Figure 9.**Experimental data of rats AEC2 survival from [47]. Data were fitted using the LQ model (in red) and the curve was used in the ABM to convert the simulated damaged fractions into the corresponding delivered doses.

Category | Type | Initial Number per Alveolus | Source |
---|---|---|---|

Epithelial | AEC1 | 41 | [60] |

AEC2 healthy | 69 | [60] | |

AEC2 damaged | 0 | ||

AEC2 senescent | 0 | ||

Macrophage | Macrophage M1 | 6 ^{1} | [60] |

Macrophage M2 | 6 ^{1} | [60] | |

Mesenchymal | Fibroblast | 24 ^{2} | [46,60] |

Myofibroblast | 36 ^{2} | [46,60] |

^{1}Contrary to our previous model, we include only alveolar macrophages (M1 and M2) and neglect interstitial macrophages. We assume that M1 and M2 are equally distributed in homeostatic conditions.

^{2}See our previous work [21].

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**MDPI and ACS Style**

Cogno, N.; Bauer, R.; Durante, M. An Agent-Based Model of Radiation-Induced Lung Fibrosis. *Int. J. Mol. Sci.* **2022**, *23*, 13920.
https://doi.org/10.3390/ijms232213920

**AMA Style**

Cogno N, Bauer R, Durante M. An Agent-Based Model of Radiation-Induced Lung Fibrosis. *International Journal of Molecular Sciences*. 2022; 23(22):13920.
https://doi.org/10.3390/ijms232213920

**Chicago/Turabian Style**

Cogno, Nicolò, Roman Bauer, and Marco Durante. 2022. "An Agent-Based Model of Radiation-Induced Lung Fibrosis" *International Journal of Molecular Sciences* 23, no. 22: 13920.
https://doi.org/10.3390/ijms232213920