# A 3D Agent-Based Model of Lung Fibrosis

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## Abstract

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## 1. Introduction

_{a}[13,14], platelet-derived growth factor, PDGF [15], interleukin 13, IL13 [16], matrix metalloproteinases, MMP [17], and tissue inhibitors of metalloproteinases, TIMPs [17] which affect several cellular species. Among these, fibroblasts are mesenchymal cells that secrete inactive TGFβ (TGFβ

_{i}) [18], the ECM [19], and are able to differentiate into myofibroblasts. The ECM is degraded by the MMP, whose effect is inhibited by TIMP [20]. TGFβ (which can be activated by damaged AEC2 [21] and whose secretion by M2 is amplified by IL13 [22]), in its active form, increases the proliferation of fibroblasts [23], stimulates their differentiation into myofibroblasts [24], and damages healthy AEC2 cells [25]. IL13 further enhances the proliferation of fibroblasts [26] together with the basic fibroblast growth factor (bFGF or FGF2) secreted by the AEC2 [27,28], while PDGF triggers the differentiation of the fibroblasts [29]. Myofibroblasts deposit additional ECM [30] (a mechanism intensified by TGFβ [31]) and have a pivotal role in the remodeling process that takes place in the alveolar region. The population of mesenchymal cells can be further expanded through the epithelial–mesenchymal transition (EMT), a process by which activated AEC2 differentiate into fibroblasts and/or myofibroblasts [32]. The overall effect is an increase in the number of ECM-secreting cells that leads to the abnormal accumulation of the ECM.

## 2. Materials and Methods

#### 2.1. Framework and Implementation

- a MacBook Pro 2018 running macOS Big Sur on a 2.3 GHz Quad-Core Intel Core i5 processor with 8 GB RAM;
- a compute node of the Lichtenberg HPC system running CentOS 8.2 on 2x 2.3 GHz Intel Cascade-Lake AP 48-cores processor (96 total cores) with 384 GB RAM.

#### 2.2. Building the Simulation Space

- Given the number of generations Ngen, define the number of segments as Nseg = 2
^{Ngen}− 1, the average segment length as avgSegLength, and the vector containing the coordinates of all the agents as Coords. - Set the coordinates of the agent 0 (the first alveolar segment) to {0., 0., 0.} (i.e., the center of the simulation space), its unique index to 0, its origin to {0., 0., −avgSegLength/2}, and its end to {0., 0., avgSegLength/2}. Add {0., 0., 0.} to Coords.
- Loop (Nseg/2) times. At each step (starting from 1):
- a.
- Define the index of the father agent (i.e., the one from which branch 1 and branch 2 stem) as father = [(step + 1)/2] − 1.
- b.
- Project the coordinates of the father agent along its axis by avgSegLength.
- c.
- For each of the two new branches:
- i.
- Generate random polar and azimuthal angles θ and φ.
- ii.
- Rotate the projected father agent by θ and φ, keeping its origin fixed.
- iii.
- If the branch doesn’t overlap with the other agents already in the tree add its coordinates to Coords.
- iv.
- If the branch overlaps with any of the other agents already in the tree, go back to step i.

- Write Coords to a file, so that it can be used for multiple simulations.

^{3}(or 1 cm

^{3}). The ranges that we choose for θ and φ are [20°, 160°] and [45°, 89°], respectively, and are the smallest that allow the tree to fully develop without overlaps. Since the alveolar ducts could be further decomposed into multiple agents (i.e., the alveoli) made up of subagents (i.e., the cells, [54]), the agents in our model can be seen as meta-agents.

#### 2.3. Extracellular Mediators

^{3}) can enclose an agent (with diameter 700 µm). Moreover, we select the scheme “DiffuseWithOpenEdge” to make BioDynaMo run our simulations with open boundaries.

- the function used to define substances at the beginning of a simulation so that both the depleting substance and the binding coefficient can be specified;
- the function that implements the central difference method by embedding the local depletion (i.e., ${\mu}_{AB}\ast {A}_{i}$, where ${\mu}_{AB}$ is the binding coefficient and ${A}_{i}$ is the concentration of the depleting substance in the i-th box where the calculation is performed) into the decay term.

#### 2.4. Hybrid Multi-Agent-Based Model

#### 2.4.1. Secretion Behaviors

- PDGF/MMP/TIMP/IL13 secretion by M2, TNFα secretion by M1, TNFα/MCP1 secretion by active AEC2 (whose activation process is described in Section 2.4.4), and TGFβ secretion by fibroblasts have similar templates and depend on constant secretion rates.
- Both FGF2 secretion by active AEC2 and ECM secretion by myofibroblasts are increased by TGFβ. Similarly, TGFβ secretion by M2 is increased by IL13. In our model, these dynamic rates are expressed by

- As in [48], the constant secretion rate of ECM by fibroblasts is multiplied by the factor in Equation (6), where $EC{M}_{sat}$ is the value at which the ECM saturates. When $ECM>EC{M}_{sat}$, secretion is stopped.

- Activated AEC2 cells transform the latent form of TGFβ secreted by fibroblasts into its active form. Within the same time step, they reduce the local concentration of TGFβ
_{i}and increase that of TGFβ_{a}by the same amount given by Equation (7), where ${K}_{AEC2}$ is a saturation constant.

#### 2.4.2. Proliferation Behaviors

- The proliferation of AEC2 is governed by a constant rate that allows for the survival of both the AEC2 and AEC1 populations. To do so, at every time step, the AEC2 population increases by a constant fraction.
- Proliferation of fibroblasts depends on the number of healthy AEC2 in homeostasis and is further increased by damage-associated mediators such as FGF2, TGFβ
_{a}, and IL13. To uncouple the two mechanisms, we implement (i) the F_addition behavior by which the number of fibroblasts is incremented according to a fixed fraction of AEC2 (represented by the parameter ${\lambda}_{F,AEC2}$), and (ii) the F_proliferate that computes the fraction of newborn fibroblasts using the rate in Equation (8).

#### 2.4.3. Differentiation Behaviors

- In AEC21_Differentiation and M12_Differentiation, the phenotypes of constant fractions of AEC2 and M1 are changed so that the AEC1 and M2 populations in homeostasis can survive.
- M0 cells act as a reservoir for M1 cells (hence indirectly for M2 cells) both in homeostasis and in inflammatory conditions. In our model, we implement two different mechanisms within the same behavior to ensure that a minimum number of M1 is always maintained. We define a constant rate ${\lambda}_{M01}$ for the M0 to M1 differentiation and use this value only if the concentration of MCP1 is too low to provide the M1 cells’ baseline. As the inflammation develops and the MCP1 can sustain the growth of M1 cells, we use the differentiation rate in Equation (9), where the last factor ensures that M1 cells never exceed M0 cells, as described in [48]. As stated before, the principle of local information exchange is not violated since each alveolar duct agent records only its number of M0 and M1 cells. Therefore, ${\lambda}_{M01}$ may assume different values for different agents.

- Fibroblast to myofibroblast and M2 to M1 differentiation are implemented in F_MF_Differentiation and M21_Differentiation. Since both are triggered by extracellular mediators, their templates are similar, and the rates that describe the transitions are outlined in the following equations

#### 2.4.4. Activation Behaviors

#### 2.4.5. Apoptosis Behaviors

#### 2.5. Initial Conditions and Input/Output System

- An operation runs through all the agents, collects the number of cells for each cell type, and stores the information in a vector.
- Another operation exploits the agents as probes: it gathers their position, uses these positions to get the local concentration of all the substances, computes the average concentration for each substance, and finally stores the information in a vector (note that there is no measurement within the diffusion grid boxes where agents are not localized).

#### 2.6. Sensitivity Analysis

## 3. Results

#### 3.1. Homeostasis

_{m}for the two quantities in Figure S1.

#### 3.2. Inflammation

#### 3.3. Sensitivity Analysis

_{sat}, ${\lambda}_{M01}$, and ${\lambda}_{F,AEC2}$(definitions are provided in Section 2.4). Each parameter is altered by ±10%, and the characteristics (both the cell numbers and the substance concentrations) are measured after a simulated time frame of 30 days so that the steady-state is reached (in homeostatic conditions). The relative sensitivity of the simulation readouts to the parameters listed above is shown in Figure 7.

_{in}(secreted by F) are mostly sensitive to ${\lambda}_{F,AEC2}$.

## 4. Discussion

_{a}, which is reflected by different ECM patterns. The curves provided in Figure 6, with $\overline{f}=6.5\%,$ show that our model can qualitatively reproduce the results of Hao et al. [48]. Further, by perturbing the steady-state stability that characterizes the homeostatic situation, the symmetry of the system is broken, and we observe its evolution through an irreversible process towards its final state. We also show that increasing the fraction of damaged AEC2 leads to a corresponding rise in the concentration of ECM for $\overline{f}\le 40\%$. Conversely, for $\overline{f}>40\%,$ we observed progressively decreasing amounts of ECM. The reason for this is the strong coupling between AEC2 cells and fibroblasts: as the fraction of damaged AEC2 cells exceeds the aforementioned threshold, the healthy AEC2 cells are no longer able to sustain the population of fibroblasts, an effect that is not counteracted by the additional FGF2 secreted by the active AEC2 cells. We further investigated the effects of different fractions of damaged AEC2 cells on the ECM distribution by measuring its concentration during the last time step of a 300-day simulation. The results, provided in Figure 6 as density distribution histograms, show a right shift of the ECM distribution for $\overline{f}\le 40\%$ and a left shift for bigger values of $\overline{f}$.

## 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.**Schematic representation of the main cells, extracellular mediators, and pathways in lung fibrosis. AEC1 = alveolar epithelial cells type 1, AEC2 = alveolar epithelial cells type 2, AEC2

_{d}= damaged AEC2, AEC2

_{ap}= apoptotic AEC2, AEC2

_{ac}= activated AEC2, F = fibroblast, MF = myofibroblast, M1 = type 1 macrophage, M2 = type 2 macrophage, M0 = monocyte, MMP = matrix metalloproteinases, ECM = extracellular matrix, TIMP = tissue inhibitors of metalloproteinases, IL13 = interleukin 13, TGFβ

_{i}= inactive form of the transforming growth factor beta, TGFβ

_{a}= active form of the transforming growth factor beta, FGF2 = basic fibroblast growth factor, TNFα = tumor necrosis factor, PDGF = platelet-derived growth factor, MCP1 = monocyte chemoattractant protein 1.

**Figure 3.**Time evolution of the average MMP (

**a**) and TIMP (

**b**) concentrations in a stationary simplified model simulation with different $\Delta t$ values.

**Figure 4.**Time evolution of the extracellular mediators’ average local concentration (

**a**) and the average number of cells per alveolus (

**b**) in homeostatic conditions. Red bands show the steady-states from the literature ±10%.

**Figure 5.**Time evolution of the extracellular mediators’ average local concentration (

**a**) and the average number of cells per alveolus (

**b**) with initial heterogeneous damage. Different colors show different initial average fractions of damaged AEC2, and the thicker lines are used for $\overline{f}=6.25\%$.

**Figure 6.**ECM distribution histograms for different initial fractions of damaged AEC2 cells (damage is heterogeneous). Shown is the ECM concentration from the cell-hosting voxels of the diffusion grid at the last time step of a 300-days simulated time frame. The ECM concentration is used as a surrogate for the Hounsfield CT units, while the voxel percentages represent the volume fraction filled with a certain amount of ECM.

**Figure 7.**Relative parameter sensitivity map showing the impact of changes in the parameters on the characteristics. For each pair of characteristic c and parameter p, ${r}_{c,p}$ indicates the ratio between the sensitivity ${S}_{c,p}$ and the maximum absolute value of ${S}_{c,p}$ for that particular characteristic (i.e., computed over an entire row).

Extracellular Substance | Initial Concentration (g cm ^{−3}) | Diffusion Coefficient (cm ^{2} day^{−1}) | Decay Coefficient (day ^{−1}) | Source |
---|---|---|---|---|

TGFβ_{a} | 2.51 × 10^{−12} | 4.32 × 10^{−2} | 3.33 × 10^{2} | [48] |

TGFβ_{i} | 2.51 × 10^{−12} | 4.32 × 10^{−2} | 3.33 × 10^{2} | Estimated |

PDGF | 3.50 × 10^{−9} | 8.64 × 10^{−2} | 3.84 | [48] |

FGF2 | 0 | 5.62 × 10^{−2} | 1.66 | [55,56] |

TIMP | 5.74 × 10^{−10} | 4.32 × 10^{−2} | 21.60 | [48] |

ECM | 3.26 × 10^{−3} | 0 | 0.37 | [48] |

MMP | 3.66 × 10^{−8} | 4.32 × 10^{−2} | 4.32 | [48] |

TNFα | 2.50 × 10^{−8} | 1.29 × 10^{−2} | 55.45 | [48] |

IL13 | 3.20 × 10^{−8} | 1.08 × 10^{−2} | 12.47 | [48] |

MCP1 | 0 | 1.73 × 10^{−1} | 1.73 | [48] |

Cell Type | Cell Number per Alveolus ^{1} | Source |
---|---|---|

AEC 1 | 41 | [57] |

AEC 2 | 69 | [57] |

M1 | 13 | [57] |

M2 | 12 | [57] |

Fibroblasts | 24 | [48,57] |

Myofibroblasts | 36 ^{2} | [48,57] |

M0 ^{3} | 65 | [48,57] |

Secretion | Proliferation | Differentiation | Activation | Apoptosis | |
---|---|---|---|---|---|

AEC2_TNFaSecretion | M2_PDGFSecretion | F_Proliferate | AEC21_Differentiation | AEC2_Activation | Apoptosis |

AEC2_MCP1Secretion | M2_MMPSecretion | F_Addition | F_MF_Differentiation | ||

AEC2_FGF2Secretion | M2_TIMPSecretion | AEC2_Proliferate | M01_Differentiation | ||

AEC2_TGFbSecretion | M2_TGFbSecretion | M12_Differentiation | |||

F_TGFbSecretion | M2_IL13Secretion | M21_Differentiation | |||

F_ECMSecretion | M1_TNFaSecretion | ||||

MF_ECMSecretion |

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**MDPI and ACS Style**

Cogno, N.; Bauer, R.; Durante, M.
A 3D Agent-Based Model of Lung Fibrosis. *Symmetry* **2022**, *14*, 90.
https://doi.org/10.3390/sym14010090

**AMA Style**

Cogno N, Bauer R, Durante M.
A 3D Agent-Based Model of Lung Fibrosis. *Symmetry*. 2022; 14(1):90.
https://doi.org/10.3390/sym14010090

**Chicago/Turabian Style**

Cogno, Nicolò, Roman Bauer, and Marco Durante.
2022. "A 3D Agent-Based Model of Lung Fibrosis" *Symmetry* 14, no. 1: 90.
https://doi.org/10.3390/sym14010090