# On the Role of Mechanics in Chronic Lung Disease

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## Abstract

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## 1. Motivation

**Figure 1.**Airway wall remodeling in chronic lung disease. In contrast to the healthy airway wall, the airway wall in asthma, left, and chronic bronchitis, right, displays airway constriction, associated with smooth muscle thickening at the outer airway wall, and airway inflammation, associated with mucosal growth at the inner airway wall. These changes manifest themselves in an increase in pressure at the outer wall and an increase in volume at the inner wall, resulting in a narrowing of the lumen and progressive airflow obstruction; adapted from [1,4].

**Figure 2.**Schematic of the lungs. The trachea, the root of the respiratory tree, branches into two main bronchi, which enter the left and right lungs. In the lungs, the bronchial tree continues to branch in humans into 23 to 27 generations, yielding approximately 17 million branches. The final generation of terminal bronchioles opens into the alveolar space, where gas transfer occurs. Airway wall remodeling affects the small airways between the fourth and 14th generation. Healthy small airways are less than 2 mm in diameter, non-cartilaginous and compliant. In chronic lung disease, the small airways experience smooth muscle thickening and mucosal growth, resulting in progressive airflow obstruction.

**Figure 3.**Schematic of the airway wall. Small airways consist of three distinct layers, the mucosa, the submucosa and the smooth muscle layer. In chronic lung disease, the smooth muscle layer thickens and creates an elevated pressure at the outer wall, while the mucosal layer experiences inflammation and increases in volume at the inner wall. The airway wall folds inwards, resulting in progressive airflow obstruction.

analysis | microstructural mechanism | analytical solution onset of failure | numerical solution post-failure regime |
---|---|---|---|

symptoms | |||

airway constriction | smooth muscle thickening → increased pressure on submucosal layer | bifurcation analysis → folding pressure ${p}^{\mathrm{fold}}$ → number of folds ${n}^{\mathrm{fold}}$ [13,14,15,17] | finite element analysis → contact pressure ${p}^{\mathrm{crit}}$ → folded configuration [14,15] |

airway inflammation | mucosal inflammation → volume growth of mucosal layer | bifurcation analysis → folding growth ${\vartheta}^{\mathrm{fold}}$ → number of folds ${n}^{\mathrm{fold}}$ [8,13,16,17,23,24] | finite element analysis → contact growth ${\vartheta}^{\mathrm{crit}}$ → folded configuration [8,16,23,25,31,32] |

## 2. Methods

#### 2.1. Continuum Modeling of Growth

#### 2.2. Computational Modeling of Growth

## 3. Results

**Figure 4.**Branching airway segment. Along the bronchial tree, the airway cross varies substantially in geometry. We characterize each cross-section through the semi-major-to-semi-minor-axis ratio ${R}_{\mathrm{I}}:{R}_{\mathrm{II}}$ and through the mucosal-thickness-to-radius ratio ${t}_{\mathrm{m}}:R$. The ratio ${R}_{\mathrm{I}}:{R}_{\mathrm{II}}$ is larger close to and smaller away from a branching region. The ratio ${t}_{\mathrm{m}}:R$ is smaller close to and larger away from the trachea.

#### 3.1. Sensitivity of Failure Mode with Respect to Ellipticity

**Figure 5.**Sensitivity of the failure mode with respect to ellipticity. Smaller ellipticity ratios represent circular cross-sections away from a branching region; larger ellipticity ratios represent elliptical cross-sections close to a branching region. With increasing ellipticity, the heterogeneity of the failure mode increases. With increasing heterogeneity, the lumen area at the first contact remains significantly larger. This might indicate that circular cross-sections away from the branching region are at a higher risk of to airflow obstruction than elliptical cross-sections close to a branching region.

#### 3.2. Sensitivity of the Failure Mode with Respect to Relative Mucosal Thickness

**Figure 6.**Sensitivity of failure mode with respect to relative mucosal thickness for a circular cross-section with ${R}_{\mathrm{I}}:{R}_{\mathrm{II}}=1.0$. Larger relative mucosal thicknesses represents smaller bronchi away from the trachea; smaller relative mucosal thicknesses represents larger bronchi close to the trachea. With increasing relative mucosal thickness, the number of folds decreases. With the decreasing number of folds, the critical growth ϑ at the first contact becomes significantly smaller. This might indicate that smaller bronchi with a larger relative mucosal thickness are at a higher risk of airflow obstruction than larger bronchi with a smaller relative mucosal thickness.

#### 3.3. Sensitivity of Failure Mode with Respect to Ellipticity and Relative Mucosal Thickness

**Figure 7.**Sensitivity of the failure mode with respect to relative mucosal thickness for moderately elliptical cross-section with ${R}_{\mathrm{I}}:{R}_{\mathrm{II}}=1.5$. The instability occurs naturally at regions of highest curvature along the semi-major axis and propagates inward until it reaches the semi-minor axis. First contact occurs at regions of highest curvature and induces a moderately localized failure mode. This might indicate that airway obstruction is less drastic in moderately elliptical cross-sections than in circular cross-sections.

**Figure 8.**Sensitivity of the failure mode with respect to relative mucosal thickness for severely elliptical cross-section with ${R}_{\mathrm{I}}:{R}_{\mathrm{II}}=2.0$. The instability occurs naturally at regions of highest curvature along the semi-major axis and propagates inward until it reaches the semi-minor axis. First contact occurs at regions of highest curvature and induces a severely localized failure mode. This might indicate that airway obstruction is less drastic in severely elliptical cross-sections than in circular and moderately elliptical cross-sections.

**Figure 9.**Number of folds and critical growth for varying ellipticities and varying relative mucosal thicknesses. With increasing relative mucosal thickness, the number of folds and the critical growth at the first contact decrease. With increasing ellipticity, the number of folds increases, while the critical growth decreases. This might indicate that smaller airways are at a higher risk of airflow obstruction than larger airways and that circular cross-sections obstruct more drastically than elliptical cross-sections.

## 4. Discussion

#### 4.1. Comparison with Previous Studies

#### 4.2. Limitations

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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Eskandari, M.; Pfaller, M.R.; Kuhl, E. On the Role of Mechanics in Chronic Lung Disease. *Materials* **2013**, *6*, 5639-5658.
https://doi.org/10.3390/ma6125639

**AMA Style**

Eskandari M, Pfaller MR, Kuhl E. On the Role of Mechanics in Chronic Lung Disease. *Materials*. 2013; 6(12):5639-5658.
https://doi.org/10.3390/ma6125639

**Chicago/Turabian Style**

Eskandari, Mona, Martin R. Pfaller, and Ellen Kuhl. 2013. "On the Role of Mechanics in Chronic Lung Disease" *Materials* 6, no. 12: 5639-5658.
https://doi.org/10.3390/ma6125639