Homothety ratio of airway diameters and site of airway resistance in healthy and COPD subjects

Highlights

• The homothety ratio, constant over generations, is the diameter of child/parent bronchus.

• A decrease in the homothety ratio explains the shift in airway resistance in COPD.

• A formula of resistance using tracheal dimensions and homothety ratio is provided.

Abstract

Our objective was to evaluate whether a decrease in the homothety ratio (h: diameter of child/parent bronchus, constant over generations) explains the shift in airway resistance toward periphery in chronic obstructive pulmonary disease (COPD). Using a validated computational model of fluid motion, we determined that reduced values of h (<0.76) were associated with a shift in resistance toward periphery. The calculated luminal diameters of terminal bronchioles using normal h (0.80–0.85) or reduced h (0.70–0.75) fitted well with measured micro-CT values obtained by McDonough et al. (N. Engl. J. Med., 2011; 365:1567–75) in control and COPD patients, respectively. A semi-analytic formula of resistance using tracheal dimensions and h was developed, and using experimental data (tracheal area and h from patients [Bokov et al., Respir. Physiol. Neurobiol., 2010; 173:1–10]), we verified the agreement between measured and calculated resistance (r = 0.42). In conclusion, the remodeling process of COPD may reduce h and explain the shift in resistance toward lung periphery.

Introduction

For some time, the measurement of airway resistance (Raw) has been considered the “gold standard” by physiologists for assessing airway obstruction due to its physiological background (Dubois et al., 1956). However, Raw measurement has recently become less popular in pulmonary function test laboratories because it requires complex equipment (body plethysmograph) and, more importantly, because the site of obstruction (i.e., the bronchial generations that are the most resistive) that is assessed is deemed to be very proximal. Along this line, the American Thoracic Society and European Respiratory Society have stated in their recommendations for pulmonary function tests that “airflow resistance is more sensitive for detecting narrowing of extrathoracic or large central intrathoracic airways than of more peripheral intrathoracic airways” (Pellegrino et al., 2005). Nevertheless, direct measurement of the distribution of resistance in the lower respiratory tract has long established that small airways (i.e., <2 mm in internal diameter) become the major sites of obstruction in patients with chronic obstructive pulmonary disease (COPD) (Hogg et al., 1968). Resistance to flow through tubes is inversely related to the reduction in the radius raised to the fourth power for fully developed laminar flow. Since loss of half of such airways will only double the total peripheral resistance because of their parallel arrangement, an increase in peripheral airway resistance by a factor of 4–40, as has been reported in patients with COPD (Hogg et al., 1968), has recently been explained by both generalized narrowing and loss of airways (McDonough et al., 2011).

On theoretical grounds, sections of the entire airway tree can be described by only two parameters, the tracheal diameter and the homothety ratio, which is a constant parameter describing the subsequent reduction in the airway lumen (h: diameter of the child/parent bronchus) at each bronchial generation. Considering the lower part of the bronchial tree (generations 6–16) and assuming that air flow in this duct system obeys Poiseuille's law (a good approximation below the sixth generation at rest), a “best” structure can be deduced by minimizing the total viscous dissipation in a finite tree volume. A purely mathematical argument, the Hess–Murray law, suggests that the best tree is fractal with a fractal dimension equal to 3. In such an ideal tree, the successive airway segments are homothetic with a size ratio equal to 0.79. Mauroy et al. have suggested that the morphology of the human bronchial tree is close to providing maximal efficiency in assuring air distribution with minimal viscous dissipation (with an average homothety ratio of 0.85) (Mauroy et al., 2004). This homothety ratio has been measured in a limited number of human casts and, more recently, using lung CT scans in a larger number of subjects (Bokov et al., 2010, Montaudon et al., 2007, Tawhai et al., 2004). Using CT scans of COPD patients, we have suggested that the remodeling process could decrease this ratio in intralobar airways, thereby increasing airway resistance (Bokov et al., 2010). We now hypothesize that this decrease in the homothety ratio, leading to a generalized narrowing of the airways, may explain the shift of airway resistance toward the lung periphery in COPD patients, as demonstrated experimentally by Hogg et al. (1968).

The objectives of our study were to provide a formula linking airway resistance, tracheal characteristics (length, section) and the homothety ratio while taking into account inertia related to non-fully developed laminar flow in the bronchial tree, and to assess the consequences of homothety ratio changes on the site of airway resistance on both theoretical and experimental (data from COPD patients) grounds.