Steady and oscillatory flow in the human bronchial tree

In this study, we investigate the steady inhalation, steady exhalation, and oscillatory flow in a realistic airway geometry for physiologically relevant regimes ranging from quiet breathing to respiration under high frequency ventilation (HFV). We use magnetic resonance velocimetry to characterize and quantify three-dimensional (3D) velocity fields in 3D printed replicas of realistic bronchial trees. Expanding on previous studies [Jalal et al., Exp. Fluids 57, 148 (2016); Jalal et al., Phys. Rev. Fluids 3, 103101 (2018)] which focused on respiration in planar double bifurcation geometries, we compare levels of axial and lateral dispersion, and find that they exceed those found in the idealized models. Furthermore, we find that the secondary flows in realistic airways propagate deep in the bronchial tree and are stronger during exhalation as compared to inhalation, while the mean flow topology does not vary significantly between the two steady regimes. Under HFV, we note significant regions of flow reversal during the inhalation-exhalation and exhalation-inhalation transitions. This is found to be due to a difference in impedance (dominated by inertance) in the different regions of the lung and results in an asynchronous ventilation between the upper and lower lobes. This phenomenon, also known as pendulluft is demonstrated experimentally for the first time, using both Eulerian velocity fields and Lagrangian pathlines. Secondary flows are stronger in exhalation compared to inhalation and at the peak of the ventilation cycle, match the steady cases although the flow topology can be significantly different. Finally, the cycle-averaged drift velocity suggests that steady streaming, while not negligible, is not the main transport mechanism during high-frequency ventilation.

Figure 1

Silhouette of the considered bronchial tree segmented from MDCT chest scan of a healthy subject, with branches grouped by lobe. The tracheal cross section where the bulk velocity $U_0$ and hydraulic diameter $D_0$ are defined (marked with a dashed line) is $3D_0$ above the carina.

Figure 2

Flow rate at the model inlet across the ventilation cycle, as measured by MRV.

Figure 3

Reynolds number (a) and Womersley number (b) as a function of generation, for steady and oscillatory cases, respectively. The black lines indicate the theoretical trends according to Eqs. (1) and (2), respectively.

Figure 4

Isosurfaces of high velocity ($U_{\mathrm{ax}}=0.6^{*}{U}_{\mathrm{ax},\max }$, red) and zero streamwise velocity ($U_{\mathrm{ax}}=0$, blue) along the airway tree for steady inhalation at (a) $\mathrm{Re}=1500$ and (b) $\mathrm{Re}=4500$, and (c) exhalation at $\mathrm{Re}=1500$.

Figure 5

(a) $D$ parameter at all generations for the different steady flow cases. (b) Two selected cross-sections at $G0$ (the trachea) and $G3$, along which normalized axial velocity contours are shown [(c)–(f)].

Figure 6

(a) $E$ parameter at all generations for the different steady flow regimes; (b) cross section with normalized out-of-plane vorticity and in-plane velocity vectors at the location shown in (c) during steady exhalation; (d) normalized secondary flow velocities averaged along $G1$ in the realistic and idealized [12] models at steady inhalation and steady exhalation.

Figure 7

(b),(c) $D$ parameter and (d),(e) $E$ parameter plotted along (a) selected paths along the bronchial tree for steady inhalation and steady exhalation.

Figure 8

Normalized magnitude of axial and secondary velocities in the entire bronchial tree under steady inhalation for (a) $\mathrm{Re}=1500$ and (b) $\mathrm{Re}=4500$.

Figure 9

Velocity isosurfaces for successive phases of the inhalation and exhalation cycle at $\mathrm{Wo}=12$. Red and blue colors here indicate flow moving in the inspiratory and expiratory direction, respectively, at 50% of the maximum axial velocity.

Figure 10

(a) Locations of two tracheal cross sections taken, and corresponding contour plots of normalized axial velocity during (b),(d) exhalation-inhalation inversion and (c),(e) inhalation-exhalation inversion.

Figure 11

(a) Selected lobes at $G2$ to highlight (b) the phase lag of axial velocity during the ventilation cycle.

Figure 12

Asynchronous flow at (a) exhalation-inhalation inversion and (b) inhalation-exhalation inversion; (c),(d) sample Lagrangian trajectories color-coded according to the temporal abscissa $t$ along the ventilation cycle of period $T$.

Figure 13

(a) Location of right and left main bronchi ($G1$) highlighted; (b) normalized secondary velocities averaged along the main bronchi for realistic and idealized [12] airway models. For both cases, the peak Reynolds number and Womersley number are $\mathrm{Re}=1500$ and $\mathrm{Wo}=12$, respectively.

Figure 14

(a) Locations of selected cross-sections at the main bronchi in HFV conditions; cross sections of ${\mathrm{\Gamma }}_2$ at (b),(c) peak inhalation and (d),(e) peak exhalation, compared with (f),(g) steady inhalation and (h),(i) steady exhalation.

Figure 15

(a) Isosurfaces of positive-negative drift velocity $U_{\mathrm{D}}$ taken at $30\%$ of the maximum velocity in the domain; (b) contour plots of drift velocity along selected cross sections at the trachea and at the main bronchi.

Figure 16

Magnitude of $U_{\mathrm{D}}$, (i.e., the cross-section average of its absolute value) normalized by the tracheal bulk velocity $U_0$, evaluated along two selected bronchial pathways. Both paths start from the trachea ($G0$); one follows the right main bronchus ($G1$ right) and continues in the second generation in the right upper lobe ($G2$ RUL); the other follows the left main bronchus ($G1$ left) and continues in the second generation in the left lower lobe ($G2$ LLL). For comparison, the magnitude of $U_{\mathrm{D}}$ for the idealized model in Ref. [12] is also reported.