Estimates of reflux velocity and pressure.

<p>A. Partial occlusion moving distally pushes fluid proximally (reflux), and creates a pressure gradient across the stenosis. B. At the stenosis, average reflux velocity </p><p></p><p></p><p></p><p></p><p><mi>v</mi><mo>¯</mo></p><p><mi>r</mi><mi>e</mi><mi>f</mi></p><p></p><mo>=</mo><p><mi>v</mi></p><p><mi>p</mi><mi>e</mi><mi>r</mi></p><p></p><mo>⋅</mo><p></p><p></p><p><mo>(</mo></p><p><mn>1</mn><mo>−</mo></p><p></p><p></p><p><mo>(</mo></p><p><mn>1</mn><mo>−</mo><mi>O</mi></p><mo>)</mo><p></p><p></p><mn>2</mn><p></p><p></p><mo>)</mo><p></p><p></p><mo>/</mo><p></p><p></p><p></p><p><mo>(</mo></p><p><mn>1</mn><mo>−</mo><mi>O</mi></p><mo>)</mo><p></p><p></p><mn>2</mn><p></p><p></p><p></p><p></p><p></p><p></p> is proportional to velocity of peristaltic wave <i>v</i>_<i>per</i>, but increases rapidly with occlusion (dashed curve). Pressure gradient across the stenosis is proportional to fluid viscosity <i>μ</i> and strongly depends on occlusion <i>O</i>: <p></p><p></p><p></p><p></p><p><mi>d</mi><mi>p</mi></p><p><mi>d</mi><mi>x</mi></p><p></p><mo>=</mo><p></p><p><mn>8</mn><mi>μ</mi></p><p><mi>v</mi></p><p><mi>p</mi><mi>e</mi><mi>r</mi></p><p></p><p></p><p></p><p><mi>a</mi><mn>2</mn></p><p></p><p></p><mo>⋅</mo><p></p><p><mn>1</mn><mo>−</mo></p><p></p><p></p><p><mo>(</mo></p><p><mn>1</mn><mo>−</mo><mi>O</mi></p><mo>)</mo><p></p><p></p><mn>2</mn><p></p><p></p><p></p><p></p><p></p><p><mo>(</mo></p><p><mn>1</mn><mo>−</mo><mi>O</mi></p><mo>)</mo><p></p><p></p><mn>4</mn><p></p><p></p><p></p><p></p><p></p><p></p>, where <i>a</i> is the relaxed lumen radius (solid curve).<p></p

Parameters and variables used in estimates and computational model.

<p>¹ No studies report stiffness of embryonic lung tissue. Range is an estimate. Lower bound: 20 Pa for amphibian embryos; upper bound 400 Pa for ASM cells in vitro.</p><p>² We assume that the viscosity of airway lumen fluid in the embryo is lower than that of neonatal airway mucus but higher than that of blood.</p><p>³ Fetal pig airway SM 1–20 kPa, highest in trachea, lowest in bronchioles. We assume this as an upper bound, and that embryonic SM will likely be weaker by 1–2 orders of magnitude. We assume a SM thickness of 15 microns.</p><p>⁴ Fetal pig, pseudoglandular stage</p><p>⁵ Fetal mouse (lowest value).</p><p>⁶ Rabbit fetus, static pressure.</p><p>⁷ Fetal sheep, static pressure.</p><p>Parameters and variables used in estimates and computational model.</p

Geometry of embryonic lung and model.

<p>A. Explanted E11.5 mouse lung showing lumen (l), epithelium (e, green), and mesenchyme (m, red). Smooth muscle (sm) not visible. B. Embryonic lung idealized as unbranched, axisymmetric tubule, with three uniform tissue layers plus lumen. Smooth muscle undergoes active circumferential contraction wave (red), propagating distally, building lumen pressure ahead of it (blue).</p

Diffusion coefficients (μm²/s) of various molecules in various fluids.

<p>Diffusion coefficients in μm²/s. Viscosities (Pa-s) of water 0.001, mouse embryonic lung lumen fluid (this paper) 0.016, neonatal mucus 0.4, adult mucus 3000.</p><p>Diffusion coefficients (μm²/s) of various molecules in various fluids.</p

Frames from model simulations of AP with partial occlusion, for open and closed trachea.

<p>Each frame shows half the symmetric tubule. A. Closed trachea. Lumen pressure is spatially uniform and increases as soon as AP begins. B. Open trachea. Lumen pressure is negligible until occlusion is almost complete. Pressure is uniform everywhere in the lumen except at stenosis, where flow is fastest. Maximal occlusion shown ~ 90%. C. Detail of open-trachea AP. Maximal occlusion precedes maximal pressure. Pressure distal to pinch forces fluid leakage and reduces occlusion as wave moves distally. Identical parameters (stiffness, viscosity, force input). Frames every 1.0 sec (A, B) and 0.5 sec (C).</p

Velocity and shear rate.

<p>Lumen fluid velocity at the midline (solid curves) and shear rate at the lumen surface (dashed curves) track each other in time (horiz. axis). Curves correspond to locations on airway at left. Red dots indicate location, relative magnitude, and time of SM force peak. Each curve shows time series of fluid velocity and shear rate. Maximal flow at a position occurs slightly after maximal SM force at that position. Flow is fastest towards trachea, opposite the direction of peristaltic SM wave; refilling flows are slower. Flow distal to SM is negligible. Flow is dramatically reduced in the closed-end airway.</p

Peristaltic wave dramatically stretches fluid layers adjacent to the occlusion, while modestly affecting distal fluid.

<p>If the trachea is open, mixing is much more dramatic than if the trachea is closed. Even for the closed trachea, fluid markers do not return precisely to their original locations despite the low Reynolds number. The spatiotemporal asymmetry of the waveform results in mixing.</p

Time scales of transport in the embryonic lung.

<p>In the absence of flow, solutes can only diffuse (dotted curve). In the absence of diffusion, solutes and particles advect with the flow (dashed curve). Advection-diffusion (solid curve) transports solutes rapidly relative to diffusion alone, and a small occlusion from weak airway peristalsis can yield a dramatic reduction in transport time. 100 kDa globular protein in lumen fluid of measured viscosity 0.016 Pa-s.</p