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Diffusion and Convection in Intrapulmonary Gas Mixing

Johannes Piiper, Peter Scheid

10.1002/cphy.cp030404

Source: Supplement 13: Handbook of Physiology, The Respiratory System, Gas Exchange

Originally published: 1987

Published online: January 2011

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Diffusion
1.1 Diffusion Laws
1.2 Binary Diffusion
1.3 Diffusion in Multicomponent Systems
2 Diffusion, Convection, and Their Interactions
2.1 Dispersion
2.2 Attempts to Quantify Dispersion in Upper Airways
2.3 Dispersion During High‐Frequency Oscillation
2.4 Convective Mixing by Mechanical Action of the Heart
3 Anatomical Basis for Models of Lung Gas Mixing
3.1 Regularly Branching Lung Model
3.2 Irrregularly Branching Lung Model
3.3 Conclusion
4 Mathematical Analysis of Lung Gas Mixing
4.1 Models Considering Diffusive Mixing Alone
4.2 Models Considering Convection and Diffusion
5 Experimental Evidence for Stratification in Lungs
5.1 Effects of Series and Parallel Inhomogeneities
5.2 Methods and Results
5.3 Attempts at Quantification
6 Critical Remarks and Conclusions
6.1 Physiological Measurements Versus Morphometric Models
6.2 Dead Space Versus Stratification
6.3 Effects of Series and Parallel Inhomogeneity
6.4 Mixing in Overall Pulmonary Gas Transfer
Figure 1. Figure 1.

Schema of lung structure shows potential for parallel [ventilation‐perfusion ratio (VA/Q)] and series (stratified) inhomogeneities and their effects on efficiency of alveolar gas exchange. Levels of partial pressure of O2 and CO2 (Po2 and Pco2) in compartments 1 and 2 [arterialized blood (a) = alveolar gas (A)] are indicated. In parallel inhomogeneity, mixed alveolar gas () and arterial blood () result from ventilation‐weighted and perfusion‐weighted mixing, respectively. In series inhomogeneity, partial pressures in arterial blood () result from mixing, but values in expired alveolar gas (A′) are those of proximal compartment 1 (A1). In both cases, there arise alveolar‐arterial Po2 and Pco2 differences (AaD).
Figure 2. Figure 2.

Dispersion in a fluid moving steadily (mean velocity, ) in circular tube under laminar‐flow conditions. Left panels: tracer distribution [dots, introduced at t = 0 at x = 0 (t, time; x, length)] is displayed at t > 0, when imaginary plane, moving with , has moved to x = t. Right panels: mean tracer concentration, (x). A: dispersion of nondiffusible tracer occurs only by convection (left panel). Thin arrows, radial distribution of flow relative to moving plane. Heavy arrow, net convective tracer transport through this plane. Axial distribution of mean concentration (right panel) decreases linearly in axial range from x = 0 to x = 2 t. B: dispersion of diffusible tracer (Taylor dispersion). Broken parabolic line (left panel), denotes radial flow distribution (equal to distribution of nondiffusible tracer, see A). Radial diffusion (open arrows) leads to net convective tracer back‐transport through moving plane (closed small arrows). Hence net convective transport through moving plane in direction of movement (closed arrow) is reduced and so is axial dispersion (right panel, thin straight line for comparison with A). C: dispersion when radial diffusion is infinitely fast and axial diffusion is significant (disregarded in B). Net diffusive transport through moving plane (left panel, open arrow) leads to axial tracer dispersion (right panel).
Figure 3. Figure 3.

Morphometric lung model according to Weibel 164. A: equivalent radius plotted against axial distance. Same length scale for both axes. B: details of terminal airways (acinus) resolved by expanding abscissa scale 100‐fold, retaining ordinate scale. Bottom schema shows representative path in acinus, which originates from one terminal bronchiole.

From Scheid and Piiper 133
Figure 4. Figure 4.

Concentration profiles during respiratory cycle in trumpet model. Linear abscissa scale for axial distance from terminal end of bronchial tree. Delimitation of generations shown on upper scale. Ordinate: O2 fraction (FO2). Single breath of O2 is inhaled into, and then exhaled from, an initially O2‐free lung with constant flow. Solid lines, concentration profiles at successive equally timed periods during inspiration (1–5) and expiration 6,7,8,9,10. Dashed‐dotted lines 1,2,3,4,5, tidal volume front (i.e., concentration front for nondiffusible tracer) during inspiration. Dashed line 6, concentration profile calculated assuming no (axial) diffusion during expiration.

Adapted from Paiva 102
Figure 5. Figure 5.

Branched‐trumpet model. Two trumpets branch off a common stem. By cutting off from end of second trumpet, ratio of both trumpet volumes (V2/V1) can be made to vary. S and s, cross‐sectional area of trumpet branches with and without alveoli, respectively.

From Paiva and Engel 106
Figure 6. Figure 6.

Profiles of O2 concentration during expiration, after single breath of O2 (see Fig. 4), calculated for branched‐trumpet model with uneven volume ratio (V2/V1 = 0.12). Abscissa: distance, x, from the terminal surface. Ordinate: fractional O2 concentration, F(x,t), as function of distance x (abscissa) and time t (parameter). Dashed lines refer to short unit; solid lines, long unit. Numbers 5–10, concentration profiles at successive equally timed periods during expiration. Inset: expirogram. Ordinate: fractional O2 concentration, F(x1v), at model exit (generation 13/14).

From Paiva and Engel 106
Figure 7. Figure 7.

Two‐compartment models of series and parallel inhomogeneities. Gas‐transfer conductance between compartments 1 and 2 of series model is indicated (0, finite, or ∞). Thickness of arrows in parallel models denotes magnitude of ventilation (A) and perfusion (). For inhomogeneities, a general case and 2 limiting cases, shunt [giving rise to shuntlike effects, () > (PaCO2 ‐ PACO2)] and alveolar dead space [producing alveolar dead space‐like effects, () ≈ (PaCO2 ‐ PACO2)] are shown.


Figure 1.

Schema of lung structure shows potential for parallel [ventilation‐perfusion ratio (VA/Q)] and series (stratified) inhomogeneities and their effects on efficiency of alveolar gas exchange. Levels of partial pressure of O2 and CO2 (Po2 and Pco2) in compartments 1 and 2 [arterialized blood (a) = alveolar gas (A)] are indicated. In parallel inhomogeneity, mixed alveolar gas () and arterial blood () result from ventilation‐weighted and perfusion‐weighted mixing, respectively. In series inhomogeneity, partial pressures in arterial blood () result from mixing, but values in expired alveolar gas (A′) are those of proximal compartment 1 (A1). In both cases, there arise alveolar‐arterial Po2 and Pco2 differences (AaD).



Figure 2.

Dispersion in a fluid moving steadily (mean velocity, ) in circular tube under laminar‐flow conditions. Left panels: tracer distribution [dots, introduced at t = 0 at x = 0 (t, time; x, length)] is displayed at t > 0, when imaginary plane, moving with , has moved to x = t. Right panels: mean tracer concentration, (x). A: dispersion of nondiffusible tracer occurs only by convection (left panel). Thin arrows, radial distribution of flow relative to moving plane. Heavy arrow, net convective tracer transport through this plane. Axial distribution of mean concentration (right panel) decreases linearly in axial range from x = 0 to x = 2 t. B: dispersion of diffusible tracer (Taylor dispersion). Broken parabolic line (left panel), denotes radial flow distribution (equal to distribution of nondiffusible tracer, see A). Radial diffusion (open arrows) leads to net convective tracer back‐transport through moving plane (closed small arrows). Hence net convective transport through moving plane in direction of movement (closed arrow) is reduced and so is axial dispersion (right panel, thin straight line for comparison with A). C: dispersion when radial diffusion is infinitely fast and axial diffusion is significant (disregarded in B). Net diffusive transport through moving plane (left panel, open arrow) leads to axial tracer dispersion (right panel).



Figure 3.

Morphometric lung model according to Weibel 164. A: equivalent radius plotted against axial distance. Same length scale for both axes. B: details of terminal airways (acinus) resolved by expanding abscissa scale 100‐fold, retaining ordinate scale. Bottom schema shows representative path in acinus, which originates from one terminal bronchiole.

From Scheid and Piiper 133


Figure 4.

Concentration profiles during respiratory cycle in trumpet model. Linear abscissa scale for axial distance from terminal end of bronchial tree. Delimitation of generations shown on upper scale. Ordinate: O2 fraction (FO2). Single breath of O2 is inhaled into, and then exhaled from, an initially O2‐free lung with constant flow. Solid lines, concentration profiles at successive equally timed periods during inspiration (1–5) and expiration 6,7,8,9,10. Dashed‐dotted lines 1,2,3,4,5, tidal volume front (i.e., concentration front for nondiffusible tracer) during inspiration. Dashed line 6, concentration profile calculated assuming no (axial) diffusion during expiration.

Adapted from Paiva 102


Figure 5.

Branched‐trumpet model. Two trumpets branch off a common stem. By cutting off from end of second trumpet, ratio of both trumpet volumes (V2/V1) can be made to vary. S and s, cross‐sectional area of trumpet branches with and without alveoli, respectively.

From Paiva and Engel 106


Figure 6.

Profiles of O2 concentration during expiration, after single breath of O2 (see Fig. 4), calculated for branched‐trumpet model with uneven volume ratio (V2/V1 = 0.12). Abscissa: distance, x, from the terminal surface. Ordinate: fractional O2 concentration, F(x,t), as function of distance x (abscissa) and time t (parameter). Dashed lines refer to short unit; solid lines, long unit. Numbers 5–10, concentration profiles at successive equally timed periods during expiration. Inset: expirogram. Ordinate: fractional O2 concentration, F(x1v), at model exit (generation 13/14).

From Paiva and Engel 106


Figure 7.

Two‐compartment models of series and parallel inhomogeneities. Gas‐transfer conductance between compartments 1 and 2 of series model is indicated (0, finite, or ∞). Thickness of arrows in parallel models denotes magnitude of ventilation (A) and perfusion (). For inhomogeneities, a general case and 2 limiting cases, shunt [giving rise to shuntlike effects, () > (PaCO2 ‐ PACO2)] and alveolar dead space [producing alveolar dead space‐like effects, () ≈ (PaCO2 ‐ PACO2)] are shown.

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Article Title: Diffusion and Convection in Intrapulmonary Gas Mixing
 

How to Cite

Johannes Piiper, Peter Scheid. Diffusion and Convection in Intrapulmonary Gas Mixing. Compr Physiol 2011, Supplement 13: Handbook of Physiology, The Respiratory System, Gas Exchange: 51-69. First published in print 1987. doi: 10.1002/cphy.cp030404