Steady-state air-blood exchange for two geometric configurations.
Steady-state air blood exchange is presented for two different hypothetical geometric configurations, i.e., for co-current and counter-current air-blood exchange. Alveoli-capillary interconnection is modeled as two co-centric cylinders with air and blood flowing in the same direction for co-current flow and in the opposite direction in the counter-current flow.
If diffusion and reaction is neglected the equations of the system can be written in the following form
where is the concentration is the permeability-surface product, is the volumetric flow rate, is the non-dimensional spatial coordinate, is the index referring to capillary, and is the index of alveoli.
The set of ODE can be solved as an initial value problem in case of co-current flow and as a boundary value problem in case of counter-current flow. Accordingly, the following analytical solutions can be obtain
in case of co-current flow and
in case of counter-current flow, where
the secondary subscripts and refers to the conditions imposed at the opposite ends of the cylinder. Integration of the difference over the length of the cylinder yields the total mass exchange rate from air to blood. Thus, the following relations between mass exchange rate and permeability-surface product can be obtained
for co-current flow and
The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.
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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.