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# Homogenous_AirBlood_Exch

Periodic airway transport and alveolus-capillary gas exchange.

Model number: 0156

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## Description

 A homogeneous model for respiratory gas exchange is presented. It contains an alveolar sac (compartment)
connected to a 1D airway (cylinder). The alveolar sac exchanges species with a capillary
(cylinder) through a single layer. The model can predict time dependent airway concentration
profiles in the presence of bidirectional air flow over a single or multiple breath time
span. Source of species can be set either in air or in blood.



## Equations

$\large \frac{{\partial C_{c}}}{\partial t}\,=\, -\, \frac{F_c}{V_c}\,\frac{\partial C_c}{\partial x} \, + \, \frac{D_c}{L^2}\,\frac{{\partial}^2 C_c}{\partial x^2} \, - \, \frac{PS_{c\,if}}{V_c} \, \left ( C_c - C_{if} \right ) \, - \, \frac{G_c}{V_c} \, C_c$

$\large \frac{{\partial C_{if}}}{\partial t}\,=\, \frac{D_{if}}{L^2}\,\frac{{\partial}^2 C_{if}}{\partial x^2} \, + \, \frac{PS_{c\,if}}{V_{if}} \, \left ( C_c - C_{if} \right ) \, - \, \frac{q_{ifx}}{V_{if}} \, - \, \frac{G_{if}}{V_{if}} \, C_{if}$

$\large \frac{{\partial C_{a}}}{\partial t}\,=\, -\, \frac{F_a}{V_a}\,\frac{\partial C_a}{\partial y} \, + \, \frac{D_a}{{L_a}^2}\,\frac{{\partial}^2 C_a}{\partial y^2} \, - \, \frac{G_a}{V_a} \, C_a$

$\large \frac{d V_{al}}{d t}\,=\, F_a \, + \, \frac{q_{if}}{C_{tot}}$

$\large {V_{al}} \, \frac{d C_{al}}{d t} \, + \, C_{al} \, \frac{d V_{al}}{d t} \,=\, q_{if} \, + \, F_a \, \left( \left(1-H\left(F_a\right)\right)\,C_{al} \, + \, H\left(F_a\right)\,C_{a}\left(t,1\right)\right)$

$\large q_{ifx} \, = \, PS_{if\,al} \, \left(C_{if}-C_{al}\right)$

$\large q_{if} \, = \, \int_0^1{q_{ifx}dx}$

where $\large C$ is the species concentration, $\large V$ is the volume, $\large F$ is the volume flow rate, $\large D$ is the diffusion coefficient, $\large L$ is the length, $\large PS$ is the pereability-surface product, $\large G$ is the reaction rate, $\large q_{ifx}$ is the specie flux per unit length of capillary-alveoli interface, $\large q_{if}$ is the integrated specie flux through capillary-alveoli interface, $\large H$ is the Heaviside function, index $\large c$ refers to the capillary, index $\large if$ refers to the capillary-alveoli interface, index $\large al$ refers to the alveolus, and index $\large a$ refers to the airway.

The equations for this model may also 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.

## References

 Hlastala MP, A Model of Fluctuating Alveolar Gas Exchange During the Respiratory Cycle.
Respir. Physiol. 15: 214-232, 1972



## Key Terms

gas exchange, alveolar exchange, pulmonary circulation, bronchial circulation, pulmonary capillaries, airway, alveolar sac, PDE

## Model History

Get Model history in CVS.

Posted by: bej

## Acknowledgements

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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.