Gas Molecule Density in the Lung Alveoli
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Diffusion is a vital process in human physiology. Diffusion of gas particles from the lung alveoli to the pulmonary capillaries enables the exchange of 
out of the blood and 
into the blood. The Smoluchowski differential equation models the probability distribution of alveolar gas particles in space and time. A 3D graph shows the results.
Contributed by: Abbey Todd, Rithika Treasa John, Bhavana Iyengar and Uma Malde (January 2024)
Open content licensed under CC BY-NC-SA
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This Demonstration considers the diffusion of gas particles from the lung alveoli to the pulmonary capillaries. An initial particle density 
is represented by a Gaussian distribution centered around 
. This provides the initial condition for the Smoluchowski differential equation:
to give the distribution 
with the potential
.
is the potential acting on the particle density at time 
along the alveoli–pulmonary capillary coordinate. You can choose the parameters 
, 
and 
; 
represents the thickness of the capillary that the molecule is diffusing into. The thicker the capillary, the less particle density will diffuse into the bloodstream. The variable 
represents the air pressure from the lungs. Note that as air pressure increases, more particles diffuse across the membrane. The variable 
is the permeability of the capillary wall. The more permeable the capillary, the more the gas molecule can diffuse into the bloodstream.
When 
, the particles are within the alveoli; when 
, the particles are within the pulmonary capillary.
Reference
[1] K. Schulten. "Smoluchowski Diffusion Equation." University of Illinois Urbana-Champaign. (Nov 20, 2023) www.ks.uiuc.edu/Services/Class/PHYS498/LectureNotes/chp4.pdf.
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