One of the questions arising in the Virtual Lung project is whether fluorescent dyes are a true indication of the velocity field within the PCL layer. Here are some results from numerical simulations.
The advection diffusion equation q_t + u q_x + v q_y = (a q_x)_x + (a q_y)_y is solved numerically with realistic values for all parameters as in the Matsui experiments.
(PDF file explaining model and algorithm)
The plots below show the transport of dye by advection and diffusion starting with a uniform distribution at t=0. The snapshots are at 5 second intervals and all dimensions are in microns. In order to reduce computational time the plots are made from a uniformly moving "window" with velocity U_ref=35 microns/sec. Velocity of mucus and top of PCL layer taken as U=39 microns/sec. The axis labels are hard to make out but are at 250 micron increments along x and at 50 micron increments along the depth (labeled y in this work). The PCL layer was taken as 7 microns thick and the mucus as 25 microns thick. Diffusivity within the mucus is a=3.6 micron^2/sec. The diffusivity in the PCL can be taken constant or varying according to some law across depth (linear, parabolic, etc.) When taken constant, the value a=160 micron^2/sec from the Matsui experiments is used. For comparison computations with no diffusivity contrast between layers are also presented.
comparison between the hypothetical situation of identical diffusivities in mucus and PCL and the true situation in which the diffusivity in PCL is ~45 times that of the diffusivity in mucus show how the enhanced diffusivity essentially masks the underlyinf advection effect
across appreciable extents of the dyed region the combined action of advection and diffusivity within the PCL layer lead to the appearance of a constant dye intensity with depth
the most plausible explanation of experimental observations is furnished by the plots showing dye flux vectors for a linear distribution of velocity in the PCL layer (as would be expected from hydrodynamical theory). At the leading edge an appreciable amount of dye is being drawn out from the mucus layer and injected into the PCL. This is then transported back through the PCL layer. The overall aspect is of constant dye intensity in the PCL except for the immediate leading/trailing edge areas which can erroneously lead to the conclusion that the PCL advection velocity is constant across the thickness of the PCL layer.
the best diagnostic to determine the PCL velocity field would be simultaneous confocal microscopy of the leading and trailing edge of the dyed region
it is fairly easy to input diverse velocity and diffusivity distributions, including time varying distributions so if any particular choices seem most relevant please send them to me and I'll carry out the computation.
Zoom in on trailing edge of dyed region showing flux vectors
Zoom in on leading edge of dyed region showing flux vectors
Zoom in on trailing edge of dyed region showing flux vectors
Zoom in on leading edge of dyed region showing flux vectors
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