Biomedical Engineering Reference
In-Depth Information
any dependence on h and z reduces the system to 1D with coordinate r : This
simplification means that the cylindrical shape of the trachea is imposed in the
model rather than arising from the emergent behaviour of the cells within it.
The present model is concerned with just one cylindrical segment of the tra-
chea, lying wholly within a single cartilage ring, and is lined internally with
epithelium and a sub-mucosal layer as shown in Fig. 1 d. The segment does not
include the inter-cartilaginous zones. For simplicity, the muscular posterior wall
(see Fig. 1 b) is considered to be part of the cartilage ring and not flattened, so the
tracheal segment can be assumed to be cylindrical. It is assumed that the tracheal
segment does not deform axially, and so the displacement of its top and bottom
edges are held fixed at z ¼ h = 2 : Also the outer curved surface of the trachea, at
r ¼ R o ; is held fixed while the inner curved surface, at r ¼ R l ; and the boundary
between the cartilage and the submucosa, at r ¼ R c ; are free to move.
To model the exchange of cells between the trachea and the surrounding tissue,
the flux of cells into the domain is prescribed by some function, I u ; on the
boundary. Continuity of the flux of cells across the boundary implies I u ¼ n J u
where n is the outward pointing unit normal on the boundary hence from Eq. ( 2 ),
assuming n r a ¼ 0 (see below), the boundary conditions for the cells are
D u n r u ¼ I u :
ð 3 Þ
This boundary condition is also valid for the diffusible chemical species but with
I u ¼ 0 ; thereby reducing to n r u ¼ 0 in those cases.
Because of the cylindrical symmetry I u need only be specified on the tracheal
surface at r ¼ R l and r ¼ R o ; indeed the flux of cells and growth factors is gen-
erally assumed to be zero there. However, a non-zero flux of cells entering the
tracheal segment at z ¼ h = 2 needs to be taken into account. This is done by
assuming negligible changes in the variables in the z direction i.e. u ¼ u ð r ; t Þ; v ¼
v ð r ; t Þ r and J u ¼ J r ð r ; t Þ r þ J z ð r ; t Þ z ; where r and z are unit vectors corresponding
to the coordinates r and z respectively. Integrating Eq. ( 1 ) from z ¼ h = 2to
z ¼þ h = 2 and dividing by h gives, in cylindrical polar coordinates,
o u
ot ¼ 1
o
or
Þ þ 2
ð
r ð uv þ J r Þ
h I u ð r ; h = 2 ; t Þþ f ð u ; r ; t Þ;
ð 4 Þ
r
where I u is assumed to be the same on both boundaries. This way of reducing the
problem to 1D is tantamount to making the assumption that species mobility in the
axial direction is large compared to that in the radial direction. In cylindrical polar
coordinates the radial component of Eq. ( 2 )is
J r ¼ D u o u
or þ uv u o a
or :
ð 5 Þ
Although I u specifies an influx of cells that may in real tissues have a strong
chemotactic causation, in the model chemotaxis of cells is neglected once they are
inside the tracheal segment hence v u ¼ 0 :
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