Multiscale Modelling of Lymphatic Drainage - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 165

o A

ot þ o

ox ð u x A Þ¼ 0 ;

ð 48 Þ

represents the equation of continuity for an incompressible fluid [ 16 , 44 ], where A
is the cross sectional area. Rewriting this in terms of the radius (A ¼ pa 2 ) and Q,

o a

ot ¼ 1

o Q
ox

: ð 49 Þ

2pa

To close this system we need an expression for the pressure p. For an incom-
pressible fluid within an elastic tube, the pressure, or rather the pressure difference
between outside and inside, is dictated by the dilation of the wall. Reddy used a
standard expression for a thin-walled tube for this:

;

p p ext ¼ h
a

R hoop þ R act ð 50 Þ

where p ext is the external imposed pressure, h the vessel wall thickness, a the
radius, R hoop the hoop stress (due to the elastic properties of the wall) and R ext the
additional stress due to the active contraction of the wall. This was represented as a
combination of a simple sinusoidal function with a fixed period and a limiting
contractile

stress representing the minimum condition necessary to activate

contraction.
With this, the equation set can now be discretised using standard finite differ-
ence techniques. Reddy and collaborators used upwind differencing and created a
simple explicit algorithm with one grid point per lymphangion, essentially creating
a lumped-parameter model based directly on the Navier-Stokes equations [ 37 ].
This was subsequently extended to model the complete left-side lymphatic net-
work for the human body, i.e. the larger subsystem draining into the thoracic duct,
including the effects of elevation in the system and including an external pressure
term representing the effect of breathing [ 38 ]. The authors took care to validate
their modelling as far as was possible by comparing average flow rates with known
values, and were able to demonstrate behaviour in the theoretical model consistent
with observed pumping behaviour in real lymphatics, but did observe somewhat
intermittent and random flow and pressure behaviour.
MacDonald and collaborators [ 25 , 26 , 28 ] used the same basic framework to
construct a more spatially refined model of an individual lymphangion and of short
series of lymphangions [ 27 ], using 4-6 grid points to resolve the axial distribution
of flow and wall properties within each lymphangion. In addition this work
attempted to improve some of the detailed modelling, in particular by introducing
a more complex model for the wall:

2 A

A
A 0

T
D 0

o

ox 2 þ c o A

p p ext ¼ U ; ð 51 Þ

ot