Multiscale Modelling of Lymphatic Drainage - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 168

5.1 Valve and Node Modelling

Valves are obviously an important element of the overall lymphatic system, but are
even less well researched than the systems behaviour of fluid flows within
lymphatics. Both lumped-parameter and Navier-Stokes based models described
above use empirically based valve models based on experimental measurements of
valve opening and closing pressure drops, and resistance values for the opened
valve. Usually these models have exhibited simple diodic on/off behaviour,
although Macdonald and collaborators [ 28 ] developed a more complicated model
with ramping of the valve resistance between the two states. The situation is made
more complex by there being at least two different types of valve in operation in
the system. In the initial lymphatics, overlapping endothelial cells in the vessel
wall form a primitive non-return valve for fluid transiting from the intrastitial
space into the vessel [ 41 , 54 ], often referred to as the primary valves; the vessels
themselves however are non-pumping, being simple drainage ducts, and so do not
possess internal valves. Mendoza and Schmid-Schönbein [ 31 ] attempted to
develop a model for the primary valves by treating the overlapping cell as a flat
flexible plate subject to a deflection w ð x Þ with governing equation

d 4 w ð x Þ
dx 4

¼ 1

EI Dp ð x Þ;

ð 54 Þ

with E as the elastic modulus and I the moment of inertia; the pressure drop across
the plate Dp ð x Þ being related to Stokes flow through the gap, giving

dp

dx ¼ 12Ql

; ð 55 Þ

w ð x Þ 3 T

for tissue thickness T. The equations were solved numerically using Mathematica.
Results showed that, mechanistically, such simple structures could indeed operate
as a valve, opening and closing in appropriate timescales and under appropriate
pressures, and gave a likely estimate of the size of the opening.
Mendoza and Schmid-Schonbein's model [ 31 ] represents the situation where
the valve leaflets are overlapping by a large distance, and hence the primary
impedance for the lymphatic drainage is the movement of fluid in the narrow
channel between two overlapping lymphatic endothelial cells. Galie and Spilker's
model [ 12 ] is essentially similar to that of Mendoza and Schmid-Schonbein with
the difference that they make the fluid dynamics around the valve more sophis-
ticated. Both of these models also neglect the overall curvature of the lymphatic
lumen.
Downstream in the network, as the vessels become larger and form the col-
lecting lymphatic system, we find valve structures between individual lymphan-
gions. However their exact structure is difficult to identify, being largely known
from 2D micrographs such as that shown in Fig. 8 . Two distinct structures have
been proposed; a bicuspid valve analogous to the tricuspid shape of the heart valve