Multiscale Modelling of Lymphatic Drainage - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 162

L ( t ) R ( t ) L ( t ) R ( t )

C t

( )

Fig. 7 Transmission line description of lymphangeon used by Quick, Venugopal and collab-
orators [ 35 , 56 ]

are actively contractile, contributing to the pumping; and in fact the determination
and modelling of this is a significant part of modelling the system as a whole.
Venugopal, Quick and collaborators [ 34 , 35 , 55 - 57 ] developed a more
sophisticated lumped parameter representation of the lymph system using the
circuit given in Fig. 7 with the parameters being time-varying. The circuit also
includes a diode to simulate the non-return property of the valve. Representation of
the pumping effect uses time-varying elastance theory, originally developed as a
simple model of the heart [ 49 - 51 ]. The elastance of the pumping vessel (heart or
lymphangion) is defined as

p ð t Þ
V ð t Þ V 0 :

E ð t Þ¼ ð 38 Þ

The chamber volume V ð t Þ varies as a result of the pressure p ð t Þ ; V 0 is the theo-
retical volume for zero pressure, or dead volume. If the chamber were passive this
would be a constant as it inflated and deflated; however the physiological changes
in the wall through the pumping cycle result in changes in the physical stiffness,
i.e. changes in E. Experimental measurements on cannulated lymphangions were
used to determine p V data through multiple cycles. As with other pumps (e.g.
the heart), the p V data forms a loop, albiet in the case of the lymphatic pump, a
somewhat eratic one [ 24 ]. Quick and collaborators developed an empirical model
of the time-varying elastance from in vitro pressure and volume measurements on
a pumping, cannulated valveless section of bovine mesenteric lymphangion [ 35 ];
the same paper also presented valve behaviour data for different pressure gradi-
ents. With the assumption of a cylindrical shape and Poiseuille's law, the time-
varying resistance and self-inductance can also be calculated:

R ð t Þ¼ Dp

Q ¼ 8ll

pr 4 ;

ð 39 Þ

L ð t Þ¼ Dp

dQ = dt ¼ ql pr 2 :

ð 40 Þ

Valve resistance is specified as a small fixed value for open valves (Dp valve [ 0)
switching to infinite for a closed valve (Dp valve \0), whilst the inlet and outlet