Multiscale Modelling of Lymphatic Drainage - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 161

R

C R C

Fig. 5 Windkessel circuit. The left circuit represents a complete system modelled as a

windkessel circuit; the right a single element in a chain

L R

C

Fig. 6 Transmission line description of a blood vessel

represent an entire system (left figure) or an element in a chain (right figure). Fluid
flowing into the system can flow out the other side, with associated pressure drop
(the circuit resistance) or the vessel can distend, storing fluid (equivalent to the
capacitor). Applying Kirchoff's laws to the circuit on the left, for instance, we can
derive the equation

Q ð t Þ¼ p ð t Þ

R þ C dp ð t Þ

; ð 37 Þ

dt

relating volumetric flow rate and pressure for the system. With only one possible
time constant this circuit cannot represent all possible frequencies in a problem,
leading to the development of more sophisticated circuits in order to improve the
detailed modelling [ 45 ]. For instance, the transmission line model represented in
Fig. 6 which includes the effect of fluid inertia via self-inductance L. All that
remains with all of these circuits is to identify values or models for the terms R, C
and so forth.
The earliest attempt to represent the lymphatic system in terms of an equivalent
circuit was that of Drake et al. [ 9 ], as a simple resistor/diode arrangement. The
value of R was determined from in vitro experimental measurements on a dog lung
lymph vessel. This was cannulated and the volumetric flow rate measured for
various back pressures with the slope of Q vs. p out giving the resistance
R. However this takes no account of the pumping action. In terms of this mod-
elling, lymph vessels differ from arteries in one very important respect; whilst
arterial walls are compliant they are also passive, so the response of the system can
be modelled by finding suitable fixed values for L, C and R. Lymphangion walls