Multiscale Modelling of Lymphatic Drainage - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 170

R V ¼ R V min
þ R V max

:

1
1 þ exp ð s open ð Dp p open ÞÞ þ

1
1 þ exp ð s fail ð Dp p fail ÞÞ 1

ð 56 Þ

The two exponential terms here represent the opening resistance and valve
failure, respectively. The authors developed a model of up to five lymphangions in
a chain using Matlab; representative parameters for a small lymph vessel were
used, and the effect of various conditions and coordination between lymphangions
explored. Most significantly, this model enabled a more detailed exploration of the
failure mode for adverse pressures; at the highest adverse pressures pumping was
found to fail through one of two failure modes; through leakback through the valve
or through the valve simply failing to open. Flow rates were also found to be
sensitive to the conditions, with benefits to coordinated pumping and maximum
flow rates and pressures sensitive to the contractile state.

5.2 Models of Lymphatic Development

There is a new and emerging area that deals with modelling lymphatic develop-
ment. This is an undoubtedly an exciting area, but suffers from a scarcity of
systematic experimental knowledge to build truly encompassing models. Thus,
only two, very specialised models have been published [ 10 , 39 ]. Roose and Fowler
[ 39 ] developed a model to describe the fluid flow channelisation within collagen
gels. This channelisation is thought to precede and guide the development of
mature lymph vessels in mouse tail. The model essentially considered the collagen
gel to be a multiphase rubber like material, and thus Flory-Huggins polymer theory
was used for modelling. This resulted in a macroscopic model in the form of
Cahn-Hilliard equation that was shown to exhibit experimentally consistent results
of collagen patterning. However, clearly, not enough is known about all the pro-
cesses to confirm all the model findings.
Lolas and Friedman [ 10 ] developed a model for lymphangiogenesis during
solid tumour growth. The model consists of eight reaction-diffusion equations that
include the effects of different growth factors (VEGF-C, plasmin etc.) on the
development of lymphatic endothelial cell and cancer density, in addition to the
effect of extracellular matrix.
All these models are clearly only first step in the longer research programs that
are likely to grow in importance as the applied developmental questions related to
lymphatics are addressed by, for example, tissue engineers who are looking to
create artificial tissue scaffolds with not only functioning vasculature, but also with
working lymphatics.