Multiscale Modelling of Lymphatic Drainage - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 167

partial differential equations into difference equations which can be solved, gen-
erally through an implicit, matrix-inversion-based algorithm. The big advantage of
the finite volume method is that the cells can be any shape necessary, thus adapting
to the needs of the geometry. Additional modelling may be introduced, e.g. tur-
bulence, non-Newtonian flow, and particle tracking. Prewritten codes are avail-
able, both commercial (e.g. ANSYS Fluent, Star-CCM) and open source (e.g.
OpenFOAM [ 58 ]). Alternative to the finite volume method is the functionally
equivalent Finite Element method (FEM), in which the domain is represented by a
collection of vertices connected by edges, and solution proceeds through numer-
ical minimisation of a cost function. Again, preexisting codes are available (e.g.
Comsol Multiphysics).
The major issue with CFD modelling of lymphatic vessels is that of treating the
wall compliance. Flow of fluid in a duct which can deform is known as fluid-
structure interaction (FSI). The FVM or FEM approaches can both handle this by
permitting the motion of boundary patches and allowing the interior mesh structure
to adapt accordingly. A full FSI calculation for a lymphangion would require the
solution of the flow in the lymphangion and the resultant wall forces (pressures and
shear stresses), together with a stress calculation in the lymphangion wall, with
coupling acting in both directions. This is theoretically achievable but computa-
tionally costly [ 29 , 46 ] and has not currently been attempted. However if we can
determine the motion of the wall (and this may be more appropriate given its
active state) the boundary motion may be imposed as a condition, simplifying the
calculation considerably. Rahbar and Moore [ 36 ] have done this, creating an
idealised model of a lymphangion using the commercial code StarCCM+, of
length 1000 lm (contracting section 500 lm) and a variety of contractions (ranging
from radii contracting from 60 to 40 lm to 120 contracting to 60 lm. Sinusoidal
and skewed-sinusoidal wall motions were imposed with a period of 3 : 24 s, and
steady and unsteady inlet velocity profiles imposed. The aim of the research was to
ascertain whether the commonly-used assumption of Poiseuille flow, appropriate
for steady flow in a straight, rigid tube, is appropriate for the case of flow in lymph
vessels. In general, the authors found that it was, with discrepancies from strict
Poiseuille flow of less than 4 % for wall shear stress, and parabolic velocity
profiles strongly suggesting Poiseuille flow throughout the whole pumping cycle.
5 Other Modelling Related to Lymphatics
In addition to the two fluid flow modelling areas discussed above there are
additional emerging fields that deal with modelling the detailed nature of how the
lymphatic primary and secondary valves function and how the lymphatic system
develops. We will discuss both very briefly below. In addition there are models
that deal with the immune response within the lymph nodes [ 2 , 17 , 32 ], but since
these do not usually have any fluid dynamics included in them we will not discuss
them in this review.