Biomedical Engineering Reference
In-Depth Information
boundaries are simulated with additional fixed resistances [ 56 ]. The model has been
validated against experimental results for a single lymphangion [ 35 ], and expanded
to explore networks of four lymphangions [ 56 ] and branching networks [ 55 ].
This lumped-parameter approach has proved very successful, allowing inves-
tigation of individual pumping behaviour [ 35 ], for example demonstrating that
whilst lymphatic pumping action is beneficial under normal (positive) pressure
gradients, it is counterproductive for reverse (negative) gradients. Such reverse
gradients occur for cases of edema, external compression or limb elevation, where
the interstitial fluid pressure is artificially raised, leading to the permanent opening
of the lymphangion valves and the lymphangion acting as a simple conduit [ 34 ]. It
has also allowed the investigation of the effect of coordination of pumping
between successive lymphangions in series [ 56 ]. Introducing a phase change into
the time-periodic functions (E ð t Þ , R ð t Þ etc.) allows the creation of contractile
waves propagating along the series, both orthograde (in the direction of flow) and
retrograde (opposing). The authors found this coordination of the contraction to
have very little impact upon the pumping, and the orthograde and retrograde waves
had similar effects, suggesting that individual lymphangions are able to function
independently and thus adapt to local conditions as necessary. It has also proved
possible to use the model to examine optimal structures of the lymph system [ 55 ].
The branching structure of the arterial system is well predicted by Murray's law
[ 33 ]: in an optimal system, where an artery bifurcates, the cubes of the radii of the
daughter vessels must sum to a constant value. This result is based on fundamental
physical principles (minimum energy loss in the system) and has been well vali-
dated experimentally. Venugopal and collaborators demonstrate a similar effect for
lymphatic networks, showing that a ratio of 1 : 26 for the upstream and downstream
lengths optimises lymph flow for symmetric networks. The same authors have also
investigated the effect of linear and non-linear contractile behaviour on the
pumping effect [ 57 ], a significant issue in understanding the response of lym-
phangions to increasing transmural pressure, for example in response to edema.
4.2 One Dimensional Models
Lumped parameter models assume a uniform distribution of the dependent vari-
ables throughout the individual elements of the model at a particular instant in
time. Although they are computationally efficient and so can be applied to large
networks, and, as demonstrated, can produce very good results, the individual
elements of the model are empirical and require fine tuning. A more fundamental
approach is to directly solve the basic equations of fluid dynamics, the Navier-
Stokes equations, for flow through the lymphangions. This has been carried out in
1D using purpose-built codes, discussed in this section, and could potentially be
investigated further in 2D and 3D (see Sect. 3 ). These models allow for the spatial
variation of the dependent variables; the main challenge is including the
mechanics of the wall, its compliance and contractility. Experimental input to
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