Multiscale Modelling of Lymphatic Drainage - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 151

Fig. 2 a Collecting lymph wall imaged using various two photon microscopy techniques
superimposed to create a composite image [ 1 ]. Lymph wall cells are shown in blue, with elastin
fibres (green) and collagen (red). b Data processed using image based meshing package ScanIP to
generate 3D representation of structure, focussing on the elastin fibres. Image courtesy of
Simpleware Ltd ( www.simpleware.com )

3 Modelling Primary Lymphatics

3.1 Derivation of Macroscale Primary Lymphatic Fluid Flow

To describe the fluid flow in the primary lymphatic structure we first have to write
down the fundamental equations describing these flows and then use multiple scale
homogenisation to derive the tissue scale equations. This homogenisation proce-
dure relies on the fact that the primary lymphatics have a very simple, repeating
periodic hexagonal structure and because of the periodicity is very amenable to
such analysis.

3.1.1 Fundamental Equations

We begin by describing the fluid flow inside the lymphatic capillary lumen. Our
starting point is to use the full Navier-Stokes equations and supplement these with
suitable boundary conditions that relate to the fluid drainage. Thus in the lymphatic
capillary domain X C we have

q ½ o t u C þð u C rÞ u C ¼r p C þ l r 2 u C ; and r u C ¼ 0 ;

ð 1 Þ

where q is the lymph fluid density and l is the viscosity. From a consideration of
the valve dynamics we get a relationship for the flux of fluid from the interstitium