Multiscale Modelling of Lymphatic Drainage - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 157

r x x j ¼ 0 ;

ð 30 Þ

with n r x x j ¼ n e j on the internal microstructure surfaces and periodic on the
outer surfaces.
The order O ð e 2 Þ interstitial equations are

r x p I 2 þr x r y p I 1 þr y r x p I 1 þr y p I 0 ¼ 0 ; ð 31 Þ

with boundary condition

n ðr x p I 2 þr y p I 1 Þ¼ W ð p I 0 p 0 Þ on oX : ð 32 Þ

As for the capillary flow equation, substituting ( 29 ) into the equation above and
integrating over the whole interstitial domain we obtain

r y ð E r y p I 0 Þ¼ S W ð p I 0 p 0 Þ; ð 33 Þ

V

where S is the surface area of lymphatic capillaries within the unit cell, V is the
volume of the unit cell, and hence S = V is the lymphatic surface area density in the
tissue. The effective tissue interstitial permeability is given by

Z

E ij ¼ V I

V d ij þ 1

x j n i dS ; ð 34 Þ

V

oX

where V I is the volume of interstitial space in the unit cell, V is the volume of the
unit cell, and d ij ¼ 1ifi ¼ j and zero otherwise.

3.1.5 Summary of the Macroscale Equations for Primary
Lymphatic Drainage

The capillary flow problem is described as

r y ð K r y p 0 Þ¼ 0 ;

ð 35 Þ

whilst the interstitital flow problem is described as

r y ð E r y p I 0 Þ¼ S W ð p I 0 p 0 Þ:

ð 36 Þ

V

These equations can now be solved in a fast and efficient manner with suitable
boundary conditions either analytically (1D equations) or numerically in higher
dimensions.