3D Multiscale Modelling of Angiogenesis and Vascular Tumour Growth - Bioanalysis Advanced Materials Methods and Devices - page 36

VEGF, which then can be removed by the vasculature. The concentration of VEGF
is determined by

2 p R

2 c VEGF

D VEGF r ð t ; x Þ P VEGF c VEGF þ k VEGF ð t ; x Þd VEGF c VEGF ¼ 0 ; (3.15)

wherein D VEGF is the diffusion coefficient of VEGF, P VEGF the permeability of the
vessels to VEGF, k VEGF the cell-type-dependent VEGF production rate and the
decay rate d VEGF . In our numerical algorithm, ( 3.14 ) and ( 3.15 ) are discretised with

a finite difference scheme, and the resulting sparse linear system of equations is
solved with a GMRES-solver.
In case of a non-periodic simulation domain, it is assumed that there is no flux of
diffusible substances over the boundary, and thus, homogeneous Neumann bound-
ary conditions are imposed. For simulations in a periodic domain, we apply periodic
boundary conditions for the calculation of diffusible substance concentrations.

3.2.4 Vascular Layer

We follow very closely the work of Secomb et al. and refer the reader to [ 26 ] for full
details. We assume a laminar Poiseuille flow in each vessel. The flux

Q i through

vessel i is given by

pR i

Q i ¼

L i D

P i ; (3.16)

8 mð R i ; H i Þ

where DP i is the pressure difference at the vessel segment i , L i the vessel length,
m ( R i , H i ) is the radius R i and haematocrit H i dependent blood viscosity [ 26 ]. In

( 3.16 ), we can identify the resistance of vessel i by Res i ¼ L i =ðpR i Þ

8 mð R i ; H i Þ .In

( 3.16 ), the blood viscosity is defined by

mð R ; H Þ¼m 0 m rel ð R ; H Þ; (3.17)

where m 0 is a positive constant,

" #

2 2

C

ð 1 H Þ 1 2 R 2 R

m rel ð R ; H Þ¼ 1 þðm 0 : 45 ð R Þ 1 Þ ;

C

2 R 1 : 1 2 R 1 : 1

ð 1 0 : 45 Þ 1

(3.18)

0 : 645

6e 0 : 17 R 44e 0 : 06 ð 2 RÞ m 0 : 45 ð R Þ¼ þ 3 : 2 2 : (3.19)

and