3D Multiscale Modelling of Angiogenesis and Vascular Tumour Growth - Bioanalysis Advanced Materials Methods and Devices - page 32

inflow and outflow nodes with prescribed pressures. We also specify the amount of
haematocrit entering the system through the inlets. The vessel network evolves via
(1) sprouting of tip cells with a probability that increases with the local VEGF
concentration, (2) tip cell movement described by a reinforced random walk, and
(3) new connections formed via anastomosis. In addition, vessel segments with low
WSS for a certain time are pruned away. Elliptic reaction-diffusion equations for
the distributions of oxygen and VEGF are implemented on the same spatial lattice
using finite difference approximations, and include source and sink terms based on
the location of vessels (which act as sources of oxygen and sinks of VEGF) and
the different cell types (e.g. cells act as sinks for oxygen and hypoxic cells as
sources of VEGF).
In summary, after initialising the system, the diffusible fields, cellular and
subcellular states are updated (including cell division and movement), before the
vessel network is updated; this process is then repeated until the simulation ends.
A more detailed description of the mathematical model is presented in the
following subsections. We start at the smallest spatial scale, namely, the subcellular
layer. Then the cellular and diffusible layers are introduced, before the vascular
layer. Interactions between these layers are highlighted in the final part of this
section where the computational algorithm is presented. The parameter values can
be found in Perfahl et al. [ 22 ].

3.2.1 Subcellular Layer

Coupled systems of non-linear ODEs are used to model progress through the cell
cycle, and changes in expression levels of p53 and VEGF [ 3 ]. In practice, the cell
cycle can be divided into four phases: during G 1 , the cell is not committed to
replication, but if conditions are favourable, it may enter the S (synthesis) phase, in
which DNA replication takes place. During the G 2 phase, further growth, and DNA
and chromatid alignment occur, before the cell divides during M (mitosis) phase.
For the cell cycle, we consider the cell mass M and the proteins cycCDK (cyclin-
CDK complex), Cdh1 (Cdh1-APC complex), p27 and npRB (non-phoshorylated
retinoblastoma protein). The cell cycle model that we use focuses on the G 1 - S
transition. It extends an earlier model due to Tyson and Novak [ 36 ] by accounting
for the p27-mediated effect that hypoxia has on the cell cycle [ 2 ]. Using square
brackets to represent intracellular protein concentrations, we have

d ½ Cdh1

¼ ð

1 þ b 3 ½ npRB Þð 1 ½ Cdh1 Þ b 4 M ½ cycCDK ½ Cdh1

; (3.1)

d t J 3 þ 1 ½ Cdh1 J 4 þ½ Cdh1

d ½ cycCDK

(3.2)

¼ a 4 ð a 1 þ a 2 ½ Cdh1 þ a 3 ½ p27 Þ½ cycCDK ;

d t