A Multiscale Approach Leading to Hybrid Mathematical Models for Angiogenesis: Role of Randomness - Mathematical Methods and Models in Biomedicine - page 99

Fig. 5 Snapshot of the vascular network ( left ), the VEGF ( center ), and the nutrient ( right ) density
field for the fully stochastic model

In a similar way, we assume that the nutrient diffuses, naturally degrades, and is
produced by type 1 cells [ 19 ], so that, for any

d , the concentration of

( , ) ∈ R + × R

t x

( , )

nutrient u x t is subject to the following evolution equation:

j = 1 ε ( ·, C j ( t )) ( x , 1 ) ∗ K N ( X j

N ( t )

∂ u ( x , t )

)+ α u 1
N

= − d u u ( x , t )+ D u Δ u ( x , t ( t )) . (30)

∂ t

D i , α i ∈ R + , ( Parameters d i , in Eqs. ( 29 )and( 30 ) represent the rates of

natural degradation, the diffusivities, and the rates of production, respectively.
By summarizing, at the finer scale of the microscale, the Lagrangian description
is based on the stochastic behavior of individual cells, given by the system of
stochastic differential equations ( 22 )-( 23 ), the branching process

i = g , u )

with rates
Eqs. ( 9 )-( 25 ), the Markov chain of the state change Eqs. ( 26 )-( 28 ), coupled with
the stochastic partial differential equation describing the evolution of the underlying
fields Eqs. ( 29 )and( 30 ).
An example of the obtained results is shown in Fig. 5 .
The geometrical complexity of the network is now recovered: there are frequent
branchings which appear in almost every area of the domain and even several
anastomoses occur. The pattern of the nutrient field u (right) reflects the structure
of the stochastic network (left). The pattern of the growth factor g (center) indicates
the location of type 2 cells: they are mostly concentrated on the boundary regions.
The stochastic branching leading to the realistic pattern of the vessel network
[Fig. 5 (left)] is a direct consequence of the stochasticity of the underlying fields
[Fig. 5 (center and right)].

Φ N ( t )

2.2.2 Modelling the Evolution of the Empirical Measures

and the Coupling with the Underlying Fields

d

∈ ([ , ] ,M (R × S))

From the empirical measure process Q N C 0 T , where for any

∈ [ , ] ( )

t isgivenbyEq.( 2 ), one can derive the empirical spatial distribution

of the cells of type s

0 T , Q N t

∈ S