A Multiscale Approach Leading to Hybrid Mathematical Models for Angiogenesis: Role of Randomness - Mathematical Methods and Models in Biomedicine - page 93

Fig. 3 Subsequent zooms of a simulation the network. Anastomosis is shown

2.1.2 Modelling the Evolution of the Empirical Measures

and the Coupling with the Underlying Fields

As discussed in the general context of Sect. 2 , an Eulerian description may be
given via an empirical measure as Eq. ( 2 ). In this case, we may associate two
fundamental random spatial measures, describing the network at time t ;givena
suitable scale parameter N , Q N , the empirical measure associated with the processes
(

X k v k

( ) , ( )) , = ,..., ( )

t t k 1 N t ,isgivenby

N )

i = 1 ε ( X k ( t ) , v k ( t ))

( t

1
N

Q N ( t )= (13)

while V N ( t ) , the empirical spatial distribution of velocities, is given by

N )

i = 1 v k ( t ) ε X k ( t ) =

( t

1
N

V N ( t )= vQ N ( t )( d ( x , v )) .

d

·× R