A Multiscale Approach Leading to Hybrid Mathematical Models for Angiogenesis: Role of Randomness - Mathematical Methods and Models in Biomedicine - page 105

σ

2
2

t

+

2 Δ x f

( x , 2 ) p 2 ( x , t )+ h ( x , 2 ) f ( x , 2 ) p 2 ( x , t )

0 B

+ β ∇ x g ) · ∇ x f

( x , t ) − ∇ x u ( x , t ( x , 2 ) p 2 ( x , t )

− m 21 f ( x , 2 ) p 2 ( x , t ) − m 23 f ( x , 2 ) p 2 ( x , t )

d x d t

+ m 12 f ( x , 2 ) p 1 ( x , t )

m 13 f d x d t

t

+ ( x , 3 ) p 1 ( x , t ) − m 23 f ( x , 3 ) p 2 ( x , t ) .

0 B

(51)

with

λ 12 λ 13

= ( , )

) ,

= ( , )=

) ,

m 12 : m 12 x 1 m 13 : m 13 x 1 (52)

u ( x , t ( p 1 ( ·, t ) ∗ K )( x

m 21 : = m 21 ( x , 2 )= λ 21 ( p 2 ( ·, t ) ∗ K )( x ) , m 23 : = m 23 ( x , 2 )= λ 23 , (53)

and, by Eq. ( 38 ), coupled with the dynamics of the (now purely deterministic)
underlying fields g and u

∂ g ( x , t )

= − ( , )+ ( , )+ α ( , ) ,

d g g x t D g Δ g x t g p 2 x t (54)

∂ t

∂ u ( x , t )

= − d u u ( x , t )+ D u Δ u ( x , t )+ α u p 1 ( x , t ) . (55)

∂ t

In particular, by considering the weak evolution Eq. ( 51 ) over the three subsets
B × s ∈ B ⊗ S ,forany s = 1 , 2 , 3, we find the weak form of the following system E

of PDEs:

∂ p 1 ( x , t )

t =(

h ( x , 1 ) − m 12 − m 13 ) p 1 ( x , t )+ m 21 p 2 ( x , t ) , (56)

∂

2
2

( , )

∂ p 2 x t

= σ

2 Δ

p 2 ( x , t ) − α 2 ∇ ( ∇ g ( x , t ) − ∇ u ( x , t )) p 2 ( x , t )

∂ t

( h ( x , 2 ) − m 21 − m 23 ) p 2 ( x , t )+ m 12 p 1 ( x , t ) , (57)

∂ p 3 ( x , t )

= m 13 p 1 ( x , t )+ m 23 p 2 ( x , t ) , (58)

∂ t

coupled with Eqs. ( 54 )and( 55 ). System ( 54 )-( 58 ) represents the continuum
description of the angiogenic network at the macroscale. It may be taken into
account only whenever a law of large numbers may be applied. It describes the
continuum densities of the different types of cells coupled with the deterministic
PDEs for the underlying fields.