Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 229

g 1

∂ g g m

T max

+

t = ∇ · (

D G ( x ) ∇ g ) − ∇ · ( χ ( x ) m ∇ w ) + α − − Φ

m + Ψ g ,

(3)

∂

g − to − m − to −

diffusion haptotaxis

logistic growth

∂ m

∂ t = ∇ · (

( ) ∇ ) + ∇ · ( χ ( ) ) + Φ

m − Ψ g − σ (

)
death

, ,

D M x m x m ∇ w g m (4)

g − to − m − to −

diffusion haptotaxis

∂ c

2 c

t =

diffusion + γ

production − β degradation ,

D C ∇ g

C c

(5)

∂

w g

∂ w

∂

+ m

t = − ρ

+ α W w ( 1 − w ) , (6)

θ + T

W

repair

degradation

where is the rate

at which migratory cells become proliferating. Eikenberry et al. [ 7 ] assumed that
the transition rate to the migratory phenotype is greatest when the cell density is
low (such as at the edges of the tumor):

Φ is the rate at which proliferating cells become migratory and Ψ

g

G

θ c

φ M +

Φ = Φ ( g , m , c )= τ . (7)

2
G

( g + m )

2

+ θ c

The stochasticity arises from how is implemented over the discretization of the

model. The probability of transition from a migrating to a proliferating phenotype
is assumed to be an exponential random variable with rate depending on the
chemorepellent concentration

Ψ

λ = λ 0

θ G

. (8)

+

θ c

G

Once is computed by finding the number of migrating cells that

transition at each grid point and converting this to a cell density. At a given grid point
the cell density must constitute at least one cell for any such transition to occur.
Cell death is assumed to result from crowding and occurs only when the cell
density reaches a critical threshold, M D :

λ is determined, Ψ

⎧
⎨

g 2

m
T max

+

, g + m > M D ,

σ ( g , m )= (9)

⎩

0 , otherwise.

Note the haptotaxis term in Eq. ( 3 ) is a modification of the original model and is
included to preserve conservation of mass.