Data Assimilation in Brain Tumor Models - Mathematical Methods and Models in Biomedicine - page 228

Tabl e 1 Representative parameters for the logistic Swanson model, Eq. ( 2 ), in two dimensions
Parameter

Meaning Value

0 . 2day − 1 α Maximum glioma

growth rate

000 cells mm − 2

,

T max Glioma carrying 10

capacity

Parameter Meaning White Corpus Gray CSF

matter callosum matter

D ( x ) Diffusion rate 0.0065 0.001 0.0013 0.001

(mm 2 day − 1 )

diffusivity. Therefore, D is taken to be piecewise constant within each tissue type,
but may vary among types. Parameters in Eq. ( 1 ) can be estimated from in vitro
studies, sequential patient MR studies, or the Einstein-Stokes relation [ 27 ].
Since the exponential growth term in Eq. ( 1 ) leads to unbounded tumor cell
densities, a more realistic approach assumes that cells grow logistically with some
carrying capacity, T max , at any given point in the model's domain. This modification
gives [ 29 ]

g 1

t = ∇ · D

g + α

∂ g g

T max

( x ) ∇ − . (2)

∂

We refer to Eq. ( 2 )asthe Logistic Swanson model . Baseline parameter values for
this model are reported in Table 1 .
A more sophisticated modeling approach by Eikenberry et al. [ 7 ] considers
two distinct phenotypic classes of tumor cells: proliferating and migrating. The
proliferating cells grow logistically and produce a generalized chemorepellent
which is assumed to induce transition to the migrating cell class at sufficiently
high concentrations. Migrating cells degrade the extracellular matrix (ECM) and
migrate away from the main tumor mass along the ECM gradient via haptotaxis.
The transition between the two cell classes is stochastic. The four dependent model
variables are

g ( x , t )= proliferating cell density

m ( x , t )= migrating cell density

c ( x , t )= chemorepellent density

w ( x , t )= ECM density

and the Eikenberry model is expressed as a coupled system of four stochastic partial
differential equations given by