Biomedical Engineering Reference
In-Depth Information
Tabl e 2
Representative parameters for the Eikenberry model, Eqs. (
3
)-(
6
), in 2 dimensions
Parameter
Meaning
Value
0
.
15 day
−
1
α
Maximum glioma growth rate
10 000 cells mm
−
2
T
max
Glioma carrying capacity
01 day
−
1
.
α
W
Maximum ECM recovery rate
0
25 day
−
1
β
C
Chemorepellent degradation rate
0
.
0mm
2
day
−
1
.
D
C
Chemorepellent diffusion rate
2
10
−
5
day
−
1
γ
Chemorepellent production rate
2
.
5
×
5day
−
1
λ
0
Maximum migrating to grow rate
2
.
000 cells mm
−
2
M
D
Crowding death threshold
5
,
5mm
−
2
φ
M
Half-maximum grow to migrating
coefficient
0
.
02 day
−
1
ρ
Maximum ECM remodeling rate
0
.
1day
−
1
τ
Maximum grow to migrating transition
rate
0
.
100 cells mm
−
2
θ
Half-maximum grow to migrating cell
density
G
100 cells mm
−
2
θ
Cell density at half-maximum ECM
degradation
W
Parameter
Meaning
White
Gray
CSF
matter
matter
Growing cell diffusion rate (mm
2
day
−
1
)
D
G
(
x
)
0.02
0.004
0.0001
Migrating cell diffusion rate (mm
2
day
−
1
)
D
M
(
x
)
0.1
0.02
0.0001
Haptotaxis coefficient (mm
4
day
−
1
)
χ
(
x
)
1.0
0.2
0
Tab le
2
displays the nominal parameter values for the Eikenberry model.
The values used here differ slightly from those in [
7
] and were chosen so that the
total tumor cell populations from both the Logistic Swanson model, Eq. (
2
), and the
Eikenberry model grow at approximately the same rate. The diffusion rates,
D
G
and
D
M
, in the Eikenberry model are smaller in regions corresponding to cerebrospinal
fluid (CSF) than those in the Swanson model. See [
7
] for further details.
Both sets of equations are integrated using a brain geometry from the BrainWeb
database, developed by the McConnel Brain Imaging Center of the Montreal
Neurological Institute at McGill University [
2
]. We use the discrete anatomical
model of a normal brain generated by McGill's MR simulator, which consists of a
181
181 isotropic grid of 1 mm
3
voxels in Talairach space [
31
]. Each voxel is
classified as background, CSF, gray or white matter, fat, muscle/skin, skin, skull, or
glial matter. The classification determines the boundary of the model domain and the
diffusion and haptotaxis coefficients at each grid point. To reduce the computational
expense, the equations are integrated over a representative coronal slice at the
center of the three-dimensional domain, from which voxels representing the skull
and similar tissue have been removed. The resulting two-dimensional domain is
a fixed 145
×
217
×
143 grid (the mass effect is not modeled). For simulation purposes,
glial matter is treated as white matter, and the domain has been edited by hand to
enumerate voxels comprising the corpus callosum (which is a favored migratory
pathway for GBM cells [
1
]).
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