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Modeling Brain Tumors for Patient-Specific Therapy
Olivier ClatzEnder Konukoglu
INRIA Sophia Antipolis, Asclepios Project-TeamMarch, 8th 2006
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Problem position• Understanding both the
mechanical influence and the diffusion process of gliomas
• Using the model for– Identify invaded area that are
not visible in the MRI – Predict future evolution of the
tumor– Characterize the tumor
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Tumor cell density
Tumor biology
T2 MRI T1 MRI + gad
Edema and infiltrating tumor cells
Active tumor cells + angiogenesis
Necrotic center
Mass effect
T2 MRI signal
Nécrosis
Observability !
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Geometric model
1. Skull. 2. Gray matter3. White Matter 4. Ventricles5. Falx cerebri
DTI (atlas)Initial position of the tumor
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Mechanical model
Summary: – Linear relationship
force | displacement
– Tumor acts as a local pressure
εµελσ 2)( += trσ = Stress
ε = Stress( )Tuu ∇+∇=
2
1ε
• Linear elasticity for the brain :
α = Coupling factor
Fe = External forces0 )div( 3 =+− FeIcασ
• Influence of tumor cells on the mechanics
λ, µ = Lamé Coefficient
u = Displacement
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Diffusion model
( ) )1(. cccDt
c −+∇∇=∂∂ ρReaction diffusion equation
X
Y
Cell diffusion
Cell multiplication
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Tumor Growth simulation
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March 2002
March 2002 + initial contour
September 2002
September 2002 + simulation contours
Results
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Analyzing Diffusion Model( ) )1(. cccD
t
c −+∇∇=∂∂ ρ
• Tumor profile
• Infiltration extent
• Extrapolation
• Not observable from images
• Growth speed
• Parameter Estimation
• Quantification
• Observable from time series of images
Cell density
Position
D1/ρρρρ1 >> D2/ρρρρ2 >> D3/ρρρρ3
t1
t3
Position
Cell density
Position
Cell density
t1 t2 t3 t1 t2 t3 Position
Cell density
D1ρρρρ1 >> D2ρρρρ2 >> D3ρρρρ3
D/ρ Dρ
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Model Based Growth Quantification
• Observables:
• Tumor fronts (CTV extent) :
• Tumor infiltrated edema extent for high grade tumors
• Bulk tumor extent for low grade tumors
• White matter segmentation and DTI.
• Only Dρ is observable from the images.
• What are growth speeds in the white matter and in the grey matter?
6 months
?][,][ gmwm DD ρρ
t1 t2
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Front Motion ApproximationT
umor
cel
l de
nsity
cm
• Again using the asymptotic traveling wave solution.
ρDvasymp 2=
• Assuming visible tumor front is an iso-density surface.
• Traveling time formulation for the motion of the tumor front gives: ρ2
1' =∇∇ TDT
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Identifying Parameters
( ) ( )[ ]
( )
=Γ=∇∇
−==Γ
ΓΓΓΓ=
0,2
1'
,)(
,,,2
1),(
1
12
2
2222
TTDT
ttxTx
distdistvvC gw
ρ
1Γ 2Γ
[ ]
day
mmD
day
mmD
gm
wm
24
22
103.4][
107.4
−
−
×=
×=
ρ
ρ
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Predicting tumor evolution
• Estimated parameters can help predicting future evolution
Day 0 Day 21 Day 67
Estimation Prediction
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Extrapolating Tumor Invasion• CT and MR have limited resolution for tumor cells.
• We do not see the whole tumor infiltration.
• Use of growth dynamics to understand the extents of the tumor.
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Tail DistributionT
umor
cel
l den
sity
cm
time
Traveling wave solution in the infinite cylinder with constant D:
)()(),( ξcvtnxutxc =−⋅=
( ) )1(. cccDt
c −+∇∇=∂∂ ρ
t = 63,90,125 days
⋅−= ξρξ
)(exp)( 0
nDnuu
( )1=
∇⋅∇u
uDu
ρ
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Comparison with the Model
After 6 months according to the model
Approximated tails
From 5% to 1% - Using same parameters
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Invasion Extent vs. Irradiation Margin
Radiotherapy margin (2cm)
Simulated probability of finding tumor cells
Invaded area targetted by radiotherapy
Area targetted by radiotherapyBUT NOT invaded
Area invaded BUT NOT targetted by radiotherapy
Visible tumor