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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

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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