The inverse ischemia problem; mathematical models and validation - - PowerPoint PPT Presentation

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The inverse ischemia problem; mathematical models and validation

Bjørn Fredrik Nielsen (Simula) Marius Lysaker (Simula) Per Grøttum (Fac. of medicine UiO & Simula) Martin Burger (University of Münster) Kristina Hermann Haugaa (Rikshospitalet) Aslak Tveito (Simula) Christian Tarrou (Kalkulo) Andreas Abildgaard (Rikshospitalet) Jan Gunnar Fjeld (Rikshospitalet)

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( )

f u K − = ∇ ⋅ ∇

Ischemia?

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The Bidomain Model

(Geselowitz, Miller, Schmitt, Tung, et al in the 70’s)

• : transmembrane potential
• : state vector representing cell properties,

transport of ions (1 to 50 entries)

• : conductivity tensors

e i

u u v − =

s

e i M

M ,

T u M H u M M v M H u M v M v s I v H v s F s

• e

e i i e i i t t

in in ) ) (( ) ( in ) ( ) ( ) , ( in ) , ( = ∇ ⋅ ∇ = ∇ + ⋅ ∇ + ∇ ⋅ ∇ ∇ ⋅ ∇ + ∇ ⋅ ∇ = + =

T

∂T ∂H

H

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Mathematical model (Heart)

Bidomain model (very CPU demanding) If v is given, we obtain a stationary model:

) ) (( ) ( ) ( ) ( ) , (

e e i i e i i t

u M M v M u M v M s v I v ∇ + ⋅ ∇ + ∇ ⋅ ∇ = ∇ ⋅ ∇ + ∇ ⋅ ∇ = +

) ( ) ) (( v M u M M

i e e i

∇ ⋅ −∇ = ∇ + ⋅ ∇

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Stationary model

• During rest

⎩ ⎨ ⎧ − − ≈ = ssue healthy ti 90 tissue ischemic 60 mV mV v v

r

• Plateau phase

⎩ ⎨ ⎧− ≈ = ssue healthy ti tissue ischemic 20 mV mV v v

p

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Stationary model

where

⎩ ⎨ ⎧ ≈ − = ssue healthy ti 90 tissue ischemic 40 mV mV v v h

r p

) ( ) ) (( h M r M M

i e i

∇ ⋅ −∇ = ∇ + ⋅ ∇

• Model for the shift r:

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• ECG d, cost-functional
• Inverse problem

subject to

) ( min h J

h α

] [ )] ( ) [( h M h r M M

i e i

∇ ⋅ −∇ = ∇ + ⋅ ∇

|| || ) ) ( ( ) (

) ( H 2 2

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H healthy B

h h ds d h r h J − + − = ∫

∂

α

α

Inverse Problem

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• Ischemic region
• Or employ level set function
• Conductivities depend on the ischemia?

} 40 ) ( | { mV x h H x

Inverse Problem, continued

≈ ∈ =

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Theoretical considerations

Split

ECG Ischemic region

into two sub problems

ECG Heart surface Ischemic region

ECG → heart surface

Classical Cauchy problem; compute ro on ∂H by solving ∇ · (M o∇ro) = in T, (M o∇ro) · n =

• n ∂B,

ro = ECG

• n Γ ⊂ ∂B.

T

∂B ∂H

H

Severely ill-posed; typical error amplification dk → ekπdk, {dk} Fourier coefficients of the ECG.

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Heart surface → ischemic region

Determine r and h such that ∇ · [(Mi + Me)∇r] + ∇ · (Mi∇h) = in H, ([Mi + Me] ∇r) · nH + (Mi∇h) · nH = −(Mo∇ro) · nT on ∂H, r = ro

• n ∂H,

Ischemic region= {x ∈ H| h(x) ≈ 40} Non-uniqueness

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Heart surface → ischemic region, cont.

Enforce uniqueness: min

h∈H1(H)

1 2h − hhealthy2

H1(H)

subject to the constraints ∇ · [(Mi + Me)∇r] + ∇ · (Mi∇h) = in H, ([Mi + Me] ∇r) · nH + (Mi∇h) · nH = −(Mo∇ro) · nT

• n ∂H,

r = ro

• n ∂H,

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Heart surface → ischemic region, cont.

Which leads to the saddle point problem: Determine (r, h) and (w, q) (dual) such that (h − hhealthy, ψ)H1(H) +

• H

Mi∇w · ∇ψ dx = ∀ψ,

• H

(Mi + Me)∇w · ∇ψ dx + (Tψ, q)H1/2(∂H) = ∀ψ,

• H

(Mi + Me)∇r · ∇ψ dx +

• H

Mi∇h · ∇ψ dx + g, Tψ = ∀ψ, (Tr − d, ψ)H1/2(∂H) = ∀ψ. (d=ECG) Unique solution which depends continuously on the data, well-posed.

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Theoretical considerations, cont.

ECG → heart surface: Uniqueness, unstable (well-known) Heart surface → ischemic region: Non-uniqueness, stable (new) Provided that you have a good geometrical model of the heart and a reasonable prior, don’t stop at the heart surface!

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Validation

Data collection

• MRI (geometrical model)
• ECG (electrical potential)
• Perfusion scintigraphy (visualize ischemic region)

Processing

• Geometrical modelling
• ECG-Analyzer, compute ischemic region

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Roughly consistent with the scintigraphy

Patient 013: Inverse solution

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Patient 013

Scintigraphy Inverse ECG Electrodes

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Patient 021

Scintigraphy Inverse ECG Electrodes

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Patient 001

Scintigraphy Inverse ECG Electrodes

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Challenges

• Fiber structure
• Further tests needed
• Quality of the ECG recordings
• Use more of the ECG recordings?
• Etc.

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Acknowledgement

We would like to thank

• Drs. Patrick A. Helm and Raimond L. Winslow at the

Center for Cardiovascular Bioinformatics and Modeling and

• Dr. Elliot McVeigh at the National Institute of Health for

provision of data. for their DTMRI data, which we used to generate fiber structures.