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

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) 1

2 Ischemia? f − = ) u ∇ K ( ⋅ ∇

The Bidomain Model (Geselowitz, Miller, Schmitt, Tung, et al in the 70’s) = ( , ) in s F s v H t ∂ T + = ∇ ⋅ ∇ + ∇ ⋅ ∇ ( , ) ( ) ( ) in v I s v M v M u H ∂ H t i i e H T ∇ ⋅ ∇ + ∇ ⋅ + ∇ = ( ) (( ) ) 0 in M v M M u H i i e e ∇ ⋅ ∇ = 0 in M u T o o = − : transmembrane potential v u u • i e s • : state vector representing cell properties, transport of ions (1 to 50 entries) • : conductivity tensors M , i M e 3

Mathematical model (Heart) Bidomain model (very CPU demanding) + = ∇ ⋅ ∇ + ∇ ⋅ ∇ ( , ) ( ) ( ) v I v s M v M u t i i e = ∇ ⋅ ∇ + ∇ ⋅ + ∇ 0 ( ) (( ) ) M v M M u i i e e If v is given, we obtain a stationary model: ∇ ⋅ + ∇ = −∇ ⋅ ∇ (( ) ) ( ) M M u M v i e e i 4

Stationary model • During rest − ⎧ 60 ischemic tissue mV = ≈ ⎨ v v r − 90 healthy ti ssue mV ⎩ • Plateau phase ⎧− 20 ischemic tissue mV = ≈ ⎨ v v p 0 healthy ti ssue mV ⎩ 5

Stationary model • Model for the shift r: ∇ ⋅ + ∇ = −∇ ⋅ ∇ (( ) ) ( ) M M r M h i e i where ⎧ 40 ischemic tissue mV = − ≈ ⎨ h v v p r ⎩ 90 healthy ti ssue mV 6

Inverse Problem • ECG d, cost-functional = ∫ − + α − 2 2 ( ) ( ( ) ) || || J h r h d ds h h 1 H ( ) H α healthy ∂ B • Inverse problem min ( ) J h α subject to h ∇ ⋅ + ∇ = −∇ ⋅ ∇ [( ) ( )] [ ] M M r h M h i e i 7

Inverse Problem, continued = ∈ ≈ • Ischemic region { | ( ) 40 } x H h x mV • Or employ level set function • Conductivities depend on the ischemia? 8

Theoretical considerations Split ECG Ischemic region into two sub problems ECG Heart surface Ischemic region 9

ECG → heart surface Classical Cauchy problem; compute r o on ∂H by solving ∇ · ( M o ∇ r o ) = 0 in T, ( M o ∇ r o ) · n = 0 on ∂B, = ECG on Γ ⊂ ∂B. r o ∂ B ∂ H H T Severely ill-posed; typical error amplification d k → e kπ d k , { d k } Fourier coefficients of the ECG. 1/5

Heart surface → ischemic region Determine r and h such that ∇ · [( M i + M e ) ∇ r ] + ∇ · ( M i ∇ h ) = 0 in H, ([ M i + M e ] ∇ r ) · n H + ( M i ∇ h ) · n H = − ( M o ∇ r o ) · n T on ∂H, = on ∂H, r r o Ischemic region = { x ∈ H | h ( x ) ≈ 40 } Non-uniqueness 2/5

Heart surface → ischemic region, cont. Enforce uniqueness: 1 2 � h − h healthy � 2 min H 1 ( H ) h ∈ H 1 ( H ) subject to the constraints in H, ∇ · [( M i + M e ) ∇ r ] + ∇ · ( M i ∇ h ) = 0 on ∂H, ([ M i + M e ] ∇ r ) · n H + ( M i ∇ h ) · n H = − ( M o ∇ r o ) · n T = on ∂H, r r o 3/5

Heart surface → ischemic region, cont. Which leads to the saddle point problem: Determine ( r, h ) and ( w, q ) (dual) such that � ( h − h healthy , ψ ) H 1 ( H ) + M i ∇ w · ∇ ψ dx = 0 ∀ ψ, H � ( M i + M e ) ∇ w · ∇ ψ dx + ( Tψ, q ) H 1 / 2 ( ∂H ) = 0 ∀ ψ, H � � ( M i + M e ) ∇ r · ∇ ψ dx + M i ∇ h · ∇ ψ dx + � g, Tψ � = 0 ∀ ψ, H H ( Tr − d, ψ ) H 1 / 2 ( ∂H ) = 0 ∀ ψ. (d=ECG) Unique solution which depends continuously on the data, well-posed. 4/5

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! 5/5

Validation Data collection - MRI (geometrical model) - ECG (electrical potential) - Perfusion scintigraphy (visualize ischemic region) Processing - Geometrical modelling - ECG-Analyzer, compute ischemic region 10

11 Roughly consistent with the scintigraphy Patient 013: Inverse solution

12 Electrodes Patient 013 Inverse ECG Scintigraphy

13 Electrodes Patient 021 Inverse ECG Scintigraphy

14 Electrodes Patient 001 Inverse ECG Scintigraphy

Challenges • Fiber structure • Further tests needed • Quality of the ECG recordings • Use more of the ECG recordings? • Etc. 15

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. 16