Why? The classical inverse ECG problem. The Bidomain model. Outside the heart

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Daniel Ford
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1 Ú Áion Úµ Ö Å ÖÚµ Ö Å ÖÙ µ Ñ Ø ¾»» The classical inverse ECG problem Is it possible to compute the electrical potential at the surface of the heart from body surface measurements? Why? Improve traditional ECG recordings Better qualitative and quantitative understanding of the heart Detect diseases and malfunctions... H T H T ½» The Bidomain model H T Outside the heart In Ì (torso): H T Ö ÅÖÙµ ¼ in Ì Ò ¼ along Ì ÅÖÙµ (Not a closed problem!) Ö Å ÖÚµ Ö Å Å µöù µ ¼ Ö Å ÖÙ ¼ in Ì in À in À H H T T Á ÓÒ :ioniccurrent Å Å :conductivitytensors Ú Ù Ù :membranepotential»

»» ECG (electrocardiogram) ECGrecording Øµalong Ì Ø Ø Focusononetimeinstance, Ø µ Briefly about the time dependent problem Outside the heart ECG Ì In (torso): ÅÖÙµ ¼ in Ì Ö Ò ¼ along Ì ÅÖÙµ

2 »» ECG (electrocardiogram) ECGrecording Øµalong Ì Ø Ø Focusononetimeinstance, Ø µ Briefly about the time dependent problem Outside the heart ECG Ì In (torso): ÅÖÙµ ¼ in Ì Ö Ò ¼ along Ì ÅÖÙµ ECGrecordingof Ì Ùalong H T Ù ÐÓÒ À H T» The Challenge, cont. Operator Ê µ Ù µ,where Ù Ù µsolves Properties Solve ÅÖÙµ ¼ in Ì Ö Ò ¼ along Ì ÅÖÙµ along À Ù Find suchthat for. Êisalinearoperator () is ill-posed ¾ ÊÒ Êµ If Ê µ ½µ Ê µ where isthedatafromtheecgrecording ÑÒ Ê µ ¾»

3 ½¼» ½¾» This lecture Fourier analysis on the unit square, stationary The general case, stationary The time dependent problem Numerical results Fourier analysis Unit square y u=g? δn δu = u = δu δn = δu δn = x u=d» Fourier analysis Unit square y The direct problem Find Ù Ù µsatisfying δh Ù ¼ in Ì ÖÙ Ò ¼ along Ì Ù along À δt x Γ ½½»

4 Ù µ Ü Ýµ Ù Ü Ýµ ½ ¼ Ô Ó Üµ ½ ½ ¼ Ó Üµ Ó Ýµ ½ ¼ Ô Ó Üµ ½» ½» The direct problem, cont. Separation of variables: The direct problem, cont. Linearity: ¼ ½ Æ Ü Ýµ Ó Üµ Ó Ýµ satisfies Ù Ü Ýµ ½ where areconstants,satisfies Ù ¼ in Ì Ù ¼ in Ì ÖÙ Ò ¼ along Ì ÖÙ Ò ¼ along Ì ½» The direct problem, cont. Fouriercosineseriesof : The direct problem, cont. Ê:heartsurface bodysurface Ê µ Ê Üµ Ù µ Ü ¼µ Solution formula for the direct problem Ô Ó Üµ Ó µ ¼ Ô Ó Üµ Ó Ýµ Ó µ ¼ ½»

5 ½ Ô Ó µ Ô ¼ Ô Ó Üµ ½ Ê Ó Üµµ ½» ¾¼» The direct problem, cont. Fouriercoeff.: Ô The direct problem, cont. δ= δ=.5 Large u u Ô Ó µ strong damping effect Ê has a strong smoothing effect y x y x.4.2 δ= δ= u 2 3 u y x y x.4.2 ½» The inverse problem Ê:heartsurface bodysurface ForagivenECGrecording,find suchthat The inverse problem, cont Consequently ½ Ó Üµ Ó µ Recall that Eigenvalues Ê µ ½ Ô Ó Üµ Ó µ ½ ¾ Ó µ Ê Zeroisaclusterpointfor Ênotcontinuouslyinvertible, Ê ½ not well-behaved ¼ ½»

6 ½ ¼ Ó Üµ ¼ Ó Üµ Ê ½ Æ µ Ê ½ µ Ä ½ Ç ½¼ µ ½ ¼ Ó µ Ó Üµ ½ Ê and ½ Ê ½ ¾ Ê ½ Ê ¾¾» The inverse problem, cont. Fourier expansion The inverse problem, cont. Consequently Üµ Ê ½ Üµµ Ê ½ ½ ¼ Ó Üµ Üµ Ê µ CanEasysolve ½ for ¼ Ô Ó Üµ: È yields ¼ Ô Ó µ for ¼ ½ Fouriercoeff.: Ó µ Evenforsmall, Ó µislarge,e.g. Ó µ ¾ ½¼ Ê µ ½ Ô Ó Üµ Ó µ ½ ¾½» Example Exactdata, Üµ Ó ½ µ Ó Üµ Error-pronedata, Üµ Æ Ó Üµ Æ Üµ Then Regularization Output least squares, minimize Ê µ ¾ Ä ¾ µ Â µ Tikhonov regularization Ê ½ Æ µ Ê ½ µ ¾ ½¼ Æ Ó Üµ Forexample, Æ Ä ½ Ç ½¼ µimpliesthat Â µ Ê µ ¾ Ä ¾ µ ¾ Ä ¾ Àµ Second order Tikhonov regularization ¾ µ Ê µ ¾ Ä ¾ µ ÜÜ ¾ Ä ¾ Àµ Â Approximations (derivedfrom ÖÂ ¼and ÖÂ ¾ ¼) ¾» ¾»

7 ½ Ê ½ ¾ Ê ½ ¼ Ó Üµ ½ ¼ Ó Üµ ½ ½ ¼ Ó µ Ó Üµ ½ Ó ¾ µ Ó Üµ Æµ Ê ½ µ Ê ½ Æ µ ¾ Ä ¾ Àµ ¾» ¾» Regularization, cont. Ê:heartsurface bodysurface No regularization ½ ½ Ó Üµ Ê ¼ ½ Regularization, cont. Forthelowfrequencycomponentsofthedata,the actionof Ê, and isalmostidentical, ½ Ê ½ ½ Ê ¾ providedthat issmall Thehighfrequencycomponentsof aredamped efficientlyby and ½ Ê ½ ¾ Ê Tikhonov Ó µ ¼ Second order Tikhonov Ó µ ½ µ Ó ¾ µ Ó Üµ ¾» ¼ Example, revisited Exactdata, Üµ Ó ½ µ Ó Üµ Error-pronedata, Üµ Æ Ó Üµ Æ Üµ Tikhonov, error Example, revisited - cont. errorontheheartsurface Ä No ¾ regularization ¾ ½¼ Æ Æµ Tikhonov(optimal regularization) 4 x 5 3 δ= δ= 2 δ= 3 Æµ ¼ ½¼ Æ Second order Tikhonov(optimal regularization) 2 ¾ Æµ ½¼ Æ 2 x 5 ¾»

8 ¼» ¾» Example, revisited - cont. Second order works better than plain Tikhonov regularization Ingeneral,difficulttofindanoptimalvalueforthe regularization parameter The general case Ê:heartsurface (partofthe)bodysurface Complex geometry Non-constant Å conductivity Fourier analysis impossible H T H T ¾» The general case, cont. Operator Ê µ Ù µ,where Ù Ù µsolves Linearity Êisalinearoperator: Ò ¼ along Ì ÅÖÙµ along À Ù and À. Find suchthat Ê ½ ½ ¾ ¾ µ ½ Ê ½ µ ¾ Ê ¾ µ foranyscalars ½ and ¾ andfunctions ½ and ¾ defined on À. We will use this fact to discretize our inverse problem Ö ÅÖÙµ ¼ in Ì Ê µ ½»

9 ¾ ÊÒ Ê Ò µ Ê Ò Ê Î Ò ÑÒ Ê Ò µ ¾ Ä ¾ µ ¾ÎÒ Ò Ê Ò µ ¾ Ä ¾ µ ¾ Ä ¾ Àµ Ó ÑÒ ¾ÎÒ Ê Ò µ Ò Ò Ê Ò µ ¾ Ä ¾ µ ¾ Ä ¾ Àµ Ò»» Discretization Linearly independent functions Discretization, cont. ¾ Î Ò : ½ Ò À ÁÊ and where Ô arescalars. Consequently, if ½ Ô for ½ Ò Î Ò ÔÒ ½ Ò Ê Ò µ Ö Ê thenthelinearityof Ò impliesthat Ô Ö ½» Discretization, cont. Original problem Discretization, cont. È Ò ½ Ô,thus Ê Ò µ Â µ Â Ô ½ Ô Ò µ Tikhonov where ½ Ö Ê Ò µ Ô Ö ¾ Ä ¾ µ Ò ½ for ½ Ò Ô ¾ Ä ¾ Àµ»

10 Ò» ¼» Discretization, cont. The condition Discretization, cont. Whichmaybewrittenontheform Â ¼ Ô givesthe Ò Òsystem for ½ Ò where Ô Ö Ö Ü Ü Ô for ½ Ò ¾ Ê Ò Ò Ö Ö Ü À Ü Ö Ü Ô Ô ½ Ô Ò µ Ì ¾ Ê Ò À ½ Ö ½ Ü Ö Ò Ü Ì ¾ Ê Ò» An algorithm a) Pick Ò linearly independent functions An algorithm, cont. d)computethematrix e)computetherighthandside definedatthesurface Àoftheheart À b)for ½ Ò,set inthedirectproblemand solveitfor Ù Ù µ c) Compute f)solvethelinearsystem Ô for Ô g)computethepotential attheheartsurfaceby Ò ½ Ô ½ Ò À ÁÊ Ö Ù µ ½ Ò For each newobservation, onlysteps e)-g) have to be carried out.(important for the time dependent problem)»

11 SecondorderTikhonov, ½¼ ¾»» Example 2 Example 2, cont. Tikhonov, ½¼ s at the heart surface cm Estimated s at the torso surface Computed s at the heart surface Estimated Epicardial node number s at the torso surface Computed Body surface node number Epicardial node number Body surface node number ½» Example 2, cont. SecondorderTikhonov, ½±noise, ½ The time dependent problem Timeinstances Ø ¼ Ø Å withdata s at the heart surface Estimated s at the torso surface Computed ¼ Å ¾ Ä ¾ µ defined at the body surface Compute the corresponding potentials at the heart surface 2 4 ¼ Å Epicardial node number Body surface node number Brute force: Solve Ô for ¼ Å»

12 ÑÒ Ê Ò µ ¾ Ä ¾ µ ¾ÎÒ ÑÒ Ê Ò µ ¾ Ä ¾ µ ¾ÎÒ ÑÒ Ê Ò µ ¾ Ä ¾ µ ¾ÎÒ ½ ¾ Ä ¾ Àµ ½ ¾ Ä ¾ Àµ À ¾ Ä ¾ À ¼¼½ ÑÒ Ê Ò µ ¾ Ä ¾ µ ¾ÎÒ»» The time dependent problem, cont. Ensure that the change in the epicardial potential is smallfromonetimesteptothenext Example 3 ½ ¾ Ä ¾ Àµ 3 s at the heart surface Estimated ½ Å for Hybrid scheme À ÙÖÐ À ÙÖÐ À -Laplace-Beltrami ( operator) More advanced schemes(f. Greensite) Epicardial node number» Example 3, cont. ¼¼½and ½ s at the heart surface Epicardial node number ½ ¾ Ä ¾ Àµ À ¾ Ä ¾ Àµ Estimated Summary Aim: To compute the potential at the heart surface from body surface measurements(ecgs) Leadstoalinearproblem Ê µ Ill-posed From a mathematical point of view, fairly simple Second order Tikhonov regularization works well Main practical problems: Noisy ECG data High quality geometrical models of the body required»

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