Decoding complex cardiac arrhythmia using mathematical optimization

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1 Sebastian Sager, Florian Kehrle, Eberhard Scholz Decoding complex cardiac arrhythmia using mathematical optimization June 3, 2014 MINLP2014, Pittsburgh 1 S. Sager Cardiac Arrhythmia

2 Outline 1 Medical Problem 2 Mathematical Approaches 3 Inverse Simulation of Multilevel AV block 4 Modeling of the Optimization Problem 5 Algorithmic Approaches 6 Numerical Results and Summary 7 Appendix 2 S. Sager Cardiac Arrhythmia

3 Reminder: the human heart [Images: Wikipedia] Pacemaker signal in sinoatrial node 3 S. Sager Cardiac Arrhythmia

4 Reminder: the human heart [Images: Wikipedia] Atrial chambers 3 S. Sager Cardiac Arrhythmia

5 Reminder: the human heart [Images: Wikipedia] P Atrial depolarization atrial contraction 3 S. Sager Cardiac Arrhythmia

6 Reminder: the human heart [Images: Wikipedia] P Atrial depolarization atrial contraction 3 S. Sager Cardiac Arrhythmia

7 Reminder: the human heart [Images: Wikipedia] P Atrial depolarization atrial contraction 3 S. Sager Cardiac Arrhythmia

8 Reminder: the human heart [Images: Wikipedia] PR Atrioventricular node 3 S. Sager Cardiac Arrhythmia

9 Reminder: the human heart [Images: Wikipedia] Q Depolarization of the interventricular septum 3 S. Sager Cardiac Arrhythmia

10 Reminder: the human heart [Images: Wikipedia] R Polarization of the ventricles 3 S. Sager Cardiac Arrhythmia

11 Reminder: the human heart [Images: Wikipedia] S (De)polarization 3 S. Sager Cardiac Arrhythmia

12 Reminder: the human heart [Images: Wikipedia] S Depolarization, contraction 3 S. Sager Cardiac Arrhythmia

13 Reminder: the human heart [Images: Wikipedia] ST Depolarization, contraction 3 S. Sager Cardiac Arrhythmia

14 Reminder: the human heart [Images: Wikipedia] ST Depolarization, contraction 3 S. Sager Cardiac Arrhythmia

15 Reminder: the human heart [Images: Wikipedia] T Secondary excitation 3 S. Sager Cardiac Arrhythmia

16 Reminder: the human heart [Images: Wikipedia] T Secondary excitation 3 S. Sager Cardiac Arrhythmia

17 Reminder: the human heart [Images: Wikipedia] T Secondary excitation 3 S. Sager Cardiac Arrhythmia

18 Reminder: the human heart [Images: Wikipedia] Rest 3 S. Sager Cardiac Arrhythmia

19 Reminder: the human heart [Images: Wikipedia] Rest 3 S. Sager Cardiac Arrhythmia

20 Atrial Fibrillation Regular electric impulses are overwhelmed by disorganized ones (non-constant frequency) 4 S. Sager Cardiac Arrhythmia

21 Atrial Flutter Regular electric impulses (constant frequency) in the atria, may be filtered Makes sense: pumping inefficient if too fast 5 S. Sager Cardiac Arrhythmia

22 Atrial Flutter But: filter may also result in irregular signal! 6 S. Sager Cardiac Arrhythmia

23 Summary of the decision problem Two possible reasons for chaotic ECG data (R waves): 1 Atrial fibrillation irregular atrial signal 2 Secondary tachycardia regular atrial signal Also different treatments!! 1 Mainly drug treatments 2 Mainly ablation More and more appearances of irregular flutter as secondary tachycardia after ablation Why should it be difficult to distinguish them from the ECG? 7 S. Sager Cardiac Arrhythmia

24 So, what do you think? Fibrillation or Flutter? 8 S. Sager Cardiac Arrhythmia

25 So, what do you think? Flutter, Flutter, Fibrillation 8 S. Sager Cardiac Arrhythmia

26 So, what do you think? 9 S. Sager Cardiac Arrhythmia

27 So, what do you think? 10 S. Sager Cardiac Arrhythmia

28 So, what do experts think? [Shiyovich et al Am J Med Sci] 11 S. Sager Cardiac Arrhythmia

29 So, what do (current) expert systems think? 12 S. Sager Cardiac Arrhythmia

30 So, what do (current) expert systems think? 12 S. Sager Cardiac Arrhythmia

31 Outline 1 Medical Problem 2 Mathematical Approaches 3 Inverse Simulation of Multilevel AV block 4 Modeling of the Optimization Problem 5 Algorithmic Approaches 6 Numerical Results and Summary 7 Appendix 13 S. Sager Cardiac Arrhythmia

Large variety of different approaches possible Based on statistics (of RR-intervals) Bayesian logic Fourier transforms Wavelets Clustering

.. Based on cellular automata Shows wave propagation patterns depending on probabilities of excitation by neighboring cells [M.

32 Large variety of different approaches possible Based on statistics (of RR-intervals) Bayesian logic Fourier transforms Wavelets Clustering of RR times [Esperer et al ANE] Machine learning Nonlinear time series analysis... Based on cellular automata Shows wave propagation patterns depending on probabilities of excitation by neighboring cells [M. Small, Dynamics of Biological Systems, CRC, 2012] Based on first-principle models PDEs, SDEs, DDEs, ODEs,... Survey [D. Noble, 2012]: 100 models with significant contributions 14 S. Sager Cardiac Arrhythmia

33 The Noble model [Noble, D Journal of Physiology] ODEs model action potential based on Hodgkin-Huxley equations [Nobel Prize 1963] The electrical potential V across the membrane changes due to ionic currents Sodium current in channels m and h, potassium current n dv dt dm dt dh dt dn dt = i Na + i K + i Leak = (4 105 m 3 h + 140)(V E Na ) Cm Cm V 90 (1200e e V (V E K ) n 4 (V E K ) + 75(V E An ) Cm 100( V 48) = exp(( V 48)/15) 1 (1 m) 120(V + 8) exp((v + 8)/5) 1 m V = 170 exp( )(1 h) exp(( V 42)/10) h 0.1( V 50) 90 = (1 n) exp( V )n exp(( V 50)/10) S. Sager Cardiac Arrhythmia

34 Simulation of Noble model [Noble, D Journal of Physiology] Successfully predicted several (unknown) phenomena Many extensions, models with 100 states or PDEs Very tough optimization problems [Lebiedz & Sager, Physical Review Letters, 2005] 16 S. Sager Cardiac Arrhythmia

35 Outline 1 Medical Problem 2 Mathematical Approaches 3 Inverse Simulation of Multilevel AV block 4 Modeling of the Optimization Problem 5 Algorithmic Approaches 6 Numerical Results and Summary 7 Appendix 17 S. Sager Cardiac Arrhythmia

36 Phenomenological approach Important & difficult to distinguish between fibrillation and flutter Exiting approaches have shortcomings, not real-time feasible 18 S. Sager Cardiac Arrhythmia

Phenomenological approach Important & difficult to distinguish between fibrillation and flutter Exiting approaches have shortcomings, not real-time feasible Idea: let us look at

37 Phenomenological approach Important & difficult to distinguish between fibrillation and flutter Exiting approaches have shortcomings, not real-time feasible Idea: let us look at simpler phenomenological models Well known in medicine: different kinds of AV blocks Type Mobitz I Type Mobitz II (Wenckebach) Linear prolongation of intervals 18 S. Sager Cardiac Arrhythmia

38 Multilevel approach Idea: consider sequential filters of this simple type! 19 S. Sager Cardiac Arrhythmia

39 Multilevel approach Idea: consider sequential filters of this simple type! 1912: Detection of ventricular arrhythmia for atrial flutter 1950: Implication of two block levels 1975: first EPU indicating localisation of block (= AV node!) 1976: Called Multilevel AV-Block [Kosowsky et al Circulation] 1982: last high-impact paper on this topic 19 S. Sager Cardiac Arrhythmia

40 Simulation of Mobitz block Input : n α incoming time points α i, transit data τ con Output: n β time points β j after Mobitz type block begin j := 1, r := 0 for i = 1... n α do /* Signal can be processed */ if α i + τ con r then β j = α i + τ con r = β j + τ ref j = j + 1 n β = j 1 end 20 S. Sager Cardiac Arrhythmia

41 Simulation of Wenckebach block Input : n α incoming signal α i, transit data τ con, τ inc, refrac time τ ref Output: n β time points β j after Wenckebach type block begin j := 1, c := 0 for i = 1... n α do /* Signal can be processed */ if τ con + c τ inc τ ref then β j = α i + τ con + c τ inc j = j + 1, c = c + 1 /* Signal can not be processed */ else c = 0 n β = j 1 end 21 S. Sager Cardiac Arrhythmia

42 Basic idea of our approach Published in [Scholz, E.P., Kehrle, F., Vossel, S., Hess, A., Zitron, E., Katus, H.A., Sager, S., Discriminating atrial flutter from atrial fibrillation using a multilevel model of atrioventricular conduction, Heart Rhythm, 2014, 11(5), ] (Heart Rhythm has impact Factor 5.2) Regard the inputs to simulation as optimization variables Regular signal α i = α in atrium Number n lvl and type π j of levels transit data τcon, j τ j inc and refrac time τ j ref for all levels Minimize deviation of forward simulation from ventricular data 22 S. Sager Cardiac Arrhythmia

43 Basic idea of our approach Published in [Scholz, E.P., Kehrle, F., Vossel, S., Hess, A., Zitron, E., Katus, H.A., Sager, S., Discriminating atrial flutter from atrial fibrillation using a multilevel model of atrioventricular conduction, Heart Rhythm, 2014, 11(5), ] (Heart Rhythm has impact Factor 5.2) Regard the inputs to simulation as optimization variables Regular signal α i = α in atrium Number n lvl and type π j of levels transit data τcon, j τ j inc and refrac time τ j ref for all levels Minimize deviation of forward simulation from ventricular data Verify / falsify hypothesis atrial flutter : Objective small indication for atrial flutter Objective high indication for atrial fibrillation 22 S. Sager Cardiac Arrhythmia

44 Visualizing example of our approach 23 S. Sager Cardiac Arrhythmia

45 Outline 1 Medical Problem 2 Mathematical Approaches 3 Inverse Simulation of Multilevel AV block 4 Modeling of the Optimization Problem 5 Algorithmic Approaches 6 Numerical Results and Summary 7 Appendix 24 S. Sager Cardiac Arrhythmia

46 Challenges in MINLP modeling What are the (state) variables? Dimensions? Depend on α! 25 S. Sager Cardiac Arrhythmia

47 Challenges in MINLP modeling What are the (state) variables? Dimensions? Depend on α! Logical equivalences x j i + τ j con r j i 1 δj i = 1 25 S. Sager Cardiac Arrhythmia

48 Challenges in MINLP modeling What are the (state) variables? Dimensions? Depend on α! Logical equivalences x j i + τ j con r j i 1 δj i = 1 How to model inner for loop with unknown length? for i = 1... n α do if τ con + c τ inc τ ref then β j = α i + τ con + c τ inc j = j + 1, c = c + 1 else c = 0 25 S. Sager Cardiac Arrhythmia

49 Challenges in MINLP modeling What are the (state) variables? Dimensions? Depend on α! Logical equivalences x j i + τ j con r j i 1 δj i = 1 How to model inner for loop with unknown length? for i = 1... n α do if τ con + c τ inc τ ref then β j = α i + τ con + c τ inc j = j + 1, c = c + 1 else c = 0 How to model objective function? Least-squares? l 1? Huber? How to penalize wrong number of signals compared to data? How to make sure signal i is compared to correct data j? 25 S. Sager Cardiac Arrhythmia

50 Variables Symbol Range Set J 1... n lvl Block levels (0 =atrium, n lvl + 1 =ventricle) I 1... n α Time signals atrium (output) V 1... n β Time signals ventricle (input) Symbol Range Meaning xk v k V Impulse times ventricle xi 0 i I Impulse times atrium x j i i I, j J Impulse times passing level j r j i i I, j J Cells are quiescent (excitation possible) τ j ref j J Refractory time τ j ref τcon j j J Transition constant τcon j τ j inc j J Transition increment τ j inc α Regular atrium signal π j {0, 1} j J Mobitz or Wenckebach? y j i N i I, j J Wenckebach Counter δ j i {0, 1} i I, j J Signal i arriving at level j? φ k,i {0, 1} k V, i I Permutation matrix 26 S. Sager Cardiac Arrhythmia

51 MINLP formulation subject to min x 0 i,xj i,rj i,τ j ref,τ j con,τ j inc, α,πj,y j i,δj i,φ k,i ( φ k,i x n lvl i k V i I x v k ) 2 Bounds and integrality x 0 i+1 = x0 i + α, i {1,..., n α 1} Permutation matrix φ k,i makes sense Signal x n lvl i is result of forward simulation 27 S. Sager Cardiac Arrhythmia

52 Permutation matrix e.g., 2:1 Mobitz φ = ( ) 28 S. Sager Cardiac Arrhythmia

53 Permutation matrix e.g., 2:1 Mobitz φ = ( Signal conducted? Only possible if at parent level δ 0 i = 1, i I δ j 1 i δi, j i I, j J Connection permutation matrix and block. Upper triangular φ k,i =δ n lvl i, i I k V φ k,i = 0, k V, i I with k > i Only 1 signal, no overtaking φ k,i 1, k V i I n 1 φ k+1,n φ k,i 0, k {1,..., n β 1}, n I i=1 ) 28 S. Sager Cardiac Arrhythmia

54 subject to Mobitz block: π j = 1 Conducted? x j i + τ j con r j i 1 δj i = 1 π j δi(x j j i + τcon j r j i 1 ) 0, i I, j J π j (1 δi)(x j j i + τcon j r j i 1 + ε) 0, i I, j J Quiescent? r j i changes δ j i = 1 r j 0 = 0, j J (1 δ j i)(r j i 1 rj i) = 0, i V, j J π j δ j i(x j+1 i + τ j ref rj i) = 0, i V, j J Set time x j i π j (δi(x j j i + τcon) j x j+1 i ) = 0, i I, j J 29 S. Sager Cardiac Arrhythmia

55 subject to Wenckebach block π j = 0 Conducted? τcon j + y j i 1 τ j inc τ j ref δj i = 1 (1 π j )δi(τ j con j + y j i 1 τ j inc τ j ref ) 0, i I, j J (1 π j )(1 δi)(τ j con j + y j i 1 τ j inc τ j ref ε) 0, i I, j J Increase Wenckebach counter y j i δ j i = 1 y j 0 = 0, j J δi(y j j i (y j i 1 + 1)) = 0, i V, j J (1 δi)y j j i = 0, i V, j J Avoid overtaking of signals δi(x j j+1 k x j+1 i + ε) 0, k > i I, j J Set time x j i (1 π j )(δi(x j j i + τcon j + y j i 1 τ j inc ) xj+1 i ) = 0, i I, j J 30 S. Sager Cardiac Arrhythmia

56 Outline 1 Medical Problem 2 Mathematical Approaches 3 Inverse Simulation of Multilevel AV block 4 Modeling of the Optimization Problem 5 Algorithmic Approaches 6 Numerical Results and Summary 7 Appendix 31 S. Sager Cardiac Arrhythmia

57 Optimization-driven simulation How to find the best values for τ j con, τ j inc, τ j ref, α, πj? We implemented 1 Enumeration on discrete grid 2 Derivative-free optimization Pattern search [Hooke, Jeeves] NEWUOA [Powell] Particle Swarm [Vaz, Vicente] 3 Dynamic Programming 4 Solution of the MINLP with Couenne and SCIP 2 levels and 10 signals in 30 seconds 5 Tailored branching algorithm 32 S. Sager Cardiac Arrhythmia

58 Tailored branching algorithm [Florian Kehrle] Work on (adaptive) grid as in enumeration Intelligent tree search, forward in time, like in [Jung, Reinelt, Sager, 2014 OMS] Remove subtrees with with unphysical behavior Remove subtrees which lead to missing ventricular signal 33 S. Sager Cardiac Arrhythmia

59 Tailored branching algorithm [Florian Kehrle] Work on (adaptive) grid as in enumeration Intelligent tree search, forward in time, like in [Jung, Reinelt, Sager, 2014 OMS] Remove subtrees with with unphysical behavior Remove subtrees which lead to missing ventricular signal evaluate simulations on nodes thus not really all-at-once (simulation and optimization) so far Merge upper and lower Mobitz block into neighbor level by clever modification of variables 33 S. Sager Cardiac Arrhythmia

60 Outline 1 Medical Problem 2 Mathematical Approaches 3 Inverse Simulation of Multilevel AV block 4 Modeling of the Optimization Problem 5 Algorithmic Approaches 6 Numerical Results and Summary 7 Appendix 34 S. Sager Cardiac Arrhythmia

61 Discrimination Examples [Scholz et al., Heart Rhythm, 2014] 35 S. Sager Cardiac Arrhythmia

62 Discrimination Results [Scholz et al., Heart Rhythm, 2014] Based on ECG data of 100 patients in Heidelberg Comparison to intracardiac measurements, verified by two experts Sensitivity 79%, specificity 100%. RR statistics only 58% / 24%! 36 S. Sager Cardiac Arrhythmia

63 Summary Challenging and important medical problem Innovative optimization-driven inverse simulation approach already 80 90% specificity / sensitivity Company mathe.medical founded for dissemination Challenging MI(N)LP formulation with open questions Structure-exploiting branching best algorithm so far Future: extend models to include rare events Future: remove symmetry Future: include uncertainty / stochastics Future: link to differential equations / automata Thank you for your attention! 37 S. Sager Cardiac Arrhythmia

64 Outline 1 Medical Problem 2 Mathematical Approaches 3 Inverse Simulation of Multilevel AV block 4 Modeling of the Optimization Problem 5 Algorithmic Approaches 6 Numerical Results and Summary 7 Appendix 38 S. Sager Cardiac Arrhythmia

65 Dynamic Programming data/kosowsky1.dat, Block 2, Mode 1 4 A Ladder diagram 2 1 B1 M 220: 212 B2 W 100: :11 V : Time in msec Dynamic Programming looks promising? 39 S. Sager Cardiac Arrhythmia

66 Dynamic Programming data/kosowsky1.dat, Block 2, Mode 1 4 A Ladder diagram 2 1 B1 M 220: 212 B2 W 100: :11 V : Time in msec Dynamic Programming looks promising? Controls α, π j, τcon, j τ j inc, τ j ref independent of i. Enumeration! 39 S. Sager Cardiac Arrhythmia

67 Dynamic Programming A B1 M 220: 212 B2 W 100: :11 V 27:11 data/kosowsky1.dat, Block 2, Mode Time in msec Ladder diagram Dynamic Programming looks promising? Controls α, π j, τcon, j τ j inc, τ j ref independent of i. Enumeration! Maybe rotate problem? Now dependent on j! 39 S. Sager Cardiac Arrhythmia

68 Dynamic Programming A B1 M 220: 212 B2 W 100: :11 V 27:11 data/kosowsky1.dat, Block 2, Mode Time in msec Ladder diagram Dynamic Programming looks promising? Controls α, π j, τcon, j τ j inc, τ j ref independent of i. Enumeration! Maybe rotate problem? Now dependent on j! But few levels compared to signals, curse of dimensionality 39 S. Sager Cardiac Arrhythmia

69 Calcium concentration in cells, [U. Kummer et al., 2002] State: a activated G-proteins State: b active phospholipase C State: c intracellular calcium State: d intra-er calcium Control: timing of inhibitors [Lebiedz, S. et. al., Physical Review Letters, 2005] min y,w,w max tf t 0 y(t) ŷ 2 dt subject to ȧ = k 1 + k 2 a k 3 a b a+k 4 k 5 b ḃ = k 7 a k 8 b+k 9 b c d ċ = k 10 a c a+k 6 d+k 11 + k 12 b + k 13 a k 14 ḋ = k 10 b c d d+k 11 + k 16 c c+k 17 d 10 c w c+k 15 k 16 y(t 0 ) = y 0, w(t) {1, w max }, y(t) = (a, b, c, d) T c c+k 17 + d S. Sager Cardiac Arrhythmia

70 Simulation w(t) = 1 41 S. Sager Cardiac Arrhythmia

71 Simulation w(t) = 1 w(t) = w max 41 S. Sager Cardiac Arrhythmia

72 Integer solution Control States 42 S. Sager Cardiac Arrhythmia

73 Integer solution - after small change Control too early States 43 S. Sager Cardiac Arrhythmia

74 State is instable 44 S. Sager Cardiac Arrhythmia

75 Objective function - scale S. Sager Cardiac Arrhythmia

76 Objective function - scale S. Sager Cardiac Arrhythmia

77 Objective function - optimum 47 S. Sager Cardiac Arrhythmia

78 Objective function - border 48 S. Sager Cardiac Arrhythmia

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