Arrhythmias By Dimension or Heart Attacks can give you Mathematics

Transcription

1 Arrhythmias By Dimension or Heart Attacks can give you Mathematics J. P. Keener Mathematics Department Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.1/26

2 The Medical Problem Sudden Cardiac Arrest kills 1/2 million people in the US each year. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.2/26

3 The Medical Problem Sudden Cardiac Arrest kills 1/2 million people in the US each year. Antiarrhythmic drugs are not reliable. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.2/26

4 The Medical Problem Sudden Cardiac Arrest kills 1/2 million people in the US each year. Antiarrhythmic drugs are not reliable. Implantable defibrillators are expensive. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.2/26

5 The Medical Problem Sudden Cardiac Arrest kills 1/2 million people in the US each year. Antiarrhythmic drugs are not reliable. Implantable defibrillators are expensive. Defibrillators are not always available when needed. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.2/26

6 The Challenge Identify the different arrhythmias and their underlying mechanisms. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.3/26

7 The Challenge Identify the different arrhythmias and their underlying mechanisms. Understand the cause or origin of each arrhythmia Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.3/26

8 The Challenge Identify the different arrhythmias and their underlying mechanisms. Understand the cause or origin of each arrhythmia Determine how to control the arrhythmia. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.3/26

9 Conduction system of the heart Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.4/26

10 Conduction system of the heart Electrical signal originates in the SA node. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.4/26

11 Conduction system of the heart Electrical signal originates in the SA node. The signal propagates across the atria, through the AV node, and throughout the ventricles. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.4/26

12 Conduction system of the heart Electrical signal originates in the SA node. The signal propagates across the atria, through the AV node, and throughout the ventricles. The muscle cells contract in unison, and then relax awaiting the next signal. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.4/26

13 Some Examples of EKG s Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.5/26

14 Theories of Arrhythmias D=0: Individual cells run amuck. D=1: There are closed (reentrant) pathways around anatomical obstacles. D=2: There are self-sustained spiral waves. D=3: There are self-sustained scroll-waves. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.6/26

15 Dimension 0: Single Cells Cellular Transmembrane potential and electrocardiogram Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.7/26

16 Modeling Cardiac Electrical Activity Transmembrane potential φ is regulated by transmembrane ionic currents and capacitive currents: C m dφ dt + I ion(φ, w) = I in where dw dt = g(φ, w), w Rn Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.8/26

17 Thinking about Excitable Media The forest fire analogy Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.9/26

18 APD Alternans Action Potential Duration Restitution Curve AP D n + DI n = BCL. where AP D n = A(DI n 1 ) is the restitution curve. It follows that DI n = BCL A(DI n 1 ), APD Map Animated Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.10/26

19 Theories of Arrhythmias D=0: Individual cells run amuck. D=1: There are closed (reentrant) pathways around anatomical obstacles. D=2: There are self-sustained spiral waves. D=3: There are self-sustained scroll-waves. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.11/26

20 1 Dimensional Reentry AV node (Show movie) Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.12/26

21 Observation I D reentrant arrhythmias are 100% curable. Why could this reentrant arrhythmia not be cured? Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.13/26

22 The APD Instability in 1D Stable Pulse on a Ring Unstable Pulse on a Ring Collapse of Unstable Pulse Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.14/26

23 The APD Instability in 1D Stable Pulse on a Ring Unstable Pulse on a Ring Collapse of Unstable Pulse Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.14/26

24 Theories of Arrhythmias D=0: Individual cells run amuck. D=1: There are closed (reentrant) pathways around anatomical obstacles. D=2: There are self-sustained spiral waves. D=3: There are self-sustained scroll-waves. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.15/26

25 Dimension 2: Spirals Atrial Flutter Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.16/26

26 Dimension 2: Spirals Spiral instability - Meander: Atrial Flutter Torsade de Pointe Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.16/26

27 Dimension 2: Spirals Spiral instability - Meander: Atrial Flutter Torsahd duh Pwahn Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.16/26

28 The APD Instability in 2D Spiral Breakup Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.17/26

29 Theories of Arrhythmias D=0: Individual cells run amuck. D=1: There are closed (reentrant) pathways around anatomical obstacles. D=2: There are self-sustained spiral waves. D=3: There are self-sustained scroll-waves. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.18/26

30 Dimension 3: Ventricular Reentrant Activity Ventricular Monomorphic Tachycardia Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.19/26

31 Dimension 3: Scroll Waves Scroll Waves: Columns in the Ionic style, 400 B.C. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.20/26

32 Dimension 3: Cardiac Scroll Wave 3 D structure of a single scroll wave Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.21/26

33 Scroll Wave Cross-Sections Lesson to learn: Target patterns are not necessarily created by an autonomous oscillator. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.22/26

34 Ventricular Fibrillation Ventricular Fibrillation (Real Data) Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.23/26

35 Ventricular Fibrillation Ventricular Fibrillation (Real Data) Why do we not see self-sustained spirals? Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.23/26

36 Ventricular Fibrillation Ventricular Fibrillation (Real Data) Why do we not see self-sustained spirals? Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.23/26

37 Transparent View of Fibrillation Surface View Movie 3D View Movie Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.24/26

38 Summary Mathematics gives us tools to "see" things that cannot be seen in other ways. Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.25/26

39 Acknowledgments Collaborators Sasha Panfilov, University of Utrecht Brad Peercy, Rice University Eric Cytrynbaum, UC Davis, University of British Columbia Funding for this research kindly provided by the NSF. This talk can be viewed at keener/lectures/arrhythmias No Microsoft Products were used or harmed during the production of this talk. The End Arrhythmias By DimensionorHeart Attacks can give you Mathematics p.26/26