1. Another propaga,on of ‘light’ in the heart n a grain of sand To see a world i And a heaven in a wild ﬂower Hold inﬁnity in the palm of your hand And eternity in an hour -‐ Auguries of Innocence, by William Blake Sehun Chun
2. Big picture To gaze up from the ruins of the oppressive present towards the stars is to recognize the indestruc,ble world of laws, to strengthen faith in reason, to realize the ‘harmonia mundi’ that transfuses all phenomena, and that never has been, nor will be, disturbed, by Hermann Weyl, 1919. DiLo in the heart, by Sehun Chun, 2010s
3. Introducing the HEART • 72 beat per minute, 2.5 billion ,mes in average (Beat rate = 100%) • Electrical signal -‐> contrac,on -‐> Pumping oxygenated blood • Heart model = Electricity + Mechanics + Elas,city + Fluids • One of the hardest Mathema,cal and Computa,onal models
4. How electric signal propagates
5. What is the electric signal? • Diﬀerent names: Facilitated diﬀusion (biology), Nonlinear diﬀusion equa,on (ODE), Diﬀusion-‐reac,on equa,on (PDE). • All-‐or-‐nothing non-‐decremental traveling wave • No conserva,onal laws of energy, momentum, charges. • Ex) Collision of two waves = canceling out. • Refractory area (In-‐excitable region right behind signal).
6. Analogy in nature, Forest Fire 1. Energy is not conserved (Otherwise, a match could be the most dangerous weapon !) 2. Temperature of the ﬁre does not depend on the distance from the origin, but depend on the media.
7. Electromagne,c ﬁeld in the microscopic cardiac cells i i i i ∂d i ∇⋅d = ρ , ∇×h = j + ∂t ∂b i At Intracellular space ∇ ⋅ bi = 0, i ∇×e + =0 ∂t e e e e ∂d e∇⋅d = ρ , ∇×h = j + ∂t e At Extracellular space ∂b∇ ⋅ b e = 0, e ∇×e + =0 ∂t i e i e i eρ + ρ = const., j + j = 0 in Ω ∪Ω
8. Microscopic To Macroscopic Collec,ng several cells + Averaging the quan,,es i i i i i i i id =D,h =H d =D,h =H i i i i i i i ib =B,e =E b =B,e =E
9. Electromagne,c ﬁeld in the macroscopic cardiac cells i i i i ∂D i∇⋅D = ρ , ∇×H = J + ∂t i ∂B∇ ⋅ Bi = 0, i ∇×E + =0 ∂t At the same space, but ∂D e Two diﬀerent equa,ons e e e e∇⋅D = ρ , ∇×H = J + ∂t e ∂B∇ ⋅ B e = 0, e ∇×E + =0 ∂t i e i e ρ + ρ = const., J + J = 0, in Ω
10. Introducing Bi-‐domain • Every points in the macroscopic domain means two separate points in the separate domains. • Energy and charges are conserved in bi-‐ domains. • Intracellular space is only our concern, then energy and charges are not conserved. • Diﬀusion-‐reac,on system is only possible in bi-‐domain.
11. Anisotropy in Cardiac ,ssue • Cable-‐like, cylindrical, 100 um long and 15um • Cardiac ,ssue is strongly anisotropic, with wave speeds that diﬀer substan,ally depending on their direc,on. • 0.5 m/s along ﬁbers and about 0.17 m/s transverse the ﬁbers.
12. Anisotropy in Mathema,cs • d: Diﬀusion tensor, variable coeﬃcient ∇ ⋅ ( d∇φ )• In the direc,on of characteris,cs • A liLle misplacement leads to the shock of waves. • Related to the way the heart is folded for contrac,on • Only God is allowed to put anisotropy in the heart -‐> Another beauty of the heart
13. How to simula,on the electrophysiology phenomena on complex anisotropic geometry • Method of Moving Frames • Originally developed by É. Cartan in 1920’s • Represent a geometry with oscula,on Euclidean planes by inheri,ng the metric tensor. • Diﬀerent Euclidean axis for each points with various length of the axis.
14. Rough meaning of Geometry Op-cs Electrophysiology Role of Passive role for the Ac,ve delivering geometry wave (deﬂec,on, of the wave reﬂec,on, absorp,on) Meaning Mass distribu,on Cell distribu,on? (what is equivalent to mass in biology?) Propaga,on Anything to change Anything to the propaga,on of change the the light cardiac electric signal
15. Deﬁni,on of geometry in electrophysiology • Geometry = the combina,on of the followings: 1) Loca,on of the star,ng point 2) Conduc,ng proper,es of the media 3) 3D shape of the heart -‐> Geometry is taken in the sense of the Field theory. • Electric signal only depends on geometry. • Electric signal has negligible kine,c energy. • Heart works 100% according to its design. • Analogous to the light as a signal
16. Good geometry and Bad geometry • Good geometry = to guide the electrical signal propaga,on for “eﬃcient” cardiac contrac,on, i.e., star,ng from one point to converge to one or two points in the atrium. • Bad geometry = To change its original design • Examples of bad geometry: change of loca,on or turn-‐ oﬀ of SAN, scar ,ssue (infarc,on), change of ,ssue proper,es, enlarged 3D shape, but the worst of all is the mul,ple self-‐ini,ators (though they are all correlated)
17. How atrial ﬁbrilla,on (AF) happens? § Normal propaga,on starts from a point and converges to a point. § AF only happens when the propaga,on deviates from its original track and comes back to excite cardiac ,ssue again (Reentry) § Reentry requires unidirec,onal pathway. Unidirec,onal pathway
18. Clinical observa,ons and conjectures • The role of PVs on AF: -‐ Spontaneous Ini,a,on of atrial ﬁbrilla,on by ectopic beats origina,ng in the pulmonary vein, The New England Journal of Medicine, 1998, by M. Haissaguerre et al -‐ Arrhythmogenic substrate of the Pulmonary Veins Assessed by High-‐Resolu,on Op,cal Mapping, Circula,on 2003, by R. Arora et al. -‐ Atrial Fibrilla,on Begets Atrial Fibrilla,on in the Pulmonary Veins: On the Impact of Atrial Fibrilla,on on the Electrophysiological Proper,es of the Pulmonary Veins in Humans, J. Am. Coll. Cardiol. 2008, by T. Rostock et al. • Observa,ons and theories of AF: -‐ New ideas about atrial ﬁbrilla,on 50 years on, Nature 2002, by S. NaLel -‐ Atrial Remodeling and Atrial Fibrilla,on: Mechanisms and Implica,ons, Circula,on 2008, by S. NaLel et al -‐ Rotors and Spiral waves in Atrial Fibrilla,on, J. Cardiovasc Electrophysiol 2003, by J. Jalife -‐ Circula,on movement in rabbit atrial muscle as a mechanism of tachycardia. III. The “leading circle” concept: a new model of circus movement in cardiac ,ssue without the involvement of an anatomical obstable, Circ. Res. 1977, by M.A. Allessie et al. • Structures of ler atrium and PVs: -‐ The importance of Atrial Structure and Fibers, Clinical Anatomy 2009, by S. Y. Ho -‐ The structure and components of the atrial chambers, Europace 2007, by R. H. Anderson -‐ Normal atrial ac,va,on and voltage during sinus rhythm in the human heart: An endocardial and epicardial mapping study in pa,ents with a history of atrial ﬁbrilla,on, J. Cardiovasc. Electrophysiol. 2007, by R. Lemery.
19. Blocking Pulmonary Veins by lesions A surgical procedure is to insert a catheter through the vein into the atrium and to burn cardiac cells around the PV to prevent the self-‐ini,ators around the PVs.
20. Summary of the proper,es of cardiac electric signal propaga,on (1) In bi-‐domain with none of conserva,on laws (2) Anisotropy and inhomogeneous is everywhere. (3) 3D shape is smooth, but non-‐ uniform with various curvature.
21. Theories of electrophysiology in the heart • Regarded as a kind of nerve system. • Quick transfer from ODE to PDE. • Analogous jump from a 1D cable to mul,dimensional space. • Alterna,ve is the Kinema,cs approach (1990s~): Study of the wave front to ﬁnd the cri,cal curvature to ﬁnd K * > K in ∂K ∂K ∂V 2 ( ∂ ) ∫ 0 KV dξ + C + ∂t + K V + ∂2 = −ΓV 2
22. Inspira,ons for rela,ve accelera,on approach Rule of games: (1) Excited person has 3L of water. (2) Given 1L of water, an excitable person get 2L more water and get excited. (3) A player wins if it has more children than the compe,tor. (4) Game stops if any one of the column can not receive 1L.
23. Hypothesis and Proposi,on • Hypothesis: If the rela,ve accelera,on of cardiac excita,on propaga,on becomes suﬃciently large, then the propaga,on stops. • Proposi,on: If the following is suﬃciently large, then the propaga,on stops # i & i 1 % ∂E ∂v ∂G i ∂v ∂ ( cv ) ( i − + v +G + E % ∂n ∂λ ∂n $ ∂n ∂n ( 2 2 1 ∂( gΛ k d k g kk ) where, E = ∑ (Λ k )2 d k g kk G =∑ k=1 k=1 g ∂x k
24. Diﬀusion-‐Reac,on Tensor • A DR-‐tensor, is obtained as k k kk C = d Λkg Anisotropic coeﬃcient × Propaga,onal direc,on × the 3cD geometric shape for onjugate metric tensor • Coincide with the deﬁni,on of geometry: • k (1) d : type of media. • (2) Λ k : Loca,on of SAN and surface 2D geometries. • (3) g kk : 3D shape. € € € • Large varia,on of the tensor means the break-‐up of the wave
25. Applying to the PVs θ d 4
26. Various Anisotropy on the PV A shape of PV junc,on No anisotropy Circumferen,al anisotropy Longitudinal anisotropy (Perez-‐Lugones et al, 2003)
27. Construct a model
28. Modelling of Re-‐entrance on a spherical shell with a PV-‐like column 1) Normal condi,on T=0, Front T=10, Front T=20, Back T=25, Back 2) Deteriorated myocardial cells on the PVs Mul,ple reentrant waves Overdrive suppression T=20, Back T=22, Back T=37, Front T=100, Front
29. Consequences (1) Unidirec,onal pathways are the PVs with weakened anisotropy toward which the wave approaches with an oblique angle due to some other factors, for example, the presence of scar ,ssue, the change of SAN, etc. (2) Cardiac excita,on propaga,on can be represented as the ﬁeld of the trajectories. (3) Something is moving along the trajectory
30. What is actually moving in electric signal propaga,on ? S,mula,ng moving body consist of many s,mula,on par,cles that lower the res,ng poten,al to ini,ate cardiac ac,on poten,al. Ex) For forest ﬁre, it is equivalent to a group of par,cles of high-‐temperature to ini,ate ﬁre on a unburned tree.
31. S,mula,ng moving body
32. Maxwell’s equa,ons in the universe vs. Maxwell’s equa,ons in the heart Maxwell’s in the universe in the heart equa-ons How to Ray Diﬀusion propagate Gauge choice 1 ∂Φ ∇⋅ A+ 2 =0 ∇⋅ A+Φ = 0 c ∂tGauge func,on 2 1∂Λ 2 ∂Λ 2 ∇ Λ− 2 2 =0 ∇ Λ− =0 c ∂t ∂t
33. Diﬀusion-‐reac,on equa,ons from Maxwell’s equa,ons • The diﬀusion-‐reac,on equa,ons are one projec,on of Maxwell’s equa,ons in bi-‐domain, so E and B are under-‐determined. • The normal heart is designed to generate the minimum degree of the magne,c ﬁeld (conjecture 1). • Increasing magne,c ﬁeld in the speciﬁc area of the heart may mean AF (conjecture 2). • It may explain why the external electric shock can resuscitate temporarily non-‐moving heart arer CPR or can cure AF temporarily.
34. Thank you for aLen,on ! Now I know Ive got a heart, cause its breaking