Introduction to Physiology V - Coupling and Propagation

Transcription

1 Introduction to Physiology V - Coupling and Propagation J. P. Keener Mathematics Department Coupling and Propagation p./33

2 Spatially Extended Excitable Media Neurons and axons Coupling and Propagation p.2/33

3 Spatially Extended Excitable Media Mechanically stimulated Calcium waves Coupling and Propagation p.3/33

4 Conduction system of the heart Coupling and Propagation p.4/33

5 Conduction system of the heart Electrical signal originates in the SA node. Coupling and Propagation p.4/33

6 Conduction system of the heart Electrical signal originates in the SA node. The signal propagates across the atria (2D sheet), through the AV node, along Purkinje fibers (D cables), and throughout the ventricles (3D tissue). Coupling and Propagation p.4/33

7 Spatially Extended Excitable Media The forest fire analogy Coupling and Propagation p.5/33

8 Spatial Coupling Conservation Law: d (stuff in Ω) = rate of transport + rate of production dt d udv = J nds + fdv dt becomes Ω u t Ω = (D u) + f(u) Ω Flux production f(u) Ω J Coupling and Propagation p.6/33

9 Coupled Cells Cardiac Cells φ R φ e e 2 e ie C m I ion C m I ion gap junctions C m dφ dt + I ion(φ, w) = i i = φ i i g R g φ 2 i R e + R g (φ 2 φ ) C m dφ 2 dt + I ion(φ 2, w) = i i = R e + R g (φ φ 2 ) Question: Can anything interesting happen with coupled cells that does not happen with a single cell? Take a wild guess. p.7/33

10 Coupled Cells Normal cell and cell with slightly elevated potassium - uncoupled normal cell "ischemic" cell Who could have guessed? p.8/33

11 Coupled Cells Normal cell and cell with slightly elevated potassium - coupled normal cell "ischemic" cell Who could have guessed? p.8/33

12 Coupled Cells Normal cell and cell with moderately elevated potassium - uncoupled normal cell "ischemic" cell Who could have guessed? p.8/33

13 Coupled Cells Normal cell and cell with moderately elevated potassium - coupled normal cell "ischemic" cell Who could have guessed? p.8/33

14 Coupled Cells Normal cell and cell with greatly elevated potassium - uncoupled normal cell "ischemic" cell Who could have guessed? p.8/33

15 Coupled Cells Normal cell and cell with greatly elevated potassium - coupled normal cell "ischemic" cell Who could have guessed? p.8/33

16 Axons and Fibers V e (x) r e dx V e (x+dx) Extracellular space I t dx I e (x) I t dx C m dx I ion dx C m dx I ion dx Cell membrane I i (x) V i (x) r i dx V i (x+dx) Intracellular space From Ohm s law V i (x+dx) V i (x) = I i (x)r i dx, V e (x+dx) V e (x) = I e (x)r e dx, In the limit as dx 0, I i = r i dv i dx, I e = r e dv e dx. Coupling and Propagation p.9/33

17 The Cable Equation V e (x) r e dx V e (x+dx) Extracellular space I t dx I e (x) I t dx C m dx I ion dx C m dx I ion dx Cell membrane I i (x) V i (x) r i dx V i (x+dx) Intracellular space From Kirchhoff s laws I i (x) I i (x + dx) = I t dx = I e (x + dx) I e (x) In the limit as dx 0, this becomes I t = I i x = I e x. Coupling and Propagation p.0/33

18 The Cable Equation Combining these I t = x ( r i + r e V x ), and, thus, C m V t + I ion = I t = x ( r i + r e V x ). This equation is referred to as the cable equation. Coupling and Propagation p./33

19 Modelling Cardiac Tissue Cardiac Tissue - The Bidomain Model: At each point of the cardiac domain there are two comingled regions, the extracellular and the intracellular domains with potentials φ e and φ i, and transmembrane potential φ = φ i φ e. Coupling and Propagation p.2/33

20 Modelling Cardiac Tissue Cardiac Tissue - The Bidomain Model: At each point of the cardiac domain there are two comingled regions, the extracellular and the intracellular domains with potentials φ e and φ i, and transmembrane potential φ = φ i φ e. These potentials drive currents, i e = σ e φ e, i i = σ i φ i, where σ e and σ i are conductivity tensors. Coupling and Propagation p.2/33

21 Modelling Cardiac Tissue Cardiac Tissue - The Bidomain Model: At each point of the cardiac domain there are two comingled regions, the extracellular and the intracellular domains with potentials φ e and φ i, and transmembrane potential φ = φ i φ e. These potentials drive currents, i e = σ e φ e, i i = σ i φ i, where σ e and σ i are conductivity tensors. Total current is i T = i e + i i = σ e φ e σ i φ i. Coupling and Propagation p.2/33

22 Kirchhoff s laws: Total current is conserved: (σ i φ i + σ e φ e ) = 0 Coupling and Propagation p.3/33

23 Imagine the Kirchhoff s laws: Total current is conserved: (σ i φ i + σ e φ e ) = 0 Transmembrane current is balanced: φ Extracellular Space χ ( C m τ + I ion ) = (σ i φ i ) C I φ = φ φ m ion i e Intracellular Space φ e φ i Coupling and Propagation p.3/33

24 Imagine the Kirchhoff s laws: Total current is conserved: (σ i φ i + σ e φ e ) = 0 Transmembrane current is balanced: φ Extracellular Space χ ( C m τ + I ion ) = (σ i φ i ) C I φ = φ φ m ion i e Intracellular Space φ e surface to volume ratio, φ i Coupling and Propagation p.3/33

25 Imagine the Kirchhoff s laws: Total current is conserved: (σ i φ i + σ e φ e ) = 0 Transmembrane current is balanced: φ Extracellular Space χ ( C m τ + I ion ) = (σ i φ i ) C I φ = φ φ m ion i e Intracellular Space φ e surface to volume ratio, capacitive current, φ i Coupling and Propagation p.3/33

26 Imagine the Kirchhoff s laws: Total current is conserved: (σ i φ i + σ e φ e ) = 0 Transmembrane current is balanced: φ Extracellular Space χ ( C m τ + I ion ) = (σ i φ i ) C I φ = φ φ m ion i e Intracellular Space φ e surface to volume ratio, capacitive current, ionic current, φ i Coupling and Propagation p.3/33

27 Imagine the Kirchhoff s laws: Total current is conserved: (σ i φ i + σ e φ e ) = 0 Transmembrane current is balanced: φ Extracellular Space χ ( C m τ + I ion ) = (σ i φ i ) C I φ = φ φ m ion i e Intracellular Space surface to volume ratio, capacitive current, ionic current, and current from intracellular space. φ e φ i Coupling and Propagation p.3/33

28 Imagine the Kirchhoff s laws: Total current is conserved: (σ i φ i + σ e φ e ) = 0 Transmembrane current is balanced: φ Extracellular Space χ ( C m τ + I ion ) = (σ i φ i ) C I φ = φ φ m ion i e Intracellular Space surface to volume ratio, capacitive current, ionic current, and current from intracellular space. φ e φ i Boundary conditions: n σ i φ i = 0, n σ e φ e = I(t, x) and Ω I(t, x)dx = 0 on Ω. + Coupling and Propagation p.3/33

29 Imagine the Consequences of the Bidomain Model-I: With current applied at the boundary of the domain, there is depolarization and hyperpolarization at the boundaries. For a homogeneous medium, in the interior (several space constants from the boundary), the transmembrane potential is unaffected. Extracellular Space + Intracellular Space Coupling and Propagation p.4/33

30 Consequences of the Bidomain Model-II: Resistive inhomogeneities lead to sources and sinks of transmembrane current (virtual electrodes) in the interior of the tissue domain: Coupling and Propagation p.5/33

31 Imagine the Consequences of the Bidomain Model-II: Resistive inhomogeneities lead to sources and sinks of transmembrane current (virtual electrodes) in the interior of the tissue domain: With large scale resistive inhomogeneities: Extracellular Space + Intracellular Space Coupling and Propagation p.5/33

32 /. -,+* ( (' ) '& ) &%$ $ " " " " " " " " " " " "# "# "# "# "# "# "# "# "# "# "# "# "# "# "# "# "# "# "# "# "# "#! Imagine the Consequences of the Bidomain Model-II: Resistive inhomogeneities lead to sources and sinks of transmembrane current (virtual electrodes) in the interior of the tissue domain: With small scale resistive inhomogeneities: Extracellular Space Intracellular Space Coupling and Propagation p.5/33

33 Consequences of the Bidomain Model-III: Response to a point stimulus in tissue with unequal anisotropy Virtual Electrodes Coupling and Propagation p.6/33

34 Traveling Waves u t = D 2 u x 2 + f(u) with f(0) = f(a) = f() = 0, 0 < a <. There is a unique traveling wave solution u = U(x ct), The solution is stable up to phase shifts, The speed scales as c = c 0 D, U is a homoclinic trajectory of DU + cu + f(u) = U(x) du/dx x U Coupling and Propagation p.7/33

35 Discreteness Cardiac Cells Gap junctional coupling gap junctions Cardiac Cell Calcium release sites Calcium Release through CICR Receptors Coupling and Propagation p.8/33

36 Discrete Effects Discrete Cells Discrete Calcium Release Discrete Release Sites dv n dt = f(v n) + d(v n 2v n + v n ) u t = D 2 u x 2 + g(x)f(u) Coupling and Propagation p.9/33

37 Fire-Diffuse-Fire Model Suppose a diffusible chemical u is released from a long line of evenly spaced release sites; L Release of full contents C occurs when concentration u reaches threshold θ. u t = D 2 u x 2 + n Source(x nh)δ(t t n ) Coupling and Propagation p.20/33

38 Fire-Diffuse-Fire-II Recall that the solution of the heat equation with δ-function initial data at x = x 0 and at t = t 0 is u(x, t) = 4π(t t0 ) exp( (x x 0) 2 4D(t t 0 ) ) u.5 u x t with x fixed Coupling and Propagation p.2/33

39 Fire-Diffuse-Fire-III Suppose known firing times are t j at position x j = jh, j =,, n. Find t n. At x = x n = nh, u(nh, t) = n X j= C p 4π(t tj ) exp( (nh jh)2 4D(t t j ) ) C p 4π(t tn ) exp( h 2 4D(t t n ) ) = C D t f( h h 2 ) F(D t/h 2 ) D t/h 2 Coupling and Propagation p.22/33

40 Fire-Diffuse-Fire-IV Solve the equation This is easy to do graphically: θh C = f(d t h 2 ) F(D t/h 2 ) vh/d D t/h θ h/c Conclusion: Propagation fails for θh C > θ 0.25 (i.e. if h is too large, θ is too large, or C is too small.) Coupling and Propagation p.23/33

41 With Recovery Including recovery variables Solitary Pulse Periodic Waves Skipped Beats v t = D 2 v w + f(v, w), x2 t = g(v, w) Coupling and Propagation p.24/33

42 Periodic Ring Coupling and Propagation p.25/33

43 The APD Instability in D Stable Pulse on a Ring Unstable Pulse on a Ring Collapse of Unstable Pulse Coupling and Propagation p.26/33

44 The APD Instability in D Stable Pulse on a Ring Unstable Pulse on a Ring Collapse of Unstable Pulse Coupling and Propagation p.26/33

45 Dimension 2: Spirals Atrial Flutter Coupling and Propagation p.27/33

46 Dimension 2: Spirals Spiral instability - Meander: Atrial Flutter Torsade de Pointe Coupling and Propagation p.27/33

47 Dimension 2: Spirals Spiral instability - Meander: Atrial Flutter Torsahd duh Pwahn Coupling and Propagation p.27/33

48 The APD Instability in 2D Spiral Breakup Coupling and Propagation p.28/33

49 Dimension 3: Ventricular Reentrant Activity Ventricular Monomorphic Tachycardia Coupling and Propagation p.29/33

50 Dimension 3: Cardiac Scroll Wave 3 D structure of a single scroll wave Coupling and Propagation p.30/33

51 Ventricular Fibrillation Ventricular Fibrillation Surface View Movie 3D View Movie Still unresolved: What is the mechanism for maintenance of fibrillation? (APD instability? Mother rotor hypothesis?) Coupling and Propagation p.3/33

52 Hypothesized Mechanisms for Initiation of Reentrant Activity How is a dynamical system moved from one state (the normal heartbeat) to another (reentry)? Remark: This is a spatio-temporal system; Single cell explanations are not sufficient. Coupling and Propagation p.32/33

53 Hypothesized Mechanisms for Initiation of Reentrant Activity How is a dynamical system moved from one state (the normal heartbeat) to another (reentry)? Remark: This is a spatio-temporal system; Single cell explanations are not sufficient. Anatomical - One way block on a closed D loop. (movie) Coupling and Propagation p.32/33

54 Hypothesized Mechanisms for Initiation of Reentrant Activity How is a dynamical system moved from one state (the normal heartbeat) to another (reentry)? Remark: This is a spatio-temporal system; Single cell explanations are not sufficient. Anatomical - One way block on a closed D loop. (movie) Vulnerable Period - Winfree (S-S2) mechanism (D) (2D) Coupling and Propagation p.32/33

55 Hypothesized Mechanisms for Initiation of Reentrant Activity How is a dynamical system moved from one state (the normal heartbeat) to another (reentry)? Remark: This is a spatio-temporal system; Single cell explanations are not sufficient. Anatomical - One way block on a closed D loop. (movie) Vulnerable Period - Winfree (S-S2) mechanism (D) (2D) Early After Depolarizations during Vulnerable Period. Coupling and Propagation p.32/33

56 Hypothesized Mechanisms for Initiation of Reentrant Activity How is a dynamical system moved from one state (the normal heartbeat) to another (reentry)? Remark: This is a spatio-temporal system; Single cell explanations are not sufficient. Anatomical - One way block on a closed D loop. (movie) Vulnerable Period - Winfree (S-S2) mechanism (D) (2D) Early After Depolarizations during Vulnerable Period. Dispersion (i.e. spatial/temporal inhomogeneity) of refractoriness. Coupling and Propagation p.32/33

57 More Unresolved Issues Why is calcium overload arrhythmogenic? Coupling and Propagation p.33/33

58 More Unresolved Issues Why is calcium overload arrhythmogenic? Why is long QT syndrome arrhythmogenic? Coupling and Propagation p.33/33

59 More Unresolved Issues Why is calcium overload arrhythmogenic? Why is long QT syndrome arrhythmogenic? Why are most anti-arrhythmic drugs actually proarrhythmic? Coupling and Propagation p.33/33

60 More Unresolved Issues Why is calcium overload arrhythmogenic? Why is long QT syndrome arrhythmogenic? Why are most anti-arrhythmic drugs actually proarrhythmic? What is the mechanism of EAD s and are they truly proarrhythmic? Coupling and Propagation p.33/33