Cardiac Electrophysiology on the Moving Heart [.1Cm] (... Or a Short

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Cardiac Electrophysiology on the Moving Heart (... or a short story on shape analysis)

Felipe Arrate

FOCUS PROGRAM ON GEOMETRY, MECHANICS AND DYNAMICS The Legacy of Jerry Marsden

The Fields Institute July 12, 2012 Introduction Model Electrophysiology ‘Tracking’ MeshFree Outline

1 Introduction

2 Model: Images + Electrophysiology

3 Electrophysiology

4 Image Model: Tracking shapes → Tracking Fibers Shape matching Tracking the ﬁbers

5 MeshFree Model

2 CHAPTER 2. HUMAN HEART 5

Introduction Model Electrophysiology ‘Tracking’ MeshFree Cardiovascular diseases

Heart diseases are today responsible for the 28% of deaths in western countries (Cancer is 30%).

Figure 2.1: Anatomical structure of the human heart. The illustration shows the heart in a anterior-oblique viewHeart after having failure removed is the responsible anterior ventricular of wall 400,000 and the right deaths per year in Europe atrial appendage. fatigue insufficient ventricular contraction Sudden Death affects 1 in 10, 000 per year in developed countries electrical disorder ventricular fibrillation

3 CHAPTER 2. HUMAN HEART 5

Introduction Model Electrophysiology ‘Tracking’ MeshFree Cardiovascular diseases

Heart diseases are today responsible for the 28% of deaths in western countries (Cancer is 30%).

Figure 2.1: Anatomical structure of the human heart. The illustration shows the heart in a anterior-obliqueIndividualized view after having removed model the anterior for ventricular cardiac wall and electrophysiology the right atrial appendage. Diagnosis. Eﬃcacy of deﬁbrillation in infarcted hearts. Ablation location for correcting arrhythmias.

3 Introduction Model Electrophysiology ‘Tracking’ MeshFree Outline

1 Introduction

2 Model: Images + Electrophysiology

3 Electrophysiology

4 Image Model: Tracking shapes → Tracking Fibers Shape matching Tracking the ﬁbers

5 MeshFree Model

4 Introduction Model Electrophysiology ‘Tracking’ MeshFree Model: Images + Electrophysiology, but...

Where is the mechanic model? The Heart Mechanics involves the detailed knowledge of elastic constants at diﬀerent levels inside the myocardium. No-negligible importance of internal ﬁbers. and ...

5 Introduction Model Electrophysiology ‘Tracking’ MeshFree Model: Images + Electrophysiology, but...

Where is the mechanic model? The Heart Mechanics involves the detailed knowledge of elastic constants at diﬀerent levels inside the myocardium. No-negligible importance of internal ﬁbers. and ... Can we infer more from a study mainly based on medical images? Is it possible to infer good/bad behaviour from motion itself?. Idea: Given a geometry, the wave is going to move like this, contracting the muscle like this, but is actually doing this... The solution requires less modeling.

5 Introduction Model Electrophysiology ‘Tracking’ MeshFree Outline

1 Introduction

2 Model: Images + Electrophysiology

3 Electrophysiology

4 Image Model: Tracking shapes → Tracking Fibers Shape matching Tracking the ﬁbers

5 MeshFree Model

6 CHAPTER 2. HUMAN HEART 7

Introduction Model Electrophysiology ‘Tracking’ MeshFree

Maxwell’s equations, but... Figure 2.3: Anatomical and fibrous structure of the human heart shown in a posterior- anterior oblique view (from [64]). Note the piece of epicardium cut off the left ventricle, demonstrating the counterclockwise change of the myocardial fiber structure with in- 1 Electrical field strengths are not toocreasing high depth. ⇒ biological tissue is assumed to behave linear with regard to its electrical properties. 2 Cardiac electrical activity is reflected by low frequency components only (≤ several kHz) ⇒ derivatives with respect to time can be neglected (‘quasi-static’ approximation of Maxwell’s equations)

7 CHAPTER 2. HUMAN HEART 7

Introduction Model Electrophysiology ‘Tracking’ MeshFree

Maxwell: Figure 2.3: Anatomical and fibrous structure of the human heart shown in a posterior- anterior oblique view (from [64]). Note the piece of epicardium cut off the left ventricle, ∂B Monodomaindemonstrating the counterclockwise Model: change of the myocardial fiber structure with in- ∇ × E = − creasing depth. ∂t du 1) = c f(u, w) + ∇ · (D∇u) J = −DE dt 1 −→ dw and... 2) = (u − γw) dt ∂B = 0 ⇒ E = −∇u 3)f(u, w)= c1u(u − α)(u − 1) + c2uw ∂t FitzHugh-Nagumo Model

Iion=f(u, w) dw =(u − γw) dt 7 Introduction Model Electrophysiology ‘Tracking’ MeshFree Outline

1 Introduction

2 Model: Images + Electrophysiology

3 Electrophysiology

4 Image Model: Tracking shapes → Tracking Fibers Shape matching Tracking the ﬁbers

5 MeshFree Model

8 Introduction Model Electrophysiology ‘Tracking’ MeshFree Outline

1 Introduction

2 Model: Images + Electrophysiology

3 Electrophysiology

4 Image Model: Tracking shapes → Tracking Fibers Shape matching Tracking the ﬁbers

5 MeshFree Model

9 Introduction Model Electrophysiology ‘Tracking’ MeshFree The Method of coordinates: On Growth and Form (1917) “ In a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each; and the deformation of a complicated figure may be a phenomenon easy of comprehension, though the figure itself may have to be left unanalyzed and undefined. ...This method is the Method of Coordinates, on which is based the Theory of Transformations.”

10 Introduction Model Electrophysiology ‘Tracking’ MeshFree The Method of coordinates: On Growth and Form (1917) “ In a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each; and the deformation of a complicated figure may be a phenomenon easy of comprehension, though the figure itself may have to be left unanalyzed and undefined. ...This method is the Method of Coordinates, on which is based the Theory of Transformations.”

10 Introduction Model Electrophysiology ‘Tracking’ MeshFree Our Method of coordinates Target

Template

2 E(ϕ · T emp, T arg) = dV (id, ϕ) + U(ϕ · T emp, T arg) | {z } | {z } Regulariz. Dissimilarity

11 Introduction Model Electrophysiology ‘Tracking’ MeshFree Our Method of coordinates Target

Template

2 E(ϕ · T emp, T arg) = dV (id, ϕ) + U(ϕ · T emp, T arg) | {z } | {z } Regulariz. Dissimilarity or R 1 2 Ee(v) = 0 kvkV dt + λU(ϕ(1))

11 Introduction Model Electrophysiology ‘Tracking’ MeshFree Our Method of coordinates Target

Template

... or ( µ(t)| w) := ρ | [Dϕ(t, x)]−1 w 0 x d X Z 1 E(µ(t)) = µ(t)| ( µ(t)| K(ϕ(t, x), ϕ(t, y))e ) e dt + λU(ϕ(1)) i x i y i=1 0

d dϕ(t, y) X i = µ(t)| K (ϕ(t, x), ϕ(t, y)) ei dt x i=1 d dµ(t) X i w = − µ(t)| µ(t)| D2K (ϕ(t, z), ϕ(t, x)) . w ei dt z x i=1

11 Introduction Model Electrophysiology ‘Tracking’ MeshFree Our Method of coordinates Target

Template

For points... N 1 X T E(x, α) = α (t) α (t) γ(x (t), x (t)) 2 i j i j i,j=1

N dxs(t) X = γ (x (t), x (t)) α (t) dt k s k k=1 N dαs(t) X n T o = − α (t) α (t) ∇ γ (x (t), x (t)) dt s k 2 k s k=1

11 Introduction Model Electrophysiology ‘Tracking’ MeshFree Gradient Descent on the initial momentum

Energy on the initial momentum: E(ρ0) = ( ρ0| Kρ0) + λU(ϕ(1)) Variation on the initial momentum: ρ0 → ρ0 + δρ0 δU δE(ρ0) = 2 ( δρ0| Kρ0) + λ (ϕ(1)) δϕ(1) (1) δϕ 1) From EPDiﬀ ... Linearized model:

δϕ δϕ ∂ = J δϕ(0) = 0, δµ(0) = δρ t δµ ϕ,µ δµ 0

∗ ∗ 2) Adjoint system: ξϕ := δϕ , ξµ := δµ

ξϕ ∗ ξϕ δU ∂t = −Jϕ,µ ξϕ(1) = (ϕ(1)), ξµ(1) = 0 ξµ ξµ δφ

3) So (1) becomes

δE(ρ0) = ( δρ0| 2Kρ0 + λξµ(0))

−1 ⇒∇ E(ρ0) = 2ρ0 + λK ξµ(0)

12 Introduction Model Electrophysiology ‘Tracking’ MeshFree ... Gradient Descent on the initial momentum

13 Introduction Model Electrophysiology ‘Tracking’ MeshFree Outline

1 Introduction

2 Model: Images + Electrophysiology

3 Electrophysiology

4 Image Model: Tracking shapes → Tracking Fibers Shape matching Tracking the ﬁbers

5 MeshFree Model

14 Introduction Model Electrophysiology ‘Tracking’ MeshFree Tracking the in-vivo fibers: Piecewise Affine + Diffeomorphic

1 Aﬃne Registration: Translations + Rotations + Scale. Dissimilarity measure: Normalized correlation.

Pipeline: Performs an affine registration, including scaling. Follows a PPD procedure to affine register the tensors. It makes easier (faster, more accurate registration) for lddmm to perform the nonlinear registration once the affine deformed template is much closer to the target. It allows the user to input the obtained deformed tensors into DtiStudio.

2 Nonlinear Registration: Diﬀeomorphic matching.

15 Introduction Model Electrophysiology ‘Tracking’ MeshFree Tracking the in-vivo fibers: Piecewise Affine + Diffeomorphic

3 Results: Tensor Translation + Rotation + Nonlinear Deformation.

1Vadakkumpadan, Trayanova, N., et al., “Image-based models of cardiac structure in health and disease”, Wiley Interdisciplinary Reviews: Systems Biology and Medicine, Vol. 2, 2010. (No Scale, Closest Neighbor Interpolation, Reorienting Vectors not complete tensors) 15 Introduction Model Electrophysiology ‘Tracking’ MeshFree Outline

1 Introduction

2 Model: Images + Electrophysiology

3 Electrophysiology

4 Image Model: Tracking shapes → Tracking Fibers Shape matching Tracking the ﬁbers

5 MeshFree Model

16 Introduction Model Electrophysiology ‘Tracking’ MeshFree Mesh or ... Meshless Method? Finite element methods have been extensively used for the spatial discretization of the myocardium. Complicated meshing procedures and element-based interpolation functions often result in algorithms which are either easy to implement, but numerically inaccurate, or accurate but labor-intensive The meshfree platform is more adaptive to diﬀerent cardiac geometries and thus beneﬁcial to individualized analysis. AND...

17 Introduction Model Electrophysiology ‘Tracking’ MeshFree

Ω 4 ϕ 1 2

ϕ(Ω) 1 2 3

18 Introduction Model Electrophysiology ‘Tracking’ MeshFree

Ω 4 ϕ 1 2

ϕ(Ω) 1 2 3

18 Introduction Model Electrophysiology ‘Tracking’ MeshFree

Ω 4 ϕ 1 2

ϕ(Ω) 1 2 3

Particle Methods Complicated volume meshing procedures are excluded. No re-meshing is needed for improving spatial accuracy when deformation occurs.

18 Introduction Model Electrophysiology ‘Tracking’ MeshFree Moving Least Squares (MLS) Approximation

3 1. {x1(t), . . . , xN (t)} nodes (particles) in Ω ⊂ R T 2.p (x) = [p1(x), . . . , pm(x)] polynomial basis. Moving Least Squares Governing equation Given: Approximate the solution by: m locations xi(t), for i = 1, ..., N, X u(x) = pk(x)ak(x) values u(xi, t), for i = 1, ..., N k=1 Solve the associated ODE system Minimizing the functional

N d u 2 = Φ(u, w) X h T i dt w J = w(x−xi) p (xi)a(x) − ui i=1 and obtain (w(x − x ) weighing function with new locations x (t + ∆t), for i i compact support) i = 1, ..., N, Solve the MLS problem new values u(xi, t + ∆t), for i = 1, ..., N A(x)a(x) = B(x)u

19 Introduction Model Electrophysiology ‘Tracking’ MeshFree MLS Approximation: Monodomain Model

1. Monodomain model 3. Meshfree approximation u - membrane potential Φ = [φ1(x), . . . , φN (x)] - shape function w - recovery variable. u ∼ Φu w ∼ Φw ∂u Z Z = c1f(u, w) + ∇ · (D∇u) T ∂u T ∂t Φ Φ = Φ Φ f(u, w) ∂t ∂w Ω Ω = (u − γw) Z ∂t − ∇ΦT D∇Φ u Ω f(u, w) = c1u(u − α)(u − 1) − c2uw Z ∂w Z ΦT Φ = ΦT Φ (u − γw) Ω ∂t Ω 2. Weak formulation 4. ODE system φ - regular test function Z Z Z ∂u −1 ∂u T = f(u, w) + M K u φ = φ c1f(u, w) − ∇φ (D∇u) ∂t Ω ∂t Ω Ω ∂w Z ∂v Z = (u − γw) φ = φ (u − γw) ∂t Ω ∂t Ω f(u, w) = c1u ◦ (u − α) ◦ (u − 1) − c2u ◦ w where Z Z M = ΦT Φ K = ∇ΦT D∇Φ Ω Ω 20 Introduction Model Electrophysiology ‘Tracking’ MeshFree MLS Approximation: Monodomain Model

1. Monodomain model u - membrane potential w - recovery variable. W ∂u = c1f(u, w) + ∇ · (D∇u) ∂t R ∂w = (u − γw) ∂t O f(u, w) = c1u(u − α)(u − 1) − c2uw N

2. Weak formulation G! φ - regular test function ! Z Z Z ∂u T φ = φ c1f(u, w) − ∇φ (D∇u) Ω ∂t Ω Ω ! Z ∂v Z φ = φ (u − γw) Ω ∂t Ω

20 Introduction Model Electrophysiology ‘Tracking’ MeshFree MLS Approximation: Monodomain Model

1. Monodomain model 3. Meshfree Approximation u - membrane potential ∂u Z w - recovery variable. M + ΦT [(JΦ)x˙ ]T u = Mf(u, w) − Ku ∂t ∂u Ω = c1f(u, w) + ∇ · (D∇u) ∂w Z ∂t M + ΦT [(JΦ)x˙ ]T w = M(u − γw) ∂w ∂t Ω = (u − γw) ∂t f(u, w) = c1u(u − α)(u − 1) − c2uw

2. Weak formulation φ - regular test function Z Z Z ∂u T φ = φ c1f(u, w) − ∇φ (D∇u) Ω ∂t Ω Ω Z ∂v Z φ = φ (u − γw) Ω ∂t Ω

20 Introduction Model Electrophysiology ‘Tracking’ MeshFree ...Some preliminary results... ﬁxed heart

21 Introduction Model Electrophysiology ‘Tracking’ MeshFree

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