Towards Curvature-Based Prediction of Spiral Breakup in Cardiac - - PowerPoint PPT Presentation

Towards Curvature-Based Prediction of Spiral Breakup in Cardiac Tissue

Abhishek Murthy

Stony Brook University (SBU) amurthy@cs.sunysb.edu Joint Work with Ezio Bartocci, Prof. Radu Grosu and Prof. Scott Smolka

April 28, 2011

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Outline

1

Introduction

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Outline

1

Introduction

2

Wave Breaks and Atrial Fibrillation

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Outline

1

Introduction

2

Wave Breaks and Atrial Fibrillation

3

Curvature of Cardiac Excitation Waves

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Outline

1

Introduction

2

Wave Breaks and Atrial Fibrillation

3

Curvature of Cardiac Excitation Waves

4

Case Studies

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Outline

1

Introduction

2

Wave Breaks and Atrial Fibrillation

3

Curvature of Cardiac Excitation Waves

4

Case Studies

5

Future Work

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Why Atrial Fibrillation Matters Atrial Fibrillation (AF) - the quivering of heart muscles of atrial chambers, is the most common cardiac arrhythmia. Prevalent in 2.66 Million Americans, AF responsible for 14,490 deaths in 2010. As an independent risk factor for ischemic strokes, responsible for at least 15% to 20% cases.

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Cardiac Excitation Waves Modelling electrical excitation of cardiac tissue as a reaction-diffusion system - Minimal Model Simulating model under Isotropic Diffusion (ID)

Figure: One time step of simulating cardiac electrical conduction under ID

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Wave Breaks and AF Spatio-temporal description of the fibrillating cardiac tissue involves wave breaks or phase singularities. Curved waves break up near regions of high curvature.

(Loading breakupExample.avi)

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Wave Breaks and AF Spatio-temporal description of the fibrillating cardiac tissue involves wave breaks or phase singularities. Curved waves break up near regions of high curvature.

(Loading breakupExample.avi)

Predicting wave break-ups will help predict the onset of AF.

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Wave Breaks and AF - A closer look Wave propagation speed and curvature are related.

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Wave Breaks and AF - A closer look Wave propagation speed and curvature are related. If V is propagation speed and K, the curvature, then V(K) = V0 − DK V0 = speed of a planar wave D = diffusion co-efficient

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Wave Breaks and AF - A closer look Wave propagation speed and curvature are related. If V is propagation speed and K, the curvature, then V(K) = V0 − DK V0 = speed of a planar wave D = diffusion co-efficient

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Wave Breaks and AF - A closer look Wave propagation speed and curvature are related. If V is propagation speed and K, the curvature, then V(K) = V0 − DK V0 = speed of a planar wave D = diffusion co-efficient Curved waves break near regions of high curvature - wave propagation velocity decreases with increasing convexity. Thus wave breaks up at critical curvature Kcr = V0/D

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Curvature of Cardiac Excitation Waves Requirements for estimating and analysing the curvature of excitation waves (for prediction purposes):

1

Curvature estimation must be accurate.

2

Curvature should be estimated continuously along the length of the wave.

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Curvature of Cardiac Excitation Waves Requirements for estimating and analysing the curvature of excitation waves (for prediction purposes):

1

Curvature estimation must be accurate.

2

Curvature should be estimated continuously along the length of the wave.

Figure: Curvature Estimation of Cardiac Excitation Waves

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Curvature - Estimating Excitation Waves Given a simulation of a grid G of m × n cells, wave W(c, t) can be written as W(c, t) = {(x, y)|x, y ∈ R F(x, y) = c at time t} Where F(x, y) = interpolation of the simulation results onto R2

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Curvature - Estimating Excitation Waves Given a simulation of a grid G of m × n cells, wave W(c, t) can be written as W(c, t) = {(x, y)|x, y ∈ R F(x, y) = c at time t} Where F(x, y) = interpolation of the simulation results onto R2 Check for intersection of wave and an edge of the grid.

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Curvature - Estimating Excitation Waves Given a simulation of a grid G of m × n cells, wave W(c, t) can be written as W(c, t) = {(x, y)|x, y ∈ R F(x, y) = c at time t} Where F(x, y) = interpolation of the simulation results onto R2 Check for intersection of wave and an edge of the grid. Intersection point is obtained by linear interpolation. Implemented using contour function of Matlab

Figure: Contour estimation on grid

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Curvature - Estimating Excitation Waves Given a simulation of a grid G of m × n cells, wave W(c, t) can be written as W(c, t) = {(x, y)|x, y ∈ R F(x, y) = c at time t} Where F(x, y) = interpolation of the simulation results onto R2 Check for intersection of wave and an edge of the grid. Intersection point is obtained by linear interpolation. Implemented using contour function of Matlab

Figure: Contour estimation on grid

Track the same wave across different time steps of the simulation.

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Curvature - Cubic Bézier Fits of Waves Obtain C2 continuous Bézier fit that can be used for symbolic curvature estimation.

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Curvature - Cubic Bézier Fits of Waves Obtain C2 continuous Bézier fit that can be used for symbolic curvature estimation. Fit each of the overlapping strip with cubic Bézier curves of the form: Xj(t) = (1−t)3P0

j +3t(1−t)2P1 j +3t2(1−t)P2 j +t3P3 j . t ∈ [0, 1] (1)

Yj(t) = (1−t)3Q0

j +3t(1−t)2Q1 j +3t2(1−t)Q2 j +t3Q3 j . t ∈ [0, 1] (2)

where j is the strip index.

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Curvature - Cubic Bézier Fits of Waves Obtain C2 continuous Bézier fit that can be used for symbolic curvature estimation. Fit each of the overlapping strip with cubic Bézier curves of the form: Xj(t) = (1−t)3P0

j +3t(1−t)2P1 j +3t2(1−t)P2 j +t3P3 j . t ∈ [0, 1] (1)

Yj(t) = (1−t)3Q0

j +3t(1−t)2Q1 j +3t2(1−t)Q2 j +t3Q3 j . t ∈ [0, 1] (2)

where j is the strip index. In the region of overlap take weighted average of the two curves.

Figure: Weighted average based Bézier curve fitting

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Curvature - Symbolic Curvature Estimation Closed form expressions corresponding to C2 smooth Bézier fit can be processed symbolically in MATLAB’s symbolic computation toolbox.

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Curvature - Symbolic Curvature Estimation Closed form expressions corresponding to C2 smooth Bézier fit can be processed symbolically in MATLAB’s symbolic computation toolbox. If Xj(t) and Yj(t) denote the fit for a strip, then curvature is calculated as κj(t) = |r ′

j (t) × r ′′ j (t)|

|r ′

j (t)|3

(3) where rj(t) = [Xj(t), Yj(t)] is the position vector described by the Bézier curve.

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Curvature - Symbolic Curvature Estimation Closed form expressions corresponding to C2 smooth Bézier fit can be processed symbolically in MATLAB’s symbolic computation toolbox. If Xj(t) and Yj(t) denote the fit for a strip, then curvature is calculated as κj(t) = |r ′

j (t) × r ′′ j (t)|

|r ′

j (t)|3

(3) where rj(t) = [Xj(t), Yj(t)] is the position vector described by the Bézier curve. Continuous closed form of κj(t) => continuous curvature estimate along wavefront

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Curvature Estimation of Cardiac Excitation Waves - Example

(Loading circularCoreCurvatureExample.avi)

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Case Studies

1

Linear core generated with Minimal Model (1024x1024)

2

Spiral Breakup generated with Beeler Reauter Model(1024x1024)

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Linear Core

(Loading linearCoreCurvature.avi)

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Linear Core - Curvature Trend

Figure: Curvature trend for linear core till first turn

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Wave Breakup

(Loading brkupCurvature.avi)

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Future Work Collect training data by simulating different wave break scenarios. Learn patterns of wave break-ups based up morphological features. Predict the temporal behaviour using the patterns learnt.

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Overview

1

Predicting wave-breaks for predicting AF

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Overview

1

Predicting wave-breaks for predicting AF

2

Curvature and spatio-temporal behaviour of cardiac waves are related

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Overview

1

Predicting wave-breaks for predicting AF

2

Curvature and spatio-temporal behaviour of cardiac waves are related

3

Requirements for curvature estimation

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Overview

1

Predicting wave-breaks for predicting AF

2

Curvature and spatio-temporal behaviour of cardiac waves are related

3

Requirements for curvature estimation

4

Weighted average based fitting with cubic Bézier curves

Abhishek Murthy (SBU) Towards Curvature based Breakup Prediction April 28, 2011 17 / 22

Overview

1

Predicting wave-breaks for predicting AF

2

Curvature and spatio-temporal behaviour of cardiac waves are related

3

Requirements for curvature estimation

4

Weighted average based fitting with cubic Bézier curves

5

Symbolic curvature calculation

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Overview

1

Predicting wave-breaks for predicting AF

2

Curvature and spatio-temporal behaviour of cardiac waves are related

3

Requirements for curvature estimation

4

Weighted average based fitting with cubic Bézier curves

5

Symbolic curvature calculation

6

Case studies show potential of using curvature to analyse cardiac waves.

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References

1

“Heart Disease and Stroke Statistics 2011 update: A Report from the American Heart Association”, Véronique L. Roger et. al. Circulation Journal of the American Heart Association.

2

“Measuring Curvature and Velocity Vector Fields for Waves of Cardiac Excitation in 2-D Media”, Matthew W. Kay and Richard A. Gray, IEEE Transactions on Biomedical Engineering 2005.

3

“Role of Wavefront Curvature in Propagation of Cardiac Impulse”, Vladmir G. Fast and André G. Kléber, Cardiovascular Research 1997.

4

“The Fibrillating Atrial Myocardium. What can the Detection of Wave Breaks Tell Us?”, André G. Kléber, Cardiovascular Research 2000.

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Details - Contour estimation Consider an edge e on the grid G whose end points are (xg1, yg1) and (xg2, yg2) and the excitation levels at the two end points are c1 and c2. The wavefront crosses this edge if c1 ≤ c ≤ c2. Let (x, y) be the point at which the wavefront intersects this edge. x and y can be calculated using linear interpolation as follows: x = xg1 + c − c1 c2 − c1 (xg2 − xg1) y = yg1 + c − c1 c2 − c1 (yg2 − yg1) Running time for n x n grid = O(n2)

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Details - Bezier Curve fitting Bezier curve: Xj(t) = (1 − t)3P0

j + 3t(1 − t)2P1 j + 3t2(1 − t)P2 j + t3P3 j . t ∈ [0, 1] (4)

Yj(t) = (1 − t)3Q0

j + 3t(1 − t)2Q1 j + 3t2(1 − t)Q2 j + t3Q3 j . t ∈ [0, 1] (5)

Error functions Ex =

SL

• i=1

[xi − Xj(ti)]2 (6) Ey =

SL

• i=1

[yi − Yj(ti)]2 (7) which give Ex =

SL

• i=1

[xi − (1 − ti)3P0

j + 3t(1 − ti)2P1 j + 3t2 i (1 − ti)P2 j + t3 i P3 j ]2

Ey =

SL

• i=1

[yi − (1 − ti)3Q0

j + 3ti(1 − ti)2Q1 j + 3t2 i (1 − ti)Q2 j + t3 i Q3 j ]2

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Details - Bezier Curve fitting P1

j and P2 j can be obtained at the minimum value of Ex by

∂Ex ∂P1

j

= 0 ∂Ex ∂P2

j

= 0 Solving the above two equations we obtain the following expressions for P1

j and P2 j :

P1

j = αj 2βj 1 − αj 3βj 2

αj

1αj 2 − αj 3 2

(8) P2

j = αj 1βj 2 − αj 3βj 1

αj

1αj 2 − α2 3 j

(9)

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Details - Bezier Curve fitting where α1, α2, α3, β1 and β2 for each segment are given by: α1 = 9

SL

• i=1

[t2

i (1 − ti)4]

α2 = 9

SL

• i=1

[t4

i (1 − ti)2]

α3 = 9

SL

• i=1

[t3

i (1 − ti)3]

β1 = 3

SL

• i=1

[ti(xi − (1 − ti)3P0 − t3

i P3)(1 − ti)2]

β2 = 3

SL

• i=1

[t2

i (xi − (1 − ti)3P0 − t3 i P3)(1 − ti)]

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