Radu Grosu SUNY at Stony Brook

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Modeling and Analysis of Atrial Fibrillation

Joint work with

Ezio Bartocci, Flavio Fenton, Robert Gilmour, James Glimm and Scott A. Smolka

Emergent Behavior in Heart Cells

Arrhythmia afflicts more than 3 million Americans alone

Surface

Modeling

Tissue Modeling: Triangular Lattice CellExcite and Simulation

Communication by diffusion

Tissue Modeling: Square Lattice

CellExcite and Simulation

Communication by diffusion

Single Cell Reaction: Action Potential

Membrane’s AP depends on: • Stimulus (voltage or current):

– External / Neighboring cells • Cell’s state

failed initiation

Threshold

Resting potential

Schematic Action Potential

AP has nonlinear behavior!• Reaction diffusion system:

∂u∂t

= R(u) +∇(D∇u)

BehaviorIn time

Reaction Diffusion

Frequency Response

APD90: AP > 10% APm DI90: AP < 10% APm BCL: APD + DI

Frequency Response

S1-S2 Protocol: (i) obtain stable S1; (ii) deliver S2 with shorter DI

Frequency Response

S1S2 Protocol: (i) obtain stable S1; (ii) deliver S2 with shorter DIRestitution curve: plot APD90/DI90 relation for different BCLs

Existing Models

• Detailed ionic models: – Luo and Rudi: 14 variables– Tusher, Noble2 and Panfilov: 17 variables – Priebe and Beuckelman: 22 variables – Iyer, Mazhari and Winslow: 67 variables

• Approximate models:– Cornell: 3 or 4 variables – SUNYSB: 2 or 3 variable

Stony Brook’s Cycle-Linear Model

Objectives

• Learn a minimal mode-linear HA model:– This should facilitate analysis

• Learn the model directly from data:– Empirical rather than rational approach

• Use a well established model as the “myocyte”:– Luo-Rudi II dynamic cardiac model

• Training set: for simplicity 25 APs generated from the LRd– BCL1 + DI2: from 160ms to 400 ms in 10ms intervals

• Stimulus: step with amplitude -80μA/cm2, duration 0.6ms

• Error margin: within ±2mV of the Luo-Rudi model

• Test set: 25 APs from 165ms to 405ms in 10ms intervals

HA Identification for the Luo-Rudi Model(with P. Ye, E. Entcheva and S. Mitra)

Stimulated

Action Potential (AP) Phases

Stimulated

off∧u <θ

u ≥θU

u ≥θE

u ≤θP

u ≤θR

u ≤θF

Identifying a Mode-Linear HA for One AP

Null Pts: discrete 1st Order deriv. Infl. Pts: discrete 2nd Order deriv. Thresholds: Null Pts and Infl. Pts Segments: Between Seg. Pts

Problem: too many Infl. Pts Problem: too many segments?

Identifying the Switching for one AP

Solution: use a low-pass filter- Moving average and spline LPF: not satisfactory- Designed our own: remove pts within trains of inflection points

Null Pts: discrete 1st Order deriv. Infl. Pts: discrete 2nd Order deriv. Thresholds: Null Pts and Infl. Pts Segments: Between Seg. Pts

Problem: too many Infl. Pts Problem: too many segments?

Identifying the Switching for one AP

Problem: somewhat different inflection points

Identifying the Switching for all AP

Solution: align, move up/down and remove inflection points- Confirmed by higher resolution samples

Identifying the Switching for all AP

Stimulated

off∧u <θ

u ≥θU

u ≥θE

u ≤θR

&u=&xi + &xo + Is&xi =bixi

&xo =boxo

u ≥θP /

xi =ai

xo =ao

Identifying the HA Dynamics for One APM

Stimulated

off∧u <θ

i) son

/ di=t

u ≥θU(d

u ≥θE(d

u ≤θ

u ≤θP(d

u ≤θF(d

Summarizing all HA

&u=&xi + &xo + Is&xi =bi(di )xi

&xo =bo(di )xo

u ≥θP(di ) /

xi =ai(di )

xo =ao(di )

Finding Parameter Dependence on DI

Solution: apply mProny once again on each of the 25 points

Stimulated

off∧u <θ

i) son

/ di=t

u ≥θU(d

u ≥θE(d

u ≤θ

u ≤θP(d

u ≤θF(d

Summarizing all HA

&u=&xi + &xo + Is&xi =bi(di)xi

&xo =bo(di)xo

u ≥θP(di ) /

xi =ai(di )

xo =ao(di )

bi (di ) =a i1ebi1di + a i2e

bo(di)=ao1ebo1di + ao2e

Frequency Response on Test Set

AP on test set: still within the accepted error margin Restitution on test set: follows very well the nonlinear trend

Cornell’s Nonlinear Minimal Model

Objectives

• Learn a minimal nonlinear model:– This should facilitate analysis

• Approximate the detailed ionic models:– Rational rather than empirical approach

• Identify the parameters based on: – Data generated by a detailed ionic model– Experimental, in-vivo data

us =0.5

ks =16

Switching Control

S(ks (u−us))=1

1+ e−ks (u−us )

H (u−us)=0 u < us

1 u≥us

⎧⎨⎪⎩⎪

R(u,us1,us2 ) =

0 u < us1

u−us1

us2 −us1

1 u≥us2

⎨⎪⎪

⎩⎪⎪

&u =∇(D∇u)−(Jfi + Jsi + Jso)

Jfi =−H(u−θv)(u−θv)(uu −u)v/ t fi

Cornell’s Minimal Model

Fast inputcurrent

DiffusionLaplacia

nvoltage Slow input

currentSlow output

current

&v = (1−H(u−θv)) (v∞ −v) / tv−−H(u−θv)v / tv

&w = (1−H(u−θw ))(w∞ −w) / tw−−H(u−θw)w / tw

&s = (S(2ks(u−us))−s) / t s

Jfi =−H(u−θv)(u−θv)(uu −u)v/ t fi

&u =∇(D∇u)−(Jfi + Jsi + Jso)

Jfi =−H(u−θv)(u−θv)(uu −u)v/ t fiJ fi = −H(u−θv) (u−θv)(uu −u)v/ t fi

Jsi = −H(u−θw) ws/ t si

Jso = (1−H(u−θw)) (u−uo) / t o + H(u−θw) / t so

PiecewiseNonlinear

Heaviside(step)

Sigmoid(s-step)

PiecewiseNonlinear

PiecewiseBilinear

PiecewiseLinear

Nonlinear

ActivationThreshol

Fast inputGateSlow Input

GateSlow Output

GateResistanceTime Cst

t v− = (1 − H (u −θv

− )) τ v1− + H (u −θv

− ) τ v2−

τ s = (1 − H (u −θw )) τ s1 + H (u −θw ) τ s2τ o = (1 − H (u −θo )) τ o1 + H (u −θo ) τ o2

w∞tw

− = τ w1− + (τ w2

− − τ w1− ) S(2kw

− (u − uw− ))

τ so = τ so1 + (τ so2 − τ so1) S(2kso(u − uso ))

Time Constants and Infinity Values

PiecewiseConstant

Sigmoidal

v∞ = (1−H(u−θv−))

w∞ = (1−H(u−θo)) (1−u / tw∞) + H(u−θo) w∞*

t so = (1−H(u−θo)) t o1 + H(u−θo) t o2

PiecewiseLinear

Single Cell Action Potential

u ≥θo

u ≥θv

u ≥θw

θo ≤ u < θw&u = ∇(D∇u) − u / τ o2

&v = −v / τ v2−

&w = (w∞* − w) / τ w1

&s = (S(2ks (u − us )) − s) / τ s

θw ≤ u < θ v&u = ∇(D∇u) + ws / τ si −1 / τ so&v = −v / τ v2

&w = −w / τ w+

&s = (S(2ks (u − us )) − s) / τ s2

u < θo =θv− =0.006

u < θw =0. 13

u < θv =0.3

u < θo

&u =∇(D∇u)−u / t o1

&v = (1−v) / tv1−

&w = (1−u / tw∞ −w) / tw−

&s = (S(2ks(u−us))−s) / t s

θv ≤ u

&u =∇(D∇u)+ (u−θv)(uu −u)v/ t fi +ws/ t fi −1 / t so

&v =−v/tv+

&w =−w / tw+

&s = (S(2ks(u−us))−s) / t s2

u ≥θo

u ≥θv

u ≥θw

u < θo =θv− =0.006

u < θw =0. 13

u < θv =0.3

v < vc

Partition with Respect to v

u ≥θo

u ≥θv

u ≥θw

u < θo =θv− =0.006

u < θw =0. 13

u < θv =0.3

v < vc

θv ≤ u

&u =∇(D∇u)+ (u−θv)(uu −u)v/ t fi +ws/ t fi −1 /t so

&v =−v/tv+

&w =−w /tw+

&s = (S(2ks(u−us))−s) / t s2

(θv ≤ u) ∧ (v < vc)

&u =∇(D∇u)+ ws/ t fi −1 /t so

&v =−v/tv+

&w =−w /tw+

&s = (S(2ks(u−us))−s) / t s2

Partition with Respect to v

Superposed Action Potentials

u ≥θo

u ≥θw

&w = −w / τ w+

&s = (S(2ks (u − us )) − s) / τ s2

u < θo

u < θw

u < θv

HA for the Model

(θv ≤ u) ∧ (v ≥ vc)

&u =∇(D∇u)+ (u−θv)(uu −u)v/ t fi +ws/ t fi −1 /t so

&v =−v/tv+

&w =−w / tw+

&s = (S(2ks(u−us))−s) / t s2

u < θo

&u =∇(D∇u)−u / t o1

&v = (1−v) / tv1−

&w = (1−u / tw∞ −w) / tw−

&s = (S(2ks(u−us))−s) / t s

u ≥θv

∧v< vc

(θv ≤ u) ∧ (v < vc)

&u =∇(D∇u)+ ws/ t fi −1 / t so

&v =−v/tv+

&w =−w / tw+

&s = (S(2ks(u−us))−s) / t s2

θo ≤ u < θw&u = ∇(D∇u) − u / τ o2

&v = −v / τ v2−

&w = (w∞* − w) / τ w1

&s = (S(2ks (u − us )) − s) / τ s

u ≥θv

∧v≥vc

tw− = τ w1

− + (τ w2− − τ w1

− ) S(2kw− (u − uw

− ))

τ so = τ so1 + (τ so2 − τ so1)S(2kso(u − uso ))&s = (S(2ks (u − us )) − s) / τ s

Analysis of Sigmoidal Switching

tw− = (1 − H (u − uw

− ))τ w1− + H (u − uw

− )τ w2−

&s = (rsR(u,θv)−s) / t s

u ≥uw−

u ≥θw

&w = −w / τ w+

&s = −s / τ s2 u < uw−

u < θw

u < θv

Current HA of Cornell’s Model

(θv ≤ u) ∧ (v ≥ vc)

&u =∇(D∇u)+ (u−θv)(uu −u)v/ t fi +ws/ t fi −1 / t so

&v =−v/tv+

&w =−w /tw+

&s = ((u−θv) / (2rsus)−s) / t s2

u ≥θv

∧v< vc

(θv ≤ u) ∧ (v < vc)

&u =∇(D∇u)+ ws/ t fi −1 / t so

&v =−v/tv+

&w =−w /tw+

&s = ((u−θv) / (2rsus)−s) / t s2

uw− ≤ u < θw

&u =∇(D∇u)−u / t o2

&v =−v/tv2−

&w = (w∞* −w) / tw2

&s =−s/t s1

u ≥θv

∧v≥vc

θo ≤ u < uw−

&u =∇(D∇u)−u / t o1

&v = (1−v) / tv1−

&w = (w∞* −w) / tw1

&s =−s/ t s1

u < θo

&u =∇(D∇u)−u / t o1

&v = (1−v) / tv1−

&w = (1−u / tw∞ −w) / tw1−

&s =−s/t s1u ≥θo

u < θo

Analysis of 1/τso ?

t so = τ so1 + (τ so2 − τ so1)S(2kso(u − uso ))

Jso = (1 − H (u −θw ))(u − uo ) + H (u −θw ) / τ so

Cubic Approximation of 1/τso ?

t so = τ so1 + (τ so2 − τ so1)S(2kso(u − uso ))

Jso = (1 − H (u −θw ))(u − uo ) + H (u −θw ) / τ so

Very sensitive!

Summary of Models

• Both models are nonlinear– Stony Brook’s: Linear in each cycle– Cornell’s: Nonlinear in specific modes

• Both models are deterministic

• Both models require identification– Stony Brook’s: On a mode-linear basis– Cornell’s: On an adiabatically approximated model

Modeling Challenges

• Identification of atrial models– Preliminary work: Already started at Cornell

• Dealing with nonlinearity– Analysis: New nonlinear techniques? Linear approx?

• Parameter mapping to physiological entities– Diagnosis and therapy: To be done later on

Analysis

Atrial Fibrillation (Afib)

• A spatial-temporal property– Has duration: it has to last for at least 8s– Has space: it is chaotic spiral breakup

• Formally capturing Afib– Multidisciplinary: CAV, Computer Vision, Fluid Dynamics– Techniques: Scale space, curvature, curl, entropy, logic

Spatial Superposition

• Detection problem: – Does a simulated tissue

contain a spiral ?

• Specification problem:– Encode above property as a

logical formula?– Can we learn the formula?

How? Use Spatial Abstraction

Superposition Quadtrees (SQTs)

i ij jj=1

1p (m) = p (m )4l!m {s,u,p,r}. p (m) = 1

Abstract position and compute PMF p(m) ≡ P[D=m]

Linear Spatial-Superposition Logic

Syntax

Semantics

The Path to the Core of a Spiral

21 3 4

Click the core to determine the quadtree

Overview of Our Approach

Emerald: Learning LSSL Formula

Emerald: Bounded Model Checking

Curvature Analysis

• Some properties of the curvature:– The curvature of a straight line is identical to 0– The curvature of a circle of radius R is constant– Where the curve undergoes a tight turn, the curvature is large

• Measuring the curvature:– Adapting Frontier Tool [Glimm et.al]: MPI code on Blue Gene– Also corrects topological errors

N - NormalT - TangentdT - Curvature

Edge Detection

Scalar field Front waveCanny algorithm

Normal Vectors Computation

Compute the Gradient

Tangent Vectors Computation

Based on the Gradient

The Curl of the Tangent Field

Curl = infinitesimal rotation of a vector field (circulation density of a fluid)

Verification Setup

• Models are deterministic with one initial state:– A spiral: induced with a specific protocol

• Verification becomes parameter estimation/synthesis: – In normal tissue: no fibrillation possible– Diseased tissue: brute force gives parameter bounds– Parameter space search: increases accuracy

• Parameters are mapped to the ionic entities:– Obtained mapping: used for diagnosis and therapy

Possible Collaborations

• Pancreatic cancer group:– Spatial properties: also a reaction diffusion system– Nonlinear models: approximation, diff. invariants, statistical MC– Parameter estimation: information theory, statistical MC

• Aerospace / Automotive groups: – Monitoring & Control: low energy defibrillation, stochastic HA – Machine learning: of spatial temporal patterns

Radu Grosu SUNY at Stony Brook

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