Radu Grosu SUNY at Stony Brook

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

EKG

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

time

volta

geSt

imul

us

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

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

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

Frequency Response

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

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

s

off∧u <θ

U son

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

s

off∧u <θ

Uons

u ≥θU

u ≥θE

Pv V

u ≤θR

Fv V

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

&xo =boxo

u ≥θP /

xi =ai

xo =ao

Identifying the HA Dynamics for One APM

odifi

ed P

rony

Met

hod

Stimulated

s

off∧u <θ

U(d

i) son

/ di=t

u ≥θU(d

i)

u ≥θE(d

i)

u ≤θ

R(d

i)

/t =0

u ≤θP(d

i)

u ≤θF(d

i)

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

s

off∧u <θ

U(d

i) son

/ di=t

u ≥θU(d

i)

u ≥θE(d

i)

u ≤θ

R(d

i)

/t =0

u ≤θP(d

i)

u ≤θF(d

i)

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

bi2di

bo(di)=ao1ebo1di + ao2e

bo2di

Cyc

le L

inea

r

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

else

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

Cornell’s Minimal Model

PiecewiseNonlinear

Heaviside(step)

Sigmoid(s-step)

PiecewiseNonlinear

PiecewiseBilinear

PiecewiseLinear

Nonlinear

ActivationThreshol

d

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 ))

w∞

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

Cornell’s Minimal Model

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

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

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

Superposed Action Potentials

u ≥uw−

u ≥θw

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

−

&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

Superposed Action Potentials

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)

4

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

Root

21 3 4

21 3 4

21 3 4

21 3 4

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

T

T

N N

dT

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