Radu Grosu SUNY at Stony Brook
description
Transcript of Radu Grosu SUNY at Stony Brook
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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
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Emergent Behavior in Heart Cells
Arrhythmia afflicts more than 3 million Americans alone
EKG
Surface
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Modeling
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Tissue Modeling: Triangular Lattice CellExcite and Simulation
Communication by diffusion
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Tissue Modeling: Square Lattice
CellExcite and Simulation
Communication by diffusion
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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
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Frequency Response
APD90: AP > 10% APm DI90: AP < 10% APm BCL: APD + DI
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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
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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
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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
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Stony Brook’s Cycle-Linear Model
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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
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• 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)
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Stimulated
Action Potential (AP) Phases
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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
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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
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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
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Problem: somewhat different inflection points
Identifying the Switching for all AP
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Solution: align, move up/down and remove inflection points- Confirmed by higher resolution samples
Identifying the Switching for all AP
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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
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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 )
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Finding Parameter Dependence on DI
Solution: apply mProny once again on each of the 25 points
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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
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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
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Cornell’s Nonlinear Minimal Model
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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
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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
⎧
⎨⎪⎪
⎩⎪⎪
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&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
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&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
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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
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Single Cell Action Potential
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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
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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
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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
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Superposed Action Potentials
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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
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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
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Superposed Action Potentials
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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
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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
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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
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Superposed Action Potentials
Very sensitive!
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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
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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
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Analysis
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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
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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
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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]
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Linear Spatial-Superposition Logic
Syntax
Semantics
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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
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Overview of Our Approach
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Emerald: Learning LSSL Formula
Emerald: Bounded Model Checking
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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
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Edge Detection
Scalar field Front waveCanny algorithm
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Normal Vectors Computation
Compute the Gradient
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Tangent Vectors Computation
Based on the Gradient
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The Curl of the Tangent Field
Curl = infinitesimal rotation of a vector field (circulation density of a fluid)
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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
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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