Radu Grosu SUNY at Stony Brook
description
Transcript of Radu Grosu SUNY at Stony Brook
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