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Modeling approaches to characterizing irregular conduction in
micro-heterogeneous tissues
Tanmay Gokhale, Huda Asfour, Nenad Bursac and Craig Henriquez
MD-PhD StudentDuke University
Durham, North Carolina
Motivating Question
How do microscopic changes in the structure of the heart tissue, such as those caused by aging and diseases, affect the
electrical conduction of the heart?
Outline
• Background• Methodology of Paired Computational and
Experimental Studies• Effects of Microheterogeneities on
Conduction
Fibrosis Distorts Conduction
Zhang et al. 2013; Adapted from de Jong et al. 2011
Heal
thy
Cont
rol
Atria
l Fib
rilla
tion
Red: Normal cardiac muscleBlue: Collagen
Arrow: Muscle breakdown
Tissue Geometry Alters Conduction
Actio
n Po
tent
ial
Actio
n Po
tent
ial
Stimulus Site Recording Electrodes
Adapted from Rohr and Kucera 1997
What about more complex fibrosis?
• Individual heterogeneities change geometry and modulate local conduction
• à What is the effect of numerous heterogeneities in aggregate (i.e. complex fibrosis)?
Engineered Excitable Cells for
Paired Studies
• HEK-293 cell + Nav1.5 + Kir2.1 + Cx43
– Excitable ”Ex293 cells”– Kirkton et al. 2011
• Mathematical model of Ex293 cells
– Inter-cell variability in current densities– Gokhale et al. 2017
INa
IK
INa,wt
IK,wt
Gj
Regular patterns of heterogeneityIdealized geometry of fibrotic tissue with regularly spaced and equally sized non-conductive heterogeneities
Obstacle Width
StrandWidth
Strand Width
Obstacle to Strand
Ratio
Obstacle Width
Strand Width
Obstacle Density
0 ⎯⎯ ⎯⎯ 0 %
0.3 150 µm 520 µm 5 %
0.6 150 µm 240 µm 15 %
1.5 150 µm 100 µm 35 %
3.0 300 µm 100 µm 56 %
5.0 500 µm 100 µm 69 %
7.0 700 µm 100 µm 77 %
Non-conductiveheterogeneities
Ex293 cells
Regular patterns of heterogeneity
Obstacles patterned as 150 µm in width were 151.7 µm to 162.6 µm after 4 days of culture
Control Obstacle to Strand: 1.5 Obstacle to Strand: 7.0
Scale bar: 150 µm
Non-conductiveheterogeneities
Ex293 cells
Effects of Heterogeneity
Obstacle-to-Strand Ratio: 0Relative CV: 1.0
Obstacle-to-Strand Ratio: 1.5Relative CV: 0.862
Obstacle-to-Strand Ratio: 7.0Relative CV: 0.794
CV: Conduction velocity
0.5 cm
0 0.3 0.6 1.5 3 5 7
Rel
ativ
e C
V
0
0.2
0.4
0.6
0.8
1 *
0 0.3 0.6 1.5 3 5 7R
elat
ive
APD
0
0.2
0.4
0.6
0.8
1 *
Obstacle-to-Strand Ratio0 0.3 0.6 1.5 3 5 7
Rela
tive
CV0
0.20.40.60.8
1
Obstacle-to-Strand Ratio0 0.3 0.6 1.5 3 5 7
Rela
tive
CV
00.20.40.60.8
1
Mean � se; n = 13-68 monolayersAsterisk indicates difference from 0 case
Mean � self; Asterisk indicates difference from 0 case
AP: Action potential
Effects of Heterogeneity• Heterogeneities lead to conduction slowing and
shortened action potential duration
Rel
ativ
e C
ondu
ctio
n Ve
loci
ty
Rel
ativ
e AP
Dur
atio
n
Curvature anisotropy: ratio of distance along diagonal (black) to distance along principal axes (red)
0 0.3 0.6 1.5 3 5 7
Cur
vatu
re A
niso
tropy
0.70.75
0.80.85
0.90.95
1 * # #
Obstacle-to-Strand Ratio0 0.3 0.6 1.5 3 5 7
Rela
tive
CV
00.20.40.60.8
1Round
Flat
Mean � sd; n = 10-15 monolayers* and # indicate difference from all lower
Effects of Heterogeneity• Heterogeneities cause change in the curvature of
the macroscopic wavefront
Model Specifications
• Monodomain formulation• Finite differences discretization of
spatial operator with dx = dy = 10 µm• Obstacles with no-flux boundaries• Simulated potentials processed to
make comparable to experimental optical mapping data
Expe
rimen
tal
Sim
ulat
ion
Obstacle-to-Strand: 0 Obstacle-to-Strand: 7.0
Comparing Model and Experiments
Comparing Model and Experiments
Obstacle-to-Strand Ratio0 0.3 0.6 1.5 3 5 7
Rela
tive
CV
00.20.40.60.8
1
Obstacle-to-Strand Ratio0 0.3 0.6 1.5 3 5 7
Rela
tive
Curv
atur
e
0.7
0.8
0.9
1Experimental ModelExperimental Model
Obstacle-to-Strand Ratio0 0.3 0.6 1.5 3 5 7
Rela
tive
APD
00.20.40.60.8
1
• Model recapitulates conduction slowing and curvature
Limitation: model does not replicate reduction in AP duration
Rel
ativ
e C
ondu
ctio
n Ve
loci
ty
Rel
ativ
e AP
Dur
atio
n
AP: Action potential
Factors Affecting Macroscale Conduction
!"#$%&'("# )*+"&(',=
.(/'0#&* 12 !"#$%&'("# 30'ℎ5(6* '" 5708*+
Depends on total path length Depends on microscale conduction velocity
Conduction with HeterogeneitiesPresence of heterogeneities causes path tortuosity
: Stimulus
Evaluating Effect of Path Tortuosity with Automata Models• Rules-based approach• Each node exists in one of fixed # of states• Limitation: ignores effects of electrical load
– à Allows us to isolate impact of path tortuosity by removing effects of source-load mismatches on local conduction velocity
1 1 11
1 1 1 11 1 1 11 1 1 11 1 1
1 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 1
1 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 1
Effect of Path TortuosityBiop
hysic
alAu
tomata
Obstacle-to-Strand Ratio0 5
Rel
ativ
e C
V0
0.2
0.4
0.6
0.8
1
Obstacle-to-Strand Ratio0 5
Cur
vatu
re A
niso
tropy
0.70.75
0.80.85
0.90.95
1
Experimental Biophysical Model Automata Model
Rel
ativ
e C
ondu
ctio
n Ve
loci
ty
Effect of Path Tortuosity
2 mm
Error in Automata M
odel (ms)
0
2
4
6
8
10
12
∴ Path tortuosity alone does not explain the observed macroscopic changes.There must be local variation in microscopic conduction velocity that directly affects macroscale behavior
Difference in Activation Time (m
s)
Biophysical Activation Isochrone Automata Activation Isochrone Biophysical Activation Isochrone Automata Activation Isochrone(TrueConduction Behavior)
(EffectofPathTortuosityOnly)
Regions of Focus for Studying Microscale Behavior
: Stimulus
Branching Sites:Arriving wavefront branches in three
directions
Intersection Sites:Two wavefronts
arriving simultaneously
Behavior Along Principal Axes
Predicted activation without delay
Actual activation
à Branching along principal axes leads to conduction slowing
Activation Isochrones
a b
c d
(dVm/dt)max : Maximal upstroke velocity
Behavior Along Principal Axes
Net Effect of Branching Delays
à Branching related slowing is the primary mechanism of microscale conduction variation along the principal axes
Error in Automata Model Along Principal Axis
Delay Due to Slowing at Branching Sites
2 mm
: Stimulus
Predicted activation without acceleration
Actual activation
Wavefront collisions at intersections accelerate propagation
Behavior At Intersection Sites
Behavior At Intersection Sites
Recap• Source-load imbalances à Changes
in microscale conduction– Conduction slowing at branching
sites– Conduction speeding at
intersections• Path tortuosity + source-load
imbalances à macroscopic slowing and wavefront curvature changes
• How does reduced excitability affect these macro- and micro-scale behaviors?
Macroscopic Behavior Under Reduced Excitability
Obstacle-to-Strand Ratio: 3.0 Obstacle-to-Strand Ratio: 3.0With 100 µm TTX
Reduced sodium excitability results in slowing and a change in wavefront curvature
TTX: tetrodotoxin, Na+ channel blocker
Conduction Slowing due to Reduced Excitability
Obstacle-to-Strand Ratio0 0.3 0.6 1.5 3
Rela
tive
CV
0
0.2
0.4
0.6
0.8
1
Model exhibits block at OSR of 3.0Experiments show wavebreak and meandering at
OSR 5.0 and not sustained at 7.0
Exp. ControlExp. TTXModel ControlModel TTX
Obstacle-to-Strand Ratio0.3
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6 Exp. ControlExp. TTXModel ControlModel TTX
Obstacle-to-Strand Ratio0.3
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
TTX: tetrodotoxin, Na+ channel blocker
Reduced Excitability Attenuates Heterogeneity-Related Curvature Anisotropy
Obstacle-to-Strand Ratio0 0.3 0.6 1.5 3
Rela
tive
Curv
atur
e
0.7
0.8
0.9
1
0 0.3 0.6 1.5 3 5 7
Rel
ativ
e AP
D
0
0.2
0.4
0.6
0.8
1 *
0 0.3 0.6 1.5 3 5 7R
elat
ive
APD
0
0.2
0.4
0.6
0.8
1 *
* p < 0.05 by post-hoc Tukey
Cont
rol
100
µm T
TX
Obstacle-to-Strand Ratio: 3.0
Exp. ControlExp. TTXModel ControlModel TTX
Obstacle-to-Strand Ratio0.3
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
Exp. ControlExp. TTXModel ControlModel TTX
TTX: tetrodotoxin, Na+ channel blocker
Conduction Slowing due to Reduced Excitability is Anisotropic
TTX: tetrodotoxin, Na+ channel blocker
Effects of Reduced Excitability
32
Global slowing of conduction within strands
Control Reduced Excitability
CV: Conduction velocity
Effects of Reduced Excitability
33
Increased delay at branches and no change in acceleration at collisions drives change in wavefront curvature
Control Reduced Excitability
Conclusions
• Reduced excitability à greater slowing at branches àexaggerated effect of collisions à rounder wavefronts and slowed conduction
34
Regular heterogeneities Conduction slowing &
Wavefront curvature anisotropyPath tortuosity
Effects of source-load mismatch:
Branching àSlowing
Collisions àSpeeding
Previously described by Fast, Kleber,Rohr, Kucera and others
Novel mechanism of microscaleconduction modulation
Limitations
• Highly idealized geometry of fibrosis• Complexity of real myocardium (3D
structure, fibroblasts etc)• Question of APD reduction at high
obstacle-to-strand ratios
Future Directions
• Incorporation of local anisotropy to try to understand changes in APD
• Effect of heterogeneities on dynamic properties (i.e. restitution, rate dependence, reentry)
• Transition to realistic, histologically-inspired tissue structure (fibrosis distribution, fibroblasts, anisotropy etc)
Thank you!
Henriquez LabProf. Craig HenriquezLetitia Hubbard, PhDEli MedvescekShravan Verma, MD
Bursac LabProf. Nenad BursacRob Kirkton, PhDHuda Asfour, PhDHung Nguyen, PhD
Duke Research ComputingTom MilledgeMark DeLong
Funding Sources:NHLBINIGMS (MSTP T32)
Thank you!
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