Modeling approaches to characterizing irregular conduction ... · Tanmay Gokhale, Huda Asfour,...

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


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 * # #


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


Rela

tive

CV

00.20.40.60.8

1


Rela

tive

Curv

atur

e

0.7

0.8

0.9

1Experimental ModelExperimental Model


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


0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6


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



0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6



Conduction Slowing due to Reduced Excitability is Anisotropic


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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