1. Another
propaga,on
of
‘light’
in
the
heart
n
a
grain
of
sand
To
see
a
world
i
And
a
heaven
in
a
wild
flower
Hold
infinity
in
the
palm
of
your
hand
And
eternity
in
an
hour
-­‐
Auguries
of
Innocence,
by
William
Blake
Sehun
Chun

2. Big
picture
To
gaze
up
from
the
ruins
of
the
oppressive
present
towards
the
stars
is
to
recognize
the
indestruc,ble
world
of
laws,
to
strengthen
faith
in
reason,
to
realize
the
‘harmonia
mundi’
that
transfuses
all
phenomena,
and
that
never
has
been,
nor
will
be,
disturbed,
by
Hermann
Weyl,
1919.
DiLo
in
the
heart,
by
Sehun
Chun,
2010s

3. Introducing
the
HEART
• 72
beat
per
minute,
2.5
billion
,mes
in
average
(Beat
rate
=
100%)
• Electrical
signal
-­‐>
contrac,on
-­‐>
Pumping
oxygenated
blood
• Heart
model
=
Electricity
+
Mechanics
+
Elas,city
+
Fluids
• One
of
the
hardest
Mathema,cal
and
Computa,onal
models

4. How
electric
signal
propagates

5. What
is
the
electric
signal?
• Different
names:
Facilitated
diffusion
(biology),
Nonlinear
diffusion
equa,on
(ODE),
Diffusion-­‐reac,on
equa,on
(PDE).
• All-­‐or-­‐nothing
non-­‐decremental
traveling
wave
• No
conserva,onal
laws
of
energy,
momentum,
charges.
• Ex)
Collision
of
two
waves
=
canceling
out.
• Refractory
area
(In-­‐excitable
region
right
behind
signal).

6. Analogy
in
nature,
Forest
Fire
1. Energy
is
not
conserved
(Otherwise,
a
match
could
be
the
most
dangerous
weapon
!)
2.
Temperature
of
the
fire
does
not
depend
on
the
distance
from
the
origin,
but
depend
on
the
media.

7. Electromagne,c
field
in
the
microscopic
cardiac
cells
i i i i ∂d i
∇⋅d = ρ , ∇×h = j +
∂t
∂b i At
Intracellular
space
∇ ⋅ bi = 0, i
∇×e + =0
∂t
e e e e ∂d e
∇⋅d = ρ , ∇×h = j +
∂t
e
At
Extracellular
space
∂b
∇ ⋅ b e = 0, e
∇×e + =0
∂t
i e i e i e
ρ + ρ = const., j + j = 0 in Ω ∪Ω

8. Microscopic
To
Macroscopic
Collec,ng
several
cells
+
Averaging
the
quan,,es
i i i i i i i i
d =D,h =H d =D,h =H
i i i i i i i i
b =B,e =E b =B,e =E

9. Electromagne,c
field
in
the
macroscopic
cardiac
cells
i i i i ∂D i
∇⋅D = ρ , ∇×H = J +
∂t
i
∂B
∇ ⋅ Bi = 0, i
∇×E + =0
∂t At
the
same
space,
but
∂D e Two
different
equa,ons
e e e e
∇⋅D = ρ , ∇×H = J +
∂t
e
∂B
∇ ⋅ B e = 0, e
∇×E + =0
∂t
i e i e
ρ + ρ = const., J + J = 0, in Ω

10. Introducing
Bi-­‐domain
• Every
points
in
the
macroscopic
domain
means
two
separate
points
in
the
separate
domains.
• Energy
and
charges
are
conserved
in
bi-­‐
domains.
• Intracellular
space
is
only
our
concern,
then
energy
and
charges
are
not
conserved.
• Diffusion-­‐reac,on
system
is
only
possible
in
bi-­‐domain.

11. Anisotropy
in
Cardiac
,ssue
• Cable-­‐like,
cylindrical,
100
um
long
and
15um
• Cardiac
,ssue
is
strongly
anisotropic,
with
wave
speeds
that
differ
substan,ally
depending
on
their
direc,on.
• 0.5
m/s
along
fibers
and
about
0.17
m/s
transverse
the
fibers.

12. Anisotropy
in
Mathema,cs
• d:
Diffusion
tensor,
variable
coefficient
∇ ⋅ ( d∇φ )
• In
the
direc,on
of
characteris,cs
• A
liLle
misplacement
leads
to
the
shock
of
waves.
• Related
to
the
way
the
heart
is
folded
for
contrac,on
• Only
God
is
allowed
to
put
anisotropy
in
the
heart
-­‐>
Another
beauty
of
the
heart

13. How
to
simula,on
the
electrophysiology
phenomena
on
complex
anisotropic
geometry
• Method
of
Moving
Frames
• Originally
developed
by
É.
Cartan
in
1920’s
• Represent
a
geometry
with
oscula,on
Euclidean
planes
by
inheri,ng
the
metric
tensor.
• Different
Euclidean
axis
for
each
points
with
various
length
of
the
axis.

14. Rough
meaning
of
Geometry
Op-cs
Electrophysiology
Role
of
Passive
role
for
the
Ac,ve
delivering
geometry
wave
(deflec,on,
of
the
wave
reflec,on,
absorp,on)
Meaning
Mass
distribu,on
Cell
distribu,on?
(what
is
equivalent
to
mass
in
biology?)
Propaga,on
Anything
to
change
Anything
to
the
propaga,on
of
change
the
the
light
cardiac
electric
signal

15. Defini,on
of
geometry
in
electrophysiology
• Geometry
=
the
combina,on
of
the
followings:
1)
Loca,on
of
the
star,ng
point
2)
Conduc,ng
proper,es
of
the
media
3)
3D
shape
of
the
heart
-­‐>
Geometry
is
taken
in
the
sense
of
the
Field
theory.
• Electric
signal
only
depends
on
geometry.
• Electric
signal
has
negligible
kine,c
energy.
• Heart
works
100%
according
to
its
design.
• Analogous
to
the
light
as
a
signal

16. Good
geometry
and
Bad
geometry
• Good
geometry
=
to
guide
the
electrical
signal
propaga,on
for
“efficient”
cardiac
contrac,on,
i.e.,
star,ng
from
one
point
to
converge
to
one
or
two
points
in
the
atrium.
• Bad
geometry
=
To
change
its
original
design
• Examples
of
bad
geometry:
change
of
loca,on
or
turn-­‐
off
of
SAN,
scar
,ssue
(infarc,on),
change
of
,ssue
proper,es,
enlarged
3D
shape,
but
the
worst
of
all
is
the
mul,ple
self-­‐ini,ators
(though
they
are
all
correlated)

17. How
atrial
fibrilla,on
(AF)
happens?
§
Normal
propaga,on
starts
from
a
point
and
converges
to
a
point.
§
AF
only
happens
when
the
propaga,on
deviates
from
its
original
track
and
comes
back
to
excite
cardiac
,ssue
again
(Reentry)
§
Reentry
requires
unidirec,onal
pathway.
Unidirec,onal
pathway

18. Clinical
observa,ons
and
conjectures
• The
role
of
PVs
on
AF:
-­‐
Spontaneous
Ini,a,on
of
atrial
fibrilla,on
by
ectopic
beats
origina,ng
in
the
pulmonary
vein,
The
New
England
Journal
of
Medicine,
1998,
by
M.
Haissaguerre
et
al
-­‐
Arrhythmogenic
substrate
of
the
Pulmonary
Veins
Assessed
by
High-­‐Resolu,on
Op,cal
Mapping,
Circula,on
2003,
by
R.
Arora
et
al.
-­‐
Atrial
Fibrilla,on
Begets
Atrial
Fibrilla,on
in
the
Pulmonary
Veins:
On
the
Impact
of
Atrial
Fibrilla,on
on
the
Electrophysiological
Proper,es
of
the
Pulmonary
Veins
in
Humans,
J.
Am.
Coll.
Cardiol.
2008,
by
T.
Rostock
et
al.
• Observa,ons
and
theories
of
AF:
-­‐
New
ideas
about
atrial
fibrilla,on
50
years
on,
Nature
2002,
by
S.
NaLel
-­‐
Atrial
Remodeling
and
Atrial
Fibrilla,on:
Mechanisms
and
Implica,ons,
Circula,on
2008,
by
S.
NaLel
et
al
-­‐
Rotors
and
Spiral
waves
in
Atrial
Fibrilla,on,
J.
Cardiovasc
Electrophysiol
2003,
by
J.
Jalife
-­‐
Circula,on
movement
in
rabbit
atrial
muscle
as
a
mechanism
of
tachycardia.
III.
The
“leading
circle”
concept:
a
new
model
of
circus
movement
in
cardiac
,ssue
without
the
involvement
of
an
anatomical
obstable,
Circ.
Res.
1977,
by
M.A.
Allessie
et
al.
• Structures
of
ler
atrium
and
PVs:
-­‐
The
importance
of
Atrial
Structure
and
Fibers,
Clinical
Anatomy
2009,
by
S.
Y.
Ho
-­‐
The
structure
and
components
of
the
atrial
chambers,
Europace
2007,
by
R.
H.
Anderson
-­‐
Normal
atrial
ac,va,on
and
voltage
during
sinus
rhythm
in
the
human
heart:
An
endocardial
and
epicardial
mapping
study
in
pa,ents
with
a
history
of
atrial
fibrilla,on,
J.
Cardiovasc.
Electrophysiol.
2007,
by
R.
Lemery.

19. Blocking
Pulmonary
Veins
by
lesions
A
surgical
procedure
is
to
insert
a
catheter
through
the
vein
into
the
atrium
and
to
burn
cardiac
cells
around
the
PV
to
prevent
the
self-­‐ini,ators
around
the
PVs.

20. Summary
of
the
proper,es
of
cardiac
electric
signal
propaga,on
(1) In
bi-­‐domain
with
none
of
conserva,on
laws
(2) Anisotropy
and
inhomogeneous
is
everywhere.
(3) 3D
shape
is
smooth,
but
non-­‐
uniform
with
various
curvature.

21. Theories
of
electrophysiology
in
the
heart
• Regarded
as
a
kind
of
nerve
system.
• Quick
transfer
from
ODE
to
PDE.
• Analogous
jump
from
a
1D
cable
to
mul,dimensional
space.
• Alterna,ve
is
the
Kinema,cs
approach
(1990s~):
Study
of
the
wave
front
to
find
the
cri,cal
curvature
to
find
K
*
>
K
in
∂K 
∂K ∂V 2
( ∂ )
∫ 0 KV dξ + C + ∂t + K V + ∂2 = −ΓV 2

22. Inspira,ons
for
rela,ve
accelera,on
approach
Rule
of
games:
(1) Excited
person
has
3L
of
water.
(2) Given
1L
of
water,
an
excitable
person
get
2L
more
water
and
get
excited.
(3) A
player
wins
if
it
has
more
children
than
the
compe,tor.
(4) Game
stops
if
any
one
of
the
column
can
not
receive
1L.

23. Hypothesis
and
Proposi,on
• Hypothesis:
If
the
rela,ve
accelera,on
of
cardiac
excita,on
propaga,on
becomes
sufficiently
large,
then
the
propaga,on
stops.
•
Proposi,on:
If
the
following
is
sufficiently
large,
then
the
propaga,on
stops
# i &
i
1 % ∂E ∂v ∂G i ∂v ∂ ( cv ) (
i
− + v +G +
E % ∂n ∂λ ∂n
$ ∂n ∂n ( '
2 2
1 ∂( gΛ k d k g kk )
where, E = ∑ (Λ k )2 d k g kk G =∑
k=1 k=1 g ∂x k

24. Diffusion-­‐Reac,on
Tensor
• A
DR-­‐tensor,
is
obtained
as
k k kk
C = d Λkg
Anisotropic
coefficient
× Propaga,onal
direc,on
× the
3cD
geometric
shape
for
onjugate
metric
tensor
• Coincide
with
the
defini,on
of
geometry:
•
k
(1)
d
:
type
of
media.
• (2)
Λ
k
:
Loca,on
of
SAN
and
surface
2D
geometries.
• (3)
g
kk
:
3D
shape.
€
€ €
• Large
varia,on
of
the
tensor
means
the
break-­‐up
of
the
wave

25. Applying
to
the
PVs
θ
d 4

26. Various
Anisotropy
on
the
PV
A
shape
of
PV
junc,on
No
anisotropy
Circumferen,al
anisotropy
Longitudinal
anisotropy
(Perez-­‐Lugones
et
al,
2003)

28. Construct
a
model

29. Modelling
of
Re-­‐entrance
on
a
spherical
shell
with
a
PV-­‐like
column
1)
Normal
condi,on
T=0,
Front
T=10,
Front
T=20,
Back
T=25,
Back
2)
Deteriorated
myocardial
cells
on
the
PVs
Mul,ple
reentrant
waves
Overdrive
suppression
T=20,
Back
T=22,
Back
T=37,
Front
T=100,
Front

30. Consequences
(1) Unidirec,onal
pathways
are
the
PVs
with
weakened
anisotropy
toward
which
the
wave
approaches
with
an
oblique
angle
due
to
some
other
factors,
for
example,
the
presence
of
scar
,ssue,
the
change
of
SAN,
etc.
(2) Cardiac
excita,on
propaga,on
can
be
represented
as
the
field
of
the
trajectories.
(3) Something
is
moving
along
the
trajectory

31. What
is
actually
moving
in
electric
signal
propaga,on
?
S,mula,ng
moving
body
consist
of
many
s,mula,on
par,cles
that
lower
the
res,ng
poten,al
to
ini,ate
cardiac
ac,on
poten,al.
Ex)
For
forest
fire,
it
is
equivalent
to
a
group
of
par,cles
of
high-­‐temperature
to
ini,ate
fire
on
a
unburned
tree.

32. S,mula,ng
moving
body

33. Maxwell’s
equa,ons
in
the
universe
vs.
Maxwell’s
equa,ons
in
the
heart
Maxwell’s
in
the
universe
in
the
heart
equa-ons
How
to
Ray
Diffusion
propagate
Gauge
choice
1 ∂Φ
∇⋅ A+ 2 =0 ∇⋅ A+Φ = 0
c ∂t
Gauge
func,on
2 1∂Λ 2 ∂Λ 2
∇ Λ− 2 2 =0 ∇ Λ− =0
c ∂t ∂t

34. Diffusion-­‐reac,on
equa,ons
from
Maxwell’s
equa,ons
• The
diffusion-­‐reac,on
equa,ons
are
one
projec,on
of
Maxwell’s
equa,ons
in
bi-­‐domain,
so
E
and
B
are
under-­‐determined.
• The
normal
heart
is
designed
to
generate
the
minimum
degree
of
the
magne,c
field
(conjecture
1).
• Increasing
magne,c
field
in
the
specific
area
of
the
heart
may
mean
AF
(conjecture
2).
• It
may
explain
why
the
external
electric
shock
can
resuscitate
temporarily
non-­‐moving
heart
arer
CPR
or
can
cure
AF
temporarily.

35. Thank
you
for
aLen,on
!
Now
I
know
I've
got
a
heart,
'cause
it's
breaking