Download Electrical Wave Propagation in a Minimally Realistic Fiber

Heart: From macroscopic to microscopic
Xianfeng Song
Advisor: Sima Setayeshgar
January 9, 2007
Outline
Transport Through the Myocardium of
Pharmocokinetic Agents Placed in the
Pericardial Sac: Insights From Physical
Modeling
Electrical Wave Propagation in a Minimally
Realistic Fiber Architecture Model of the Left
Ventricle
Calcium dynamics: exploring the stochastic
effect
Transport Through the Myocardium of
Pharmocokinetic Agents Placed in the
Pericardial Sac:
Insights From Physical Modeling
Xianfeng Song, Department of Physics, Indiana University
Keith L. March, IUPUI Medical School
Sima Setayeshgar, Department of Physics, Indiana University
Pericardial Delivery: Motivation

The pericardial sac is a fluid-filled self-contained
space surrounding the heart. As such, it can be
potentially used therapeutically as a “drug
reservoir.”

Delivery of anti-arrhythmic, gene therapeutic
agents to
 Coronary vasculature
 Myocardium

Recent experimental feasibility of pericardial
access

Verrier VL, et al., “Transatrial access to the normal pericardial space: a
novel approach for diagnostic sampling, pericardiocentesis and
therapeutic interventions,” Circulation (1998) 98:2331-2333.

Stoll HP, et al., “Pharmacokinetic and consistency of pericardial delivery
directed to coronary arteries: direct comparison with endoluminal
delivery,” Clin Cardiol (1999) 22(Suppl-I): I-10-I-16.
Vperi (human) =10ml – 50ml
Diffusion in biological processes
Diffusion plays a crucial role in brain
function
Nicholson, C. (2001), “Diffusion and related transport mechanisms in brain
tissue”, Rep. Prog. Phys. 64, 815-884
Diffusion is important during early
Drosophila embryonic pattern formation
Gregor, T., W. Bialek, et al. (2005). "Diffusion and scaling during early
embryonic pattern formation." Proceedings of the National Academy of
Sciences 102(51):
Protein diffusion in single cells
M. B. Elowitz et al. “Proteins diffusion in single cells”, J. Bacteriology
Pericardial Delivery: Outline
Experiments
Mathematical modeling
Comparison with data
Conclusions
Experiments
 Experimental subjects: juvenile farm pigs
 Radiotracer method to determine the spatial concentration profile
from gamma radiation rate, using radio-iodinated test agents
 Insulin-like Growth Factor (125I-IGF, MW: 7734 Da)
 Basic Fibroblast Growth Factor (125I-bFGF, MW: 18000 Da)
 Initial concentration delivered to the pericardial sac at t=0
 200 or 2000 g in 10 ml of injectate
 Harvesting at t=1h or 24h after delivery
Experimental Procedure
 At t = T (1h or 24h), sac fluid is
distilled: CP(T)
 Tissue strips are submerged in
liquid nitrogen to fix concentration.
 Cylindrical transmyocardial
specimens are sectioned into
slices: CiT(x,T) x denotes
i
CT(x,T) = i CiT(x,T)
x: depth in tissue
Mathematical Modeling
 Goals
 Determine key physical processes, and extract governing
parameters
 Assess the efficacy of drug penetration in the myocardium
using this mode of delivery
 Key physical processes
 Substrate transport across boundary layer between pericardial
sac and myocardium: 
 Substrate diffusion in myocardium: DT
 Substrate washout in myocardium
(through the intramural vascular and lymphatic capillaries): k
Idealized Spherical Geometry
Pericardial sac: R2 – R3
Myocardium: R1 – R2
Chamber: 0 – R1
R1 = 2.5cm
R2 = 3.5cm
Vperi= 10ml - 40ml
Governing Equations and Boundary
Conditions

Governing equation in myocardium: diffusion + washout
CT: concentration of agent in tissue
DT: effective diffusion constant in tissue
k: washout rate

Pericardial sac as a drug reservoir (well-mixed and no washout): drug
number conservation

Boundary condition: drug current at peri/epicardial boundary
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2005, Los Angeles
Numerical Fits to Experiments
Conce
Drug Concentration
1 Molecule per ml = 1.3 x10-11 picograms per ml
Error surface
Fit Results
Numerical values for DT, k,  consistent for
IGF, bFGF to within experimental errors
Time Course from Simulation
Parameters: DT=7×10-6cm2s-1 k=5×10-4s-1 =3.2×10-6cm2s2
Effective Diffusion,D*, in Tortuous Media

Stokes-Einstein relation
D: diffusion constant
R: hydrodynamic radius
: viscosity
T: temperature

Diffusion in tortuous medium
D*: effective diffusion constant
D: diffusion constant in fluid
: tortuosity
For myocardium, = 2.11.
(from M. Suenson, D.R. Richmond, J.B. Bassingthwaighte, “Diffusion of sucrose, sodium, and water in ventricular myocardium,
American Joural of Physiology,” 227(5), 1974 )

Numerical estimates for diffusion constants
 IGF : D ~ 4 x 10-7 cm2s-1
 bFGF: D ~ 3 x 10-7 cm2s-1

Our fitted values are in order of 10-6 - 10-5 cm2sec-1, 10 to 50 times larger !!
Transport via Intramural Vasculature
Drug permeates into vasculature from extracellular space at high
concentration and permeates out of the vasculature into the extracellular
space at low concentration, thereby increasing the effective diffusion
constant in the tissue.
Epi
Endo
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2005, Los Angeles
Diffusion in Active Viscoelastic Media
Heart tissue is a porous medium consisting of extracellular space and muscle
fibers. The extracellular space consists of an incompressible fluid (mostly
water) and collagen.
Expansion and contraction of the fiber bundles and sheets leads to changes
in pore size at the tissue level and therefore mixing of the extracellular volume.
This effective "stirring" results in larger diffusion constants.
Conclusion

Model accounting for effective diffusion and washout is consistent with
experiments despite its simplicity.

Quantitative determination of numerical values for physical parameters
 Effective diffusion constant
IGF: DT = (9±3) x 10-6 cm2s-1, bFGF: DT = (6±3) x 10-6 cm2s-1
 Washout rate
IGF: k = (8±3) x 10-4 s-1, bFGF: k = (9±3) x 10-4 s-1
 Peri-epicardial boundary permeability
IGF:  = (2.7±0.8) x 10-6 cm s-1, bFGF:  =
(6.0±1.6) x 10-6 cm s-1

Enhanced effective diffusion, allowing for improved transport

Feasibility of computational studies of amount and time course of
pericardial drug delivery to cardiac tissue, using experimentally derived
values for physical parameters.
Electrical Wave Propagation in a
Minimally Realistic Fiber Architecture
Model of the Left Ventricle
Xianfeng Song, Department of Physics, Indiana University
Sima Setayeshgar, Department of Physics, Indiana University
Outline
Motivation
Model Construction
Numerical Results
Conclusions and Future Work
Motivation

Ventricular fibrillation (VF) is the main cause of
sudden cardiac death in industrialized nations,
accounting for 1 out of 10 deaths.

Strong experimental evidence suggests that selfsustained waves of electrical wave activity in
cardiac tissue are related to fatal arrhythmias.

Mechanisms that generate and sustain VF are
poorly understood.

Conjectured mechanism:
Breakdown of a single spiral (scroll) wave
into a disordered state, resulting from various
mechanisms of spiral wave instability.
W.F. Witkowksi, et al., Nature 392, 78 (1998)
Patch size: 5 cm x 5 cm
Time spacing: 5 msec
Focus of this work
Distinguish the role in the generation of electrical wave instabilities of the
“passive” properties of cardiac tissue as a conducting medium
 geometrical factors (aspect ratio and curvature)
 rotating anisotropy (rotation of mean fiber direction through heart wall)
 bidomain description (intra- and extra-cellular spaces treated separately)
from its “active” properties, determined by cardiac cell electrophysiology.
From idealized to fully realistic
geometrical modeling
Rectangular slab
J.P. Keener, et al., in Cardiac Electrophysiology, eds.
D. P. Zipes et al. (1995)
Anatomical canine ventricular model
Courtesy of A. V. Panfilov, in Physics Today,
Part 1, August 1996
Minimally realistic model of LV for studying electrical wave propagation in three
dimensional anisotropic myocardium that adequately addresses the role of geometry and
fiber architecture and is:
 Simpler and computationally more tractable than fully realistic models
 Easily parallelizable and with good scalability
 More feasible for incorporating realistic electrophysiology, electromechanical coupling,
bidomain description
LV Fiber Architecture
 Early dissection results revealed
nested ventricular fiber surfaces,
with fibers given approximately by
geodesics on these surfaces.
Fibers on a nested pair of surfaces in the LV,
from C. E. Thomas, Am. J. Anatomy (1957).
 Peskin asymptotic model: first
principles derivation of toroidal fiber
surfaces and fiber trajectories as
approximate geodesics
Fiber angle profile through LV thickness:
Comparison of Peskin asymptotic model and dissection results,
from C. S. Peskin, Comm. in Pure and Appl. Math. (1989).
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Model Construction
 Nested cone geometry and fiber surfaces
 Fiber paths
 Geodesics on fiber surfaces
 Circumferential at midwall
2
L   f ( ,
1
d
,  )d
d
f
d  f 



 d   ' 
z
subject to:
 0  0

Fiber trajectory:
L  0 

  1
 1 
  1  a 2 sec 1 
Fiber trajectories on nested pair of conical surfaces:
inner surface
outer surface
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Governing Equations
 Transmembrane potential propagation
Cm
u
   ( Du )  I m
t
Cm: capacitance per unit ar
D: diffusion tensor
u: transmembrane potentia
Im: transmembrane current
 Transmembrane current, Im, described by simplified
FitzHugh-Nagumo type dynamics*
I m  ku(u  a)(u  1)  uv
v 
v 
    1  v  ku(u  a  1
t 
2  u 
v: gate variable
Parameters: a=0.1, 1=0.07, 2=0.3,
k=8, =0.01, Cm=1
* R. R. Aliev and A. V. Panfilov, Chaos Solitons Fractals 7, 293 (1996)
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Numerical Implementation
 Working in spherical coordinates,
with the boundaries of the
computational domain described by
two nested cones, is equivalent to
computing in a box.
 Standard centered finite difference
scheme is used to treat the spatial
derivatives, along with first-order
explicit Euler time-stepping.
Diffusion Tensor
Transformation matrix R
Local Coordinate
Dlocal
 D//

 0
 0

0
D p1
0
0 

0 
D p 2 
Lab Coordinate
Dlab  R 1 Dlocal R
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Parallelization
 The communication can be minimized when parallelized along azimuthal
direction.
 Computational results show the model has a very good scalability.
CPUs
Speed up
2
1.42 ± 0.10
4
3.58 ± 0.16
8
7.61 ±0.46
16
14.95 ±0.46
32
28.04 ± 0.85
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Phase Singularities
Tips and filaments are phase singularities that act as organizing centers for
spiral (2D) and scroll (3D) dynamics, respectively, offering a way to quantify
and simplify the full spatiotemporal dynamics.
Color denotes the transmembrane potential.
Movie shows the spread of excitation for 0 < t < 30,
characterized by a single filament.
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Find all tips
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Random choose a tip
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Search for the closest tip
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Make connection
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Continue doing search
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Continue
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Continue
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Continue
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
The closest tip is too far
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Reverse the search direction
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Continue
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Complete the filament
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Start a new filament
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding Algorithm
“Distance” between two tips:
If two tips are not on a same
fiber surface or on adjacent
surfaces, the distance is
defined to be infinity.
Otherwise, the distance is the
distance along the fiber
surface
Repeat until all tips are consumed
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Filament-finding result
t=2
FHN Model:
t = 999
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Numerical Convergence
The results for filament length agree
to within error bars for three different
mesh sizes.
The results for filament number agree
to within error bars for dr=0.7 and
dr=0.5. The result for dr=1.1 is slightly
off, which could be due to the filament
finding algorithm.
Filament Number and Filament Length
versus Heart size
The computation time for dr=0.7 for
one wave period in a normal heart size
is less than 1 hour of CPU time using
FHN-like electrophysiological model.
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Scaling of Ventricular Turbulence
Log(total filament length) and Log(filament number)
The average filament length, normalized by
Both filament length
versus Log(heart size)
average heart thickness, versus heart size
These results are in agreement with those obtained with the fully realistic
canine anatomical model, using the same electrophysiology.
A. V. Panfilov, Phys. Rev. E 59, R6251 (1999)
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Conclusions and Future Work
 We have constructed and implemented a minimally realistic fiber
architecture model of the left ventricle for studying electrical wave
propagation in the three dimensional myocardium.
 Our model adequately addresses the geometry and fiber architecture of the
LV, as indicated by the agreement of filament dynamics with that from fully
realistic geometrical models.
 Our model is computationally more tractable, allowing reliable numerical
studies. It is easily parallelizable and has good scalability.
 As such, it is more feasible for incorporating
 Realistic electrophysiology
 Biodomain description of tissue
 Electromechanical coupling
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore
Calcium Dynamics: Exploring the
stochastic effect
Xianfeng Song, Department of Physics, Indiana University
Sima Setayeshgar, Department of Physics, Indiana University
Outline
Brief introduction to calcium dynamics in
myocyte
Motivation: why stochastic
Future work
Overview of Calcium Signals
• Calcium sparks and waves
Ca sparks in an isolated mouse
ventricular myocyte.
Spiral Ca2+ wave in the Xenopus
oocyte. The image size is 420x420
um. The spiral has a wavelength of
about 150 um and a period of about
8 seconds. Part B is simulation.
• Calcium serves as an important signaling messenger.
–
–
–
Extracellular sensing
Ca2+ signaling during embryogenesis
The regulation of cardiac contractility by Ca2+
Mechanically stimulated
intercellular wave in
airway epithelial cells
Fundamental elements of Ca2+
signaling machinery
•Calcium Stores: external stores and internal stores: Endoplasmic Reticulum (ER), Sarcoplasmic Reticulum (SR),
Mitochondria
•Calcium pumps: Ca2+ is moved to these stores by a Ca2+/Na+ exchanger, plasma membrane Ca2+ pumps and SERCA
pumps.
•Calcium channels: Ca2+ can enter the cytoplasm via receptor-operated channels (ROC), store-operated channels
(SOC), voltage-operated channels (VOC), ryanodine receptors (RyR) and inositol trisphosphate receptors (IP3R).
Ventricular Myocyte
The typical cardiac myocyte is a cylindrical cell
approximately 100 um in length by 10um in diameter
and is surrounded by a cell membrane known as the
sarcolemma (SL)
Three physical compartments: the cytoplasm, the
sarcoplasmic reticulum (SR) and the mitochondria.
The primary function of SR is to store Ca for release
upon cellular excitation.
The junctional cleft is a very narrow space between
the SL and the SR membrane.
The SR release channel, or ryanodine receptor
(RyR) is found almost entirely within the part of the
SR membrane which communicates with the juntional
cleft.
Ventricular Myocyte and
Excitation-Contraction coupling
•
Ca-Induced Ca Release (CICR)
•
•
•
•
From the resting state (channel closed), Ca may bind rapidly to a relatively low affinity site (1), therby
activating the RyR.
Ca may then bind more slowly to a second higher affinity site (2) moving the release channel to an
inavitive state.
As cytoplasmic [Ca] decreases, Ca would be expected to dissocaiate from the lower affinity activating
site first and then more slowly from the inactivating site to return the channel to the resting state.
Excitation-Contraction Coupling (ECC)
A small amount of Ca is initiated by depolarization of the membrane, thus induce CICR, initiate
contraction.
Motivation: why stochastic
 Binding kinetics is by itself a stochastic process.
 Receptor number is small, i.e., in Xenopus oocytes,
IPR are arranged in clusters with a density of about 1
per 30μm2 with each cluster containing about 25 IPRs.
 Diffusive noise is large. The noise is limited by
Schematic representation of a cluster
ofmreceptors of size b, distributed
uniformly on a ring of size a.
l is the effective size of receptors or receptor
array. (W. Bialek, and S. Setayeshgar, PNAS 102,
10040(2005))
 The global Calcium wave are comprised by local
release events, called puffs.
From single localized Calcium
response to a global calcium wave
Future work