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Bioelectric Forward Problems Bioengineering 6460 Bioelectricity

Bioelectric Forward Problems

Bioengineering 6460 BioelectricityBioelectric Forward Problems

Roadmap• Transition from qualitative to

quantitative source descriptions – Currents, dipoles, surface potentials,

activation times

• Discuss common elements of forward problems

• Map sources to associated forward problem formulations– Discrete source formulations– Surface potentials – Activation time based formulation

0

1

2

100

0

50

cell body

source

sink

inhibitory

excitatory


Forward/Inverse Problems in Electrocardiography

Forward

Inverse

A

B

-

+

C

-+

+

-


Action Currents• Currents that flow from action potentials and propagation• Link the cellular sources with extracellular potentials• Localized to regions of potential difference• Contains a source on the front edge and a sink on the

back edge• Extracellular currents flow through volume conductor

Outside

Inside

Charging Currents

+ + +

-- -

Depolarizing Currents

Cell Membrane

Gap Junctions

+

-

+

-

+

-

+

-

Axial Intracellular

Extracellular

Transmembrane source

Transmembrane sink +-


Dipole Equivalent Sources


Dipole Sources


Dipole Source Description

φm =1

4πσeIor

x

y

z

I0

!1 !2!3

r

dA

Monopolar Source

φd =1

4πσ∇

�1r

�· �p

φd

x

y

z

d

r1r0

I0

-I0

φd =p cos θ4πσr2

Dipolar Source

(for infinite medium)

(*For derivation, see P&B)

*


Equations for Distributed Sources

For dipole source not at the origin

becomes

with

rr’

r - r’ p(x,y,z)source


Volume Dipole Sources

From the previous equation for a single dipole

Or more generally for sources at p not on the origin

Note: the equation becomes

invalid when the point p moves inside the volume.

If we assume there is a volume dipole density, we can write for the extracellular field

Φ(r) =1

4πσ

�

Vpv(�r�)

�r − r�

|�r − �r�|2�ndV �

or


Surface Dipole Distributions

- 1800

-

+

+

--

-

++

2200!V

ActivationFront

Ventricle

TorsoDipoleSurface

Φ(r) =1

4πσ

�

Vpv(�r�)

�r − r�

|�r − �r�|2�ndV �

Φ(r) =1

4πσ

�

Sps(�r�)

�r − r�

|�r − �r�|2· �ndS�

Φ(r) = − 14πσ

�

sps(�r�) dΩrr�

dΩ = −�r − r�

|�r − �r�|2· �ndS� = − (�r −

�r�)|�r − �r�|3

· �ndS�


Surface Dipole Sources

If the dipoles are distributed on a surface, we can write for the potential

which we can rewrite as

By defining the differential solid angle as

From the previous general form of dipole volume

Ω = −�

S

(�r − �r�)|�r − �r�|3

· �ndS�


Solid Anglep(x,y,z)

!

S’

n dS’(x’,y’,z’)

r - r’

r

r’

Area of unit sphere

dΩ = −�r − r�

|�r − �r�|2· �ndS� = − (�r −

�r�)|�r − �r�|3

· �ndS�

The incremental solid angle is defined as

which we can rewrite as

and then for the total solid angle write


Uniform Surface Dipole Sources

rr’

r - r’dS

ps(r’) ndS

p(x,y,z)If the dipole distribution is uniform, we can further simplify to

n

n

n

with the usual definition for solid angle

Φ(r) = − 14πσ

�

sps(�r�) dΩrr�

We wrote for the potential from a surface dipole source

Ω = −�

S

(�r − �r�)|�r − �r�|3

· �ndS�


Myocardial Ischemia

+

+

+

-

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-

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+

+

-


Solid Angle Theory and Ischemia

R.P. Holland and M.F. Arnsdorf, “Solid angle theory and the electrocardiogram: physiologic and quantitative interpretations”, Prog. Cardiovasc. Dis. 1977 19:431-457.

Normal

IschemiaLarger

Ischemic Zone

Larger Ischemic Amplitude

LV wall

MeasurementPoint


Adding Realistic Conditions

?

Φe(�r) =1

4πσo

�

V�pv(�r) ·∇

�1

|�r − �r�|

�dV �

∂Φ∂n

|So = 0


Primary Sources + Boundary Conditions

Now if we seek to determine the potential at a point on a surface to the finite, homogeneous volume conductor S0, we can write the boundary conditions as

So

"o

n

" = 0

pv

Returning to a previous expression for the potential from a dipole volume density in an infinite medium, we have

#e(r)

Φe(�r) =1

4πσo

�

V�pv(�r) ·∇

�1

|�r − �r�|

�dV �

− 14πσo

�

SσoΦ0(r�)n̂ ·∇�

�1

|�r − �r�|

�n̂dS


Secondary Sources

We can then write for the potential at any point p inside the volume conductor

To now incorporate the effect of the realistic boundary, picture the potential jump at the boundary #! as creating an equivalent dipole surface source ps = -"!#!n. This jump is necessary to ensure that potential outside the surface = 0.

"o#0

Primary source

Secondary source

*Primary source assumes infinite homogenous volume conductor

n

So

"o" = 0

pv #e(r)

Φe(�r) =1

2πσ0

�

V�pv(�r) ·∇

�1

|�r − �r�|

�dV �

− 12π

�

S−S�Φ0(r�)n̂ · dΩrr�


Potentials on the SurfaceFor a point on the surface, we have to integrate the expression for the secondary source and avoid the singularity at p.

Then find the contribution of the avoided surface S$ as !=2% and #=#o/2 so we can subtract #o/2 from both sides and write

n

S$

"o#0

n

So

"o" = 0

pv#e(r)


More Realistic Sources

Forward

Inverse

B

-

+

C

-+

+

-

A


Epicardial Potential Source

Forward

InverseB


Green’s Theorem Formulation

For any scalar functions f and g:

If we select f = 1/r and g = & and V is the region between surfaces that contains no sources

0

P

n

n

!B

SH!H

SB


Potential in the Volume

We can now take the 2% case and write the previous eqn as

Where the surface integral avoids the singularity.By splitting the surface into heart andbody surfaces, we can then rewrite this as

P

n

n

!B

SH!H

SB


BEM Formulation

P!B

n

n

SB

!BHSH

P

n

n

!HBSH

!HH

SB

Written once for a point on each of the two surfaces, we get


Numerical Solution

Converting the previous equations to matrix form, we get

Which we can solve to get


Transfer Coefficient Matrix

If we define a matrix of transfer coefficients,

We can rewrite the previous equation as

And then formulate an associated inverse problem


Meaning of Transfer Coefficients

Sensitivity vector

00100 =

Col

umn

3

=NB x NH

NH x 1 NB x 1

B = bodyH = heart


Activation Wavefront Source

-

++

--

-

+ +

- 1800 2200mV

ActivationFront

Ventricle

Torso


Activation Time Forward/Inverse Problems

Forward

Inverse

-

+ -+

+

-

Represent source as activation time driven uniform double layer.


Solid Angles and the Uniform Double Layer

= +


UDL for Different Shapes


UDL Assumptions

Endo

Epi

=+ = 0

=


Forward Formulation

Endo

EpiFor any time t, S(t) of the heart is excited and we can write

or more generally

which provides a way to compute potential as a function of activation time.

Transfer matrix


Summary of Sources

-

+ -+

+-

Dipole Epicardial Potentials

Epi-Endocardial Activation Time

TransmembranePotentials

Simple, few leads, conventional

Measurable, comprehensive, unique

Measurable, clinically directly useful

Measurable in cells or on surfaces (optical)

Unique with substantial constraints, not measurable, requires assumptions, misses details

Interpretation ambiguous, complex, ill-posed

Uniqueness unclear, tenuous assumptions, ill-posed

Not unique but well constrained, less ill posed?

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Role of Anisotropy

Inhomogeneous/Isotropic Anisotropic


Numerical Solutions ofForward Problem Models

• Geometry piecewise homogeneous and isotropic

• Integral form of equations

• Model described in terms of surfaces (numerically as triangles, quads)

• Fewer elements in the model and hence fewer equations

• Solution matrices are smaller but full

• Geometry elementwise homogenous, anisotopic

• Differential form of equations

• Model in terms of volumes ( numerically as hexahedra, tetrahedra)

• More elements and hence more equations, however each equation is simpler

• Solution matrices are larger but sparse

Surface Volume

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