Biomedical Engineering Reference
In-Depth Information
Now consider a square element of the surface of the Teledeltos paper. The net out
fl
ow
of current from the element is
J
x
∂
∂
J
x
J
y
∂
∂
J
y
δ
y
∂
J
x
δ
x
∂
J
y
δ
x
δ
yJ
x
δ
xJ
y
x
δ
y
δ
x
δ
y
δ
δ
y
x
y
x
y
y
2
. Since we are considering a steady state, there must be no
net rate of loss or accumulation of charge in the element. Thus, the expression given above
must be equal to zero. Dividing by the area of the element yields
neglecting terms in
δ
y
2
and
δ
∂
∂
J
x
∂
∂
J
y
y
x
0
The relationship, called the
equation of continuity
, applies to the
fl
flow of any
fl
fluidlike mate-
rial whose volume or quantity does not change when
fl
flowing across a region in steady con-
ditions. It applies to the
fl
flow of heat,
fl
fluids, electricity, and mass particles. In this case it
shows that the current
flows completely through the element without shedding any charge.
In the days before digital computers were able to run
fl
finite-element models, Teledeltos
paper was used to quickly obtain practical solutions to tough electromagnetic, thermody-
namic, and hydrodynamics problems.
Back to our discussion on modeling with Teledeltos, the electric
fi
fi
eld
E
across the sur-
face of the sheet may be expressed in terms of the potential
V
by
∂
V
x
∂
∂
V
y
∇
V
E
x
y
where
x
and
y
are unit vectors along the
x
and
y
axes. Then the continuity equation can be
rewritten as
∂
∂
2
x
V
2
∂
y
2
V
2
∂
∇
2
V
0
This equation, commonly referred to as
Laplace's equation
, at
fi
first glance looks di
cult
to solve. Although there are many known solutions, the problem is to
fi
find a solution that
fi
fits the given boundary conditions. Obviously, the solution given by Teledeltos paper is the
one that completely satis
es the analog boundary conditions modeled using cutouts and/or
painting conductive silver ink. Don't disregard Teledeltos just because it is “old-fashioned.”
By applying just a bit of common sense, you can use this simple, elegant method to solve
very complicated problems.
PASCO Scienti
fi
eld mapping kit.
One hundred sheets of conductive paper marked with a centimeter grid sells for $30 (part
PK-9025). You can make electrodes using the type of conductive silver ink pen that is used
for PCB repairs (which you can also buy from PASCO as part PK-9031B) or by attaching
3M's 1181 electrical tape, which is copper foil with a conductive acrylic adhesive.
Using the simple Teledeltos model, Grayzel and Lizzi [1969] were able to
fi
c sells black conductive paper to go along with its E-
fi
find that the rela-
tionship between source location and body surface, as expressed by the standard lead
fi
field,
was much more variable and complex in the inhomogeneous than in the homogeneous torso.
This led them to the conclusion that several “standard” lead systems of their day showed a
sharp deterioration in performance after adding inhomogeneities to the torso model.
Burger and van Milaan [1948] opted for a more realistic three-dimensional model and
constructed an electrolytic tank out of a michaplast shell molded on the statue of a supine
human to create a torso model. The tank split horizontally to provide access to the interior,
which was
fi
fi
filled with copper sulfate and out
fi
tted with copper foil electrodes
fi
xed to the
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