Biomedical Engineering Reference
In-Depth Information
constant is, by definition, the fixed ratio of reactant to
product concentrations, these allow us to introduce four
more equations into the model:
magnetic field. His results (
Norton 2003
) show that the
amplitude of the magnetic field generated in this fashion
is less than that of a traditional TMS approach, but there
are some distinct advantages in both spatial and temporal
characteristics of the induced field.
As one can imagine, numerical methods play a large
role in calculating the magnitude of an electric field
throughout the neural tissue in the brain. Briefly, let the
cortical tissue be represented in a cylindrical 3D co-
ordinate system (
r
, f,
z
) and assume that the ultrasonic
beam is collimated and propagating in the
z
direction.
Furthermore, assume that the profile is axially symmetric
and will be represented by
p
(
r
).
Norton (2003)
modeled
the particle velocity by:
K
H1
¼
½
P1
$
½
T1
½
H1
K
H2
¼
½
P2
½
T2
½
H2
$
(2.1a.6)
K
HT
¼
½
T1
½
T2
½
HT
$
K
D
¼
½
P1
$
½
P2
½
D
vð
r
Þ¼v
0
pðrÞe
ik
0
z
z
The efficiency of the annealing stage, 3
ann
(
n
), can be
calculated by comparing the amount of hybrids after
the
n
th
annealing stage to the total amount of template
present throughout the
n
th
annealing stage:
(2.1a.8)
where
v
0
is the peak velocity of the particles,
ˆ
is the unit
vector in the
z
direction and the wave number
k
0
¼ u
/c
0
,
i.e., the frequency of the ultrasonic wave divided by the
speed of the ultrasonic wave.
The components of the magnetic field E
s
induced in
the brain are given by:
3
ann
ðnÞ¼
0
:
5
½
H1
½
T1
T
þ
½
H2
½
T2
T
(2.1a.7)
in which [H1] and [H2] can be found by solving the
nonlinear system of eight equations with eight unknowns
in terms of the eight free parameters.
This deceivingly simple system of eight equations can
be solved analytically either by hand or using a computer
program for solving the system of equations symbolically.
E
r
ðr;
f
; zÞ¼
B
0
v
0
d
2
AðrÞ
e
ik
0
z
sinf
dr
2
E
f
ðr;
f
; zÞ¼
B
0
v
0
1
e
ik
0
z
cos
f
dAðrÞ
dr
(2.1a.9)
r
E
z
ðr;
f
; zÞ¼
B
0
v
0
ik
0
e
ik
0
z
sinf
dAðrÞ
dr
2.1a.4.2 Modeling transcranial
magnetic stimulation
where
Transcranial magnetic stimulation (TMS), the stimula-
tion of cortical tissue by magnetic induction, is poten-
tially a new diagnostic and therapeutic tool in clinical
neurophysiology. The magnetic fields are delivered to the
tissue by placing coils on the surface of the skull; TMS
shows the promise of being useful for brain mapping and
appears to show potential for treating brain disorders
(
Hallett 2000
). The advantage of techniques like TMS
and EEG (electroencephalogram) (see Chapter 2.2) is
high temporal resolution; a disadvantage of TMS is that
the 3D spatial resolution and depth penetration is not
very good. Poor spatial resolution is due to the fact that
the magnetic fields cannot easily be focused at a particu-
lar point in the brain.
Norton (2003)
proposed a different method of stim-
ulating cortical tissue, which potentially may allow
deeper penetration and better focusing in cortical tissue.
His idea is to create an electrical current by propagating
an ultrasonic wave in the presence of a strong DC
AðrÞ¼K
0
ðk
0
rÞ
ð
0
r
I
0
ðk
0
r
0
Þpðr
0
Þr
0
dr
0
þ I
0
ðk
0
rÞ
ð
N
r
K
0
ðk
0
r
0
Þpðr
0
Þr
0
dr
0
(2.1a.10)
and both
I
0
(
) are functions that model the
ultrasonic wave propagation.
This mathematical model is used to predict the dis-
tribution of the induced electric field in the brain when
generated by an ultrasonic wave. In order for this set of
equations to be used, they have to be solved numerically.
But, in order to solve the three equations, one has to solve
Eq. (2.1a.10) for
A
(
r
). Except for a very ideal case the
integrals in Eq. (2.1a.10) and the differential equations in
Eq. (2.1a.9) have to be evaluated using the numerical
integration techniques and the differential equation
solvers.
$
) and
K
0
(
$
