On The Feasibility Of Magneto-Thermo-Acoustic Imaging Using Magnetic Nanoparticles
And Alternating Magnetic Field
Daqing (Daching) Piao, PhD Associate Professor
School of Electrical and Computer EngineeringOklahoma State University, Stillwater, OK 74078-5032
Abstract• We propose a method of magnetically-induced thermo-acoustic imaging by
using magnetic nanoparticle (MNP) and alternating magnetic field (AMF).
• The heating effect of MNP when exposed to AMF by way of Neel and Brownian relaxations is well-known in the applications including hyperthermia.
• The AMF-mediated heating of MNP may be implemented for thermo-acoustic imaging in ways similar to the laser-mediated heating for photo-acoustic or opto-acoustic imaging and the microwave-mediated heating for microwave-induced thermo-acoustic imaging.
• We propose two possible ways of achieving such magneto-thermo-acoustic imaging;
– one is a time-domain method that applies a burst of alternating magnetic field to MNP,
– the other is a frequency-domain method that applies a frequency-chirped alternating magnetic field to MNP.
Outline
• Heating effect of magnetic nanoparticle (MNP) under alternating magnetic field (AMF)
• Rationale of applying short burst of AMF to MNP to induce thermo-acoustic signal generation
• Rationale of applying frequency-chirped AMF to MNP to induce thermo-acoustic signal generation
Symbol Identification UnitSpeed of sound in tissue [m s-1]Specific heat at cons. press [J kg-1 K-1]Specific heat at cons. volum. [Hz]Frequency [A m-1]Magnetic field strength [J K-1]Boltzmann constant [A m-1]Saturation magnetization [A m-1]Acoustic pressure [Pa]Volumetric power dissipation [W m-3]Specific power loss [W kg-1]Thermodynamic temperature [K]Time---duration [s]Time---instant [s]Internal energy [J]Hydrodynamic volume [m3]Magnetic volume [m3]
ac
PC
VC
f
H
Bk
SM
p
q
SLP
empT
T
t
HV
MV
U
Symbol Identification UnitIsobaric vol. ther. exp. coeff. [K-1]Grueneisen parameter [dimensionless]A change in a variable [dimensionless]Viscosity coefficient [N s m-2]Anisotropy energy density [J m-3]Permeability [V s A-1m-1]Absorption coefficient [m-1]Reduced scattering coefficient [m-1]Envelope function [dimensionless]Angular frequency [rad s-1]Mass density [kg m-3]Relaxation time [s]Néel relaxation time [s]Brownian relaxation time [s]Magnetic susceptibility [dimensionless]
0
a
s
R
N
B
Outline
• Heating effect of magnetic nanoparticle (MNP) under alternating magnetic field (AMF)
• Rationale of applying short burst of AMF to MNP to induce thermo-acoustic signal generation
• Rationale of applying frequency-chirped AMF to MNP to induce thermo-acoustic signal generation
MNP under AMF
The magnetic susceptibility of MNPs is dented as "' i
Under a time-varying magnetic fieldof an instant angular frequency
"the real part of the susceptibility and the imaginry part of the susceptibility become
20 ][1
1)(
R
20 ][1)("
R
R
empB
MS
Tk
VM 2
00
If the MNPs are in the single-size domain of super-paramagnetism and dispersed in a liquid matrix, the relaxation time R
is to be dominated by Néel and Brownian relaxations as
BNR 111
empBM
empBMN
TkV
TkV
exp
2 0 s90 10~
empB
HB Tk
V
3
Heating effect of MNP under AMF
Under an AMF of constant frequency, i.e. )exp()cos()( 0000 tiHtHtH
the resulted magnetization is )sin()cos()exp()( 00000 ttHtiHtM
)2cos()2sin()1(2
100
2000 ttH
t
U
tHttHtU 200000
200 2
1)2sin(1)2cos()1(
4
1)(
Change of the internal energy is
20 t0
2
2
cyclettAt a phase change of
20
00
200
20022 ][1
)(R
RHHtUq
the heat dissipation per unit volume
Heating effect of MNP under AMF
For MNPs exposed to a continuous-wave AMF, the instantaneous thermal energy deposited per unit volume per unit time, i.e. the volumetric power dissipation (unit: W m-3), is
22
1
tqqCW
202
0
2000
][1
][
2H
R
R
R
where the subscript “CW” denotes “continuous-wave”, and accordingly the specific-loss-power (SLP) (unit: W kg-1) is
)()(
),( rSLPrq
trSLP CWCW
CW
The initial slope of temperature-rise of the sample containing the MNPs is
V
CWemp
C
rSLP
t
rT )()(
which is used by many studies to predict and experimentally deduce the heating power of MNPs to evaluate the model-data agreement
Outline
• Heating effect of magnetic nanoparticle (MNP) under alternating magnetic field (AMF)
• Rationale of applying short burst of AMF to MNP to induce thermo-acoustic signal generation
• Rationale of applying frequency-chirped AMF to MNP to induce thermo-acoustic signal generation
Short-burst of AMF on MNP
We now consider the heating characteristics of MNPs exposed to a homogenous AMF of fixed frequency 0
and time-varying amplitude. We call this AMF a “tim-domain AMF”,
)cos( 00 tH )(twhich is equivalent to a “carrier” AMF modulated by an envelope function as
)cos(])([)cos()(),( 0000 tHtttHtrH
The simplest form of time-domain AMF may be the one obtained by turning on a “carrier” AMF repetitively at a short duration (s time-scale) over a period of µs-scale or longer, as
......,3,2,1);()()()( 2
0
ntTnttuTntut envelopeONn
envelope
When MNPs are exposed to a pulse-enveloped AMF, the time-variant heat dissipation will result in volumetric power dissipation as
)()]([1
)]([
)(2
)(),( 22
020
2000 tH
r
r
r
rtrq
R
R
RTD
Short-burst of AMF on MNP),( trpTD
),( trqTD
The acoustic pressure wave excited by
satisfies the following wave equation
),(),(1
),(2
22 trq
tCtrp
tctrp TD
pTD
aTD
Dr
Sr
The general solution of the acoustic pressure reaching a transducer at
and originating fro the source of thermo-acoustic signal generation at
in an unbounded medium is known to be
Sa
SDDTDV
SDpDTD rd
c
rrtrq
trrCtrp
3),(1
4),(
Since the local temperature rises rapidly when AMF pulse is on then falls rapidly when AMF pulse is off, this time-variant heating could produce abrupt expansion and transient contraction of the local tissue, which is the condition for thermo-acoustic signal generation
Outline
• Heating effect of magnetic nanoparticle (MNP) under alternating magnetic field (AMF)
• Rationale of applying short burst of AMF to MNP to induce thermo-acoustic signal generation
• Rationale of applying frequency-chirped AMF to MNP to induce thermo-acoustic signal generation
Frequency-chirped AMF on MNPWe now consider the heating of MNPs exposed to a homogenous AMF whose amplitude is fixed at 0H
but angular frequency is time-varying. We call this AMF a “frequency-domain AMF”. The simplest form of frequency-domain AMF may be obtained by linearly sweeping the frequency of AMF . The instantaneous field strength of this linearly frequency-modulated, or chirped, AMF is represented by
tbtHttHtH )(cos])(cos[)( 000
tHtH nn cos)( 0 nbn 0 ],0[ Nn We approximate the signal using
)exp()cos()( 00 tiHtHtH nnn
)sin()cos()exp()( 00 ttHtiHtM nnnnnnn
Short-time heat dissipation
the resulted magnetization is
nnnnnnn ttHt
U
)2cos()2sin()1(
2
1 200
tHttHtU nnnnnnn 200
200 2
1)2sin(1)2cos()1(
4
1)(
tHttHtTU nnnnnnnn 200
200 2
1)2sin(1)2cos()1(
4
1
00 T
n
m mmT
0
2
for m=[1, n]
Frequency-chirped AMF on MNP
r
202
0
2000
)]()[(1
)]()[(
)(2
)(),( H
rt
rt
r
rtrq
Rsweep
Rsweep
RFD
the volumetric power dissipation at a position can be approximated by
where btsweep is the frequency sweep rate.
),( trqFD
),(~ rqFD
),( trpFD
),(~ rpFD
),(~),(~),(~2
2 rqC
irp
crp FD
pFD
aFD
If the Fourier transform of is denoted as , and the Fourier transform of the excited acoustic pressure wave
as , we have the following Fourier-domain wave equation
the acoustic wave intercepted by an idealized point ultrasound transducer at dr can be written as
a
a
Sd
psd
dFDdFD c
rrti
Crr
rptrp
exp4
),(~),(
Summary
We predict that thermo-acoustic signal generation from MNPs is possible,
by rapid time-varying heat dissipation and cooling of the local tissue volume,
using time-domain or frequency-domain AMF on MNPs.
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