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 Engineering
Oklahoma 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 thermoacoustic imaging in ways similar to the laser-mediated heating for photoacoustic or opto-acoustic imaging and the microwave-mediated heating
for microwave-induced thermo-acoustic imaging.
• We propose two possible ways of achieving such magneto-thermoacoustic 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
ca
CP
CV
f
H
kB
MS
p
q
SLP
Temp
T
t
U
VH
VM
Identification
Speed of sound in tissue
Specific heat at cons. press
Specific heat at cons. volum.
Frequency
Magnetic field strength
Boltzmann constant
Saturation magnetization
Acoustic pressure
Volumetric power dissipation
Specific power loss
Thermodynamic temperature
Time---duration
Time---instant
Internal energy
Hydrodynamic volume
Magnetic volume
Unit
[m s-1]
[J kg-1 K-1]
[Hz]
[A m-1]
[J K-1]
[A m-1]
[A m-1]
[Pa]
[W m-3]
[W kg-1]
[K]
[s]
[s]
[J]
[m3]
[m3]
Symbol





0
a
 s



R
N
B

Identification
Isobaric vol. ther. exp. coeff.
Grueneisen parameter
A change in a variable
Viscosity coefficient
Anisotropy energy density
Permeability
Absorption coefficient
Reduced scattering coefficient
Envelope function
Angular frequency
Mass density
Relaxation time
Néel relaxation time
Brownian relaxation time
Magnetic susceptibility
Unit
[K-1]
[dimensionless]
[dimensionless]
[N s m-2]
[J m-3]
[V s A-1m-1]
[m-1]
[m-1]
[dimensionless]
[rad s-1]
[kg m-3]
[s]
[s]
[s]
[dimensionless]
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
 ( )   0

and the imaginry part of the susceptibility
1
1  [ R ]2
 R
 "( )   0
1  [ R ]2
"
become
 0  0
M S 2 VM
k BTemp
If the MNPs are in the single-size domain of super-paramagnetism and dispersed in a liquid matrix, the relaxation time
is to be dominated by Néel and Brownian relaxations as
1
R
N 

2
0
exp  VM k BTemp 
  VM k BTemp

1
N

1
B
 0 ~ 109 s
B 
3 VH
k BTemp
R
Heating effect of MNP under AMF
Under an AMF of constant frequency, i.e.
H (t )  H 0 cos(0 t )  H 0 exp(i0t )
M (t )    H 0 exp(i0t )  H 0    cos(0t )     sin(0t )
the resulted magnetization is
Change of the internal energy is
U 1
  0  0 H 02  (1   )  sin( 2 0 t )     cos( 2 0 t )   
t 2
U (t ) 
1
1
 0 H 02 (1   )  cos( 20 t )  1     sin( 20 t )  0  0 H 02    t
4
2
At a phase change of
0t  2
t 2  t cycle 
2
0
the heat dissipation per unit volume
q2  U (t2 )  0H 02    0H 02 0
0 R
1  [0  R ]2
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
qCW  q2 
1
t 2

0  0 [0   R ]2
2
H0
2
2 R 1  [0   R ]
where the subscript “CW” denotes “continuous-wave”, and
accordingly the specific-loss-power (SLP) (unit: W kg-1) is



qCW (r )
SLPCW (r , t ) 
 SLPCW (r )

The initial slope of temperature-rise of the sample
containing the MNPs is

Temp (r )
t

SLPCW (r )

CV
which is used by many studies to predict and experimentally
deduce the heating power of MNPs to evaluate the modeldata 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”,
which is equivalent to a “carrier” AMF
H 0 cos(0t )
modulated by an envelope function (t ) as

H (r , t )  H 0 (t ) cos(0t )  [(t )  H 0 ]cos(0t )
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



(t )   u(t  n  Tenvelope )  u(t  tON  n  Tenvelope )   2 (t );
n  1, 2, 3,......
n 0
When MNPs are exposed to a pulse-enveloped AMF, the time-variant heat dissipation will result in volumetric power dissipation as



0  0 (r ) [0 R (r )]2
2
2
qTD (r , t ) 
 
 2  H 0   (t )
2 R (r ) 1  [0 R (r )]
Short-burst of AMF on MNP

The acoustic pressure wave pTD (r , t ) excited by

qTD (r , t )
satisfies the following wave equation



1 2
 
 pTD (r , t ) 
pTD (r , t )  
qTD (r , t )
2
ca t
C p t
2

The general solution of the acoustic pressure reaching a transducer at rD

and originating fro the source of thermo-acoustic signal generation at rS
in an unbounded medium is known to be

pTD (rD , t ) 

4C p

V
 
rD  rS


1

qTD (rD , t 
)  d 3 rS
 
rD  rS t
ca
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 thermoacoustic 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 MNP
We now consider the heating of MNPs exposed to a homogenous AMF whose amplitude is fixed at
H0
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
H (t )  H0 cos[(t )  t ]  H0 cos(0  bt)t 
Hn (t )  H0 cosnt 
We approximate the signal using
n  0  nb
n  [0, N ]
Short-time heat dissipation
H n (t )  H 0 cos(n t )  H 0 exp(int )
the resulted magnetization is
M n (t )  n  H 0 exp(int )  H 0 n  cos(nt )  n  sin(nt )
U n 1
 n  0 H 02  (1   n )  sin( 2nt )   n  cos( 2nt )   n
t
2
U n (t ) 
1
1
 0 H 02 (1   n )  cos( 2nt )  1   n  sin( 2nt )  n  0 H 02  n  t
4
2
U n Tn  t  
1
1
0 H 02 (1   n )  cos( 2nt )  1   n  sin( 2nt )  n 0 H 02  n  t
4
2
T0  0
for m=[1, n]
n
2
m0
m
Tm  
Frequency-chirped AMF on MNP

the volumetric power dissipation at a position r
can be approximated by
 2


0  0 (r ) [(0  sweept ) R (r )]
2
qFD (r , t ) 
 
 2 H0
2 R (r ) 1  [(0  sweep t ) R (r )]
where  sweep  bt
as

qFD (r , t )

q~FD (r , ), and the Fourier transform of the excited acoustic pressure wave
is denoted as

~
pFD (r , ) , we have the following Fourier-domain wave equation
If the Fourier transform of

pFD (r , t )
is the frequency sweep rate.


 ~
p FD (r , )  
 ca
2
2
 ~ 
i  ~ 
 p FD (r , )  
q FD (r , )
Cp

the acoustic wave intercepted by an idealized point ultrasound transducer at

 
~

pFD (rd , )
rd  rS

  
pFD (rd , t ) 
expi   t 
 
4 rd  rs C p
ca

  


   a 


 



rd
can be written as
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.
References
1. Gilchrist RK, Medal R, Shorey WD, Hanselman RC, Parrott JC, Taylor CB, “Selective inductive heating of lymph nodes,” Ann Surg.; 146:596–
606 (1957).
2. Hergt R, Dutz S, Röder M, “Effects of size distribution on hysteresis losses of magnetic nanoparticles for hyperthermia,” J Phys Condens
Matter.; 20(38):385214 (2008).
3. Frenkel J, The kinetic theory of liquids, Dover Publications, New York, 1955.
4. Néel L, “Influence of thermal fluctuations on the magnetization of ferromagnetic small particles,” C. R. Acad. Sci., 228:664-668 (1949).
5. Hergt R, Dutz S, Zeisberger M, “Validity limits of the Néel relaxation model of magnetic nanoparticles for hyperthermia,” Nanotechnology,
21(1):015706 (2010).
6. Johannsen M, Gneveckow U, Eckelt L, Feussner A, Waldöfner N, Scholz R, Deger S, Wust P, Loening SA, Jordan A, “Clinical hyperthermia of
prostate cancer using magnetic nanoparticles: presentation of a new interstitial technique,” Int J Hyperthermia., 21(7):637-647 (2005).
7. Namdeo M, Saxena S, Tankhiwale R, Bajpai M, Mohan YM, Bajpai SK, “Magnetic nanoparticles for drug delivery applications,” J Nanosci
Nanotechnol., 8(7):3247-3271 (2008).
8. Amstad E, Kohlbrecher J, Müller E, Schweizer T, Textor M, Reimhult E, “Triggered release from liposomes through magnetic actuation of
iron oxide nanoparticle containing membranes,” Nano Lett. ; 11(4):1664-70 (2011).
9. Mehrmohammadi M, Oh J, Aglyamov SR, Karpiouk AB, Emelianov SY, “Pulsed magneto-acoustic imaging,” Conf Proc IEEE Eng Med Biol
Soc., 2009:4771-4774 (2009).
10. Jin Y, Jia C, Huang SW, O'Donnell M, Gao X, “Multifunctional nanoparticles as coupled contrast agents,” Nat Commun.; 1:41 (2010). .
11. Oldenburg AL, Gunther JR, Boppart SA, “Imaging magnetically labeled cells with magnetomotive optical coherence tomography,” Opt
Lett., 30(7):747-749 (2005).
12. Hu G, He B, “Magnetoacoustic imaging of magnetic iron oxide nanoparticles embedded in biological tissues with microsecond magnetic
stimulation,” Appl Phys Lett., 100(1):13704-137043 (2012).
13. Steinberg I, Ben-David M, Gannot I, “A new method for tumor detection using induced acoustic waves from tagged magnetic
nanoparticles,” Nanomedicine. 2011 Oct 22.
14. Nie L, Ou Z, Yang S, Xing D, “Thermoacoustic molecular tomography with magnetic nanoparticle contrast agents for targeted tumor
detection,” Med Phys., 37(8):4193-4200 (2010).
15. Rosensweig RE, “Heating magnetic fluid with alternating magnetic fields,” J Magn Magn Mater., 252:370-374 (2002).
16. Xu Y, Feng D, Wang LV, “Exact frequency-domain reconstruction for thermoacoustic tomography--I: Planar geometry,” IEEE Trans Med
Imaging.; 21(7):823-8 (2002).
17. Xu M, Xu Y, Wang LV, “Time-domain reconstruction algorithms and numerical simulations for thermoacoustic tomography in various
geometries,” IEEE Trans Biomed Eng.; 50(9):1086-99 (2003).
18. Telenkov S, Mandelis A, Lashkari B, Forcht M, “Frequency-domain photothermoacoustics: Alternative imaging modality of biological
tissues,” J. Appl. Phys. 105, 102029 (2009).