Daqing ( Daching ) Piao , PhD Associate Professor School of Electrical and Computer Engineering

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.

References1. 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).