Report

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 ~ 109 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(i0t ) M (t ) H 0 exp(i0t ) 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( 20 t ) 1 sin( 20 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 ) 0H 02 0H 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 ) 4C 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 cosnt 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(int ) the resulted magnetization is M n (t ) n H 0 exp(int ) H 0 n cos(nt ) n sin(nt ) U n 1 n 0 H 02 (1 n ) sin( 2nt ) n cos( 2nt ) n t 2 U n (t ) 1 1 0 H 02 (1 n ) cos( 2nt ) 1 n sin( 2nt ) n 0 H 02 n t 4 2 U n Tn t 1 1 0 H 02 (1 n ) cos( 2nt ) 1 n sin( 2nt ) n 0 H 02 n t 4 2 T0 0 for m=[1, n] n 2 m0 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 ) expi 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. 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