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The purpose of this study was to present a lock-in-amplifier model for analyzing the behavior of signal harmonics in magnetic particle imaging (MPI) and some simulation results based on this model. In the lock-in-amplifier model, the signal induced by magnetic nanoparticles (MNPs) in a receiving coil was multiplied with a reference signal, and was then fed through a low-pass filter to extract the DC component of the signal (output signal). The MPI signal was defined as the mean of the absolute value of the output signal. The magnetization and particle size distribution of MNPs were assumed to obey the Langevin theory of paramagnetism and a log-normal distribution, respectively, and the strength of the selection magnetic field (SMF) in MPI was assumed to be given by the product of the gradient strength of the SMF and the distance from the field-free region (x). In addition, Gaussian noise was added to the signal induced by MNPs using normally-distributed random numbers. The relationships between the MPI signal and x were calculated for the odd- and even-numbered harmonics and were investigated for various time constants of the low-pass filter used in the lock-in amplifier and particle sizes and their distributions of MNPs. We found that the behavior of the MPI signal largely depended on the time constant of the low-pass filter and the particle size of MNPs. This lock-in-amplifier model will be useful for better understanding, optimizing, and developing MPI, and for designing MNPs appropriate for MPI.

In 2005, a new imaging method called magnetic particle imaging (MPI) was introduced [

MPI utilizes the nonlinear response of MNPs to detect their presence in an alternating magnetic field called the drive magnetic field. Spatial encoding is accomplished by saturating the magnetization of the MNPs almost everywhere except in the vicinity of a special region called the field-free point (FFP) or field-free line (FFL) using a static magnetic field called the selection magnetic field [

Due to the nonlinear response of the MNPs to an applied drive magnetic field, the signals generated by the MNPs in a receiving coil contain not only the excitation frequency but also the harmonics of this frequency. These harmonics are used for image reconstruction in MPI [

For a better understanding and optimization of MPI, it is important to investigate the behavior of signal harmonics generated by MNPs under various conditions of the drive and selection magnetic fields and their dependence on the particle size and distribution of MNPs. We previously investigated the behavior of signal harmonics in MPI by experimental and simulation studies, and reported that it largely depended on the strength of the drive and selection magnetic fields and the particle size distribution of MNPs [

Lock-in amplifiers were invented in the early 1940s to extract electrical signals in extremely noisy environments [

The purpose of this study was to present a lock-in-amplifier model for analyzing the behavior of signal harmonics in MPI and some simulation results based on this model.

by ⨂ in

v m i x ( t ) = v i n ( t ) ⋅ v r e f ( t ) (1)

where v m i x ( t ) denotes the signal after mixing and v r e f ( t ) is given by

v r e f ( t ) = 2 e − j 2 π f r e f t (2)

In Equation (2), j = − 1 and f r e f denotes the frequency of the reference signal. The mixed signal is then fed through a low-pass filter to extract the DC component of the signal. Mathematically, this procedure is given by

V o u t ( f ) = V m i x ( f ) ⋅ H L P F ( f ) (3)

where V o u t ( f ) and V m i x ( f ) denote the Fourier transforms ( F ) of the output signal [ v o u t ( t ) ] and v m i x ( t ) , respectively, i.e., V o u t ( f ) = F [ v o u t ( t ) ] and V m i x ( f ) = F [ v m i x ( t ) ] . H L P F ( f ) denotes the transfer function of the first-order RC low-pass filter and is well approximated by

H L P F ( f ) = 1 1 + j 2 π f τ (4)

where τ = R C is the filter time constant with resistance R and capacitance C (

low-pass filter in the time domain [ R S T E P ( t ) ] is given by

R S T E P ( t ) = 1 − e − t τ (5)

v o u t ( t ) = | v o u t ( t ) | e j ϕ ( t ) (6)

where | v o u t ( t ) | and ϕ ( t ) denote the absolute value of v o u t ( t ) and phase at time t, respectively. In this study, the MPI signal ( S M P I ) is defined as the mean of | v o u t ( t ) | , i.e.,

S M P I = | v o u t ( t ) | ¯ (7)

Assuming a single receiving coil with sensitivity [ σ r x ( r ) ] at spatial position r, the changing magnetization induces a voltage according to Faraday’s law [ v r x ( t ) ] ,

which is given by [

v r x ( t ) = − μ 0 d d t ∫ Ω σ r x ( r ) C ( r ) M ( r , t ) d r (8)

where Ω denotes the volume containing MNPs, C ( r ) is the concentration of MNPs at position r, M ( r , t ) is the magnetization at position r and time t, and μ 0 is the magnetic permeability of a vacuum. σ r x ( r ) is the receiving coil sensitivity derived from the magnetic field that the coil would produce if driven with a unit current [

In the following, the receiving coil sensitivity is assumed to be constant and uniform over the volume of interest and is denoted by σ 0 . When we consider the signal generated by a point-like distribution of MNPs, that is, the MNP distribution is approximated by Dirac’s δ function such that C ( r ) = C 0 δ ( r ) with C 0 being constant, the volume integral in Equation (8) vanishes and v r x ( t ) given by Equation (8) is reduced to

v r x ( t ) = − μ 0 σ 0 C 0 d M ( t ) d t (9)

Note that M ( 0 , t ) is denoted by M ( t ) in Equation (9) for simplicity. We can neglect constant factors in Equation (9).

In addition, we assume that the signal obtained by the receiving coil includes Gaussian white noise [

v i n ( t ) = v r x ( t ) + v r x ( t ) 2 ¯ S N R ⋅ r a n d n (10)

where v r x ( t ) 2 ¯ , randn, and SNR denote the mean of v r x ( t ) 2 , a normally-distributed random number with zero mean and unit variance, and signal-to-noise ratio, respectively.

Assuming that MNPs are in equilibrium, the magnetization of MNPs in response to an applied magnetic field can be described by the Langevin function [

M ( ξ ) = M 0 ( coth ξ − 1 ξ ) (11)

where M_{0} is the saturation magnetization and ξ is the ratio of the magnetic energy of a particle with magnetic moment m in an external magnetic field H to the thermal energy given by the Boltzmann constant k_{B} and the absolute temperature T:

ξ = μ 0 m H k B T = μ 0 M d V M H k B T (12)

In Equation (12), M d is the domain magnetization of a suspended particle, and V M is the magnetic volume given by V M = π D 3 / 6 for a particle of diameter D.

In this study, we assume that the external magnetic field at position x and

time t [ H ( x , t ) ] is given by

H ( x , t ) = H s ( x ) + H D ( t ) (13)

where H s ( x ) is the strength of the selection magnetic field at position x and H D ( t ) is the strength of the drive magnetic field at time t. We also assume that H D ( t ) is given by

H D ( t ) = H D M F cos ( 2 π f D M F t ) (14)

where H D M F and f D M F denote the amplitude and frequency of the drive magnetic field, respectively. Furthermore, we assume that H s ( x ) is given by

H s ( x ) = G x ⋅ x (15)

where G_{x} and x denote the gradient strength of the selection magnetic field and the distance from the field-free region, respectively.

When the particle size distribution obeys a log-normal distribution [

〈 M 〉 = 1 2π ∫ 0 ∞ M ( D ) σ D exp [ − 1 2 ( ln ( D ) − μ σ ) 2 ] d D (16)

where M ( D ) denotes the magnetization of MNPs with diameter D. μ and σ denote the mean and standard deviation (SD) of the log-normal distribution, respectively [

μ = ln [ E ( D ) ] − 1 2 ln [ Var ( D ) E 2 ( D ) + 1 ] (17)

and

σ = ln [ Var ( D ) E 2 ( D ) + 1 ] (18)

respectively, where E ( D ) and Var ( D ) denote the expectation and variance of D, respectively.

In this study, we considered magnetite (Fe_{3}O_{4}) as MNPs, and M_{d} in Equation (11) was taken as 446 kA/m [

Unless specifically stated, E ( D ) and σ in Equation (17) and Equation (18) were assumed to be 20 nm and 0.2, respectively, and G_{x} in Equation (15) was assumed to be 2 T/m. When investigating the dependence of the odd- and even-numbered harmonics on the selection magnetic field, G_{x} in Equation (15) was varied from 1 to 5 T/m. When investigating the dependence of the third-harmonic signal on the particle size of MNPs, E ( D ) and σ in Equation (17) and Equation (18) were varied from 10 to 50 nm and from 0.05 to 0.4, respectively.

_{x} in Equation (15)] was taken as 2 T/m. As in

_{x} was varied from 1 to 5 T/m. The other parameters were the same as in _{x} in the x axis, because H s ( x ) was assumed to be proportional to G_{x} as given by Equation (15).

We previously investigated the behavior of signal harmonics in MPI and reported that the behavior of the odd- and even-numbered harmonics of MPI signals largely depends not only on the strength of the drive and selection magnetic fields but also on the particle size distribution of MNPs [

Lock-in amplifiers are often used to extract signals in MPI [

We simulated the magnetization of MNPs in response to the drive magnetic field by using the Langevin function given by Equation (11). This is one of the most extensively studied models in MPI and is based on the assumption that MNPs are in equilibrium [

Because not all particles in a certain volume have the same diameter D, the magnetization of MNPs should be averaged based on the particle size distribution. The result of a natural growth process during particle synthesis does not yield particles with a single diameter D, but particles with a polydispersed particle size distribution [

Theoretically, the odd-numbered harmonics should not be zero, whereas the even-numbered harmonics should be zero when the selection magnetic field is not applied, i.e., at the center of the field-free region such as FFP or FFL [

The relationship between the MPI signal [

As shown in

As shown in

We used the first-order RC low-pass filter for extracting the DC component of the signal after mixing (

As previously described, we defined the MPI signal as the mean of

where

We presented a lock-in-amplifier model for analyzing the behavior of signal harmonics in MPI and some simulation results based on this model. This model will be useful for better understanding, optimizing, and developing MPI and for designing MNPs appropriate for MPI.

This work was supported by Grants-in-Aid for Scientific Research (Grant Nos.: 25282131 and 15K12508) from the Japan Society for the Promotion of Science (JSPS) and Japan Agency of Science and Technology (JST).

Murase, K. and Shimada, K. (2018) Lock-in-Amplifier Model for Analyzing the Behavior of Signal Harmonics in Magnetic Particle Imaging. Open Journal of Applied Sciences, 8, 170-183. https://doi.org/10.4236/ojapps.2018.85014