⇦Chinese Physics Letters, 2018, Vol. 35, No. 1, Article code 014301 Lorentz Force Electrical Impedance Detection Using Step Frequency Technique * Zhi-Shen Sun(孙直申)1,2,3, Guo-Qiang Liu(刘国强)1,2**, Hui Xia(夏慧)1 Affiliations 1Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190 2University of Chinese Academy of Sciences, Beijing 100049 3Univ Lyon, Université Claude Bernard Lyon 1, Centre Léon Bérard, INSERM, LabTAU UMR1032, LYON F-69003, France Received 29 August 2017 *Supported by the National Natural Science Foundation of China under Grant Nos 51137004 and 61427806, the Scientific Instrument and Equipment Development Project of Chinese Academy of Sciences under Grant No YZ201507, and the China Scholarship Council Program under Grant No 201604910849.
**Corresponding author. Email: gqliu@mail.iee.ac.cn Citation Text: Sun Z S, Liu G Q and Xia H 2018 Chin. Phys. Lett. 35 014301 Abstract Lorentz force electrical impedance tomography (LFEIT) inherits the merit of high resolution by ultrasound stimulation and the merit of high contrast through electromagnetic field detection. To reduce the instantaneous peak power of the stimulating signal to the transducer, the sinusoidal pulse and step-frequency technique is investigated in LFEIT. The theory of application of step-frequency technique in LFEIT is formulated with the direct demodulation method and the in-phase quadrature demodulation method. Compared with the in-phase quadrature demodulation method, the direct demodulation method has simple experimental setup but could only detect half of the range. Experiments carried out with copper foils confirmed that LFEIT using the step-frequency technique could detect the electrical conductivity variations precisely, which suggests an alternative method of realization of LFEIT. DOI:10.1088/0256-307X/35/1/014301 PACS:43.80.Cs, 72.55.+s, 73.50.Rb © 2018 Chinese Physics Society Article Text Lorentz force electrical impedance tomography (LFEIT)[1-3] transforms the stimulating acoustic field into an induced electromagnetic field through the static quasi-homogeneous magnetic field. Therefore, LFEIT inherits the advantage of acoustic stimulation with high spatial resolution by restricting the measurement area to the focal zone of the ultrasound beam and the advantage of electrical measurement with high contrast because among different biological tissues of the human body their electrical conductivity varies much[4] and for the same biological tissue, its electrical conductivity also changes much between different pathological stages.[5] Traditional LFEIT uses high-voltage, narrow-width electrical signal as the stimulating source to the transducer to generate the acoustic stimulation. One problem of this type of stimulating signal is that its instantaneous peak power can reach as high as 3.2 kW, which can shorten the normal usage lifetime of the transducer. Other types of stimulating signal to the transducer and associated electronic signal processing techniques have been explored to overcome this problem, such as the sinusoidal bursts.[6,7] One kind of sinusoidal burst is the monotonic frequency signal and the technique associating with it is the step-frequency technique. In this study, we explore the use of the step-frequency technique in LFEIT. This technique originated from ground penetrating radar (GPR),[8-10] where it was used to overcome mutually conflicting demands of lower operating frequency and higher resolution which were both necessary for the GPR. The step-frequency technique utilizes information from both the phase and the amplitude of a target-reflected signal with the emphasis on the phase information. With accurate phase measurement, a resolution of much less than one wavelength was proven to be achievable. In recent years, in addition to its continuous usage in GPR,[11] it has been applied in microwave-induced thermo-acoustic imaging,[12] magneto-acoustic tomography[13] and accurate ultrasound ranging.[14,15] In this work, firstly, the theory of application of step-frequency technique in LFEIT was presented. Then, numerical simulations and experiments with copper foils were carried out to demonstrate the feasibility of this method.
cpl-35-1-014301-fig1.png
Fig. 1. Electrical current density $J$ is induced under the ultrasound stimulation as the Lorentz force acting on the positive and negative ions is in opposite directions and the ions deviate towards opposite directions when they move back and forth together with the sample.
In LFEIT, as shown in Fig. 1, the collected current signal can be expressed as[1] $$\begin{align} I_{\rm meas} (t)=\,&\alpha \iiint \Big[\frac{B_y}{\rho (x)}\frac{\partial \sigma (x)}{\partial x}\\ &\cdot\int_{-\infty}^t {p({\tau,x})d\tau}\Big]dxdydz,~~ \tag {1} \end{align} $$ where $\sigma(x)$ is the sample's local electrical conductivity, $\rho(x)$ is the sample's local density, $B_{y}$ is the magnetic induction density along $e_{y}$, $p(t, x)$ is the ultrasound pressure field, and $\alpha$ is the dimensionless constant. From Eq. (1), evidently the integral is mostly contributed by places where the variations of $\sigma (x)$ along $e_{x}$ exist, and inversely the existence and location of variations of $\sigma (x)$ along $e_{x}$ can be reconstructed using the detected current signal $I_{\rm meas}(t)$.
cpl-35-1-014301-fig2.png
Fig. 2. LFEIT using step frequency technique. (a) The signal transformation from the stimulating signals $T_{f_k}(t)$ and the detected LFEIT current signals $R_{f_k}(t)$ to the IF signals $I'_{f_k}(t)$ and finally to the existence function $A_n e^{-j2\pi f_0 \frac{n{\it \Delta}_z}{c}}$. (b) Schematic of LFEIT system using step-frequency technique. The restriction of $z_0=\frac{3N+1}{2}{\it \Delta}_z$ is specific to the direct demodulation method, which guarantees the artificial target mirrors with the real target with respect to $z=\frac{N}{2}{\it \Delta}_z$.
The setup of application of the step-frequency technique in LFEIT is shown in Fig. 2. A collection of monotonic signals with frequency increasing in equal step are in turn used to generate the sinusoidal ultrasound bursts. The sample placed in the quasi-uniform static magnetic field is vibrated by the generated ultrasound and the LFEIT current signal is induced in the sample. The detected LFEIT current signal is demodulated by the coherent local reference signal. The resulting dc signal contains the accumulated phase as the ultrasound propagates to the places of variations of $\sigma(x)$. Equivalently, the range information of the electrical conductivity variations is contained in the dc signal via the accumulated phases. Craftily, by using step frequencies and by assuming that the electrical conductivity variations' distances from the transducer increase in equal step, the dc signal represents the fast Fourier transform (FFT), at the specific frequency points, of the existence function of the electrical conductivity variations. Inversely, the distribution of the electrical conductivity variation can be deduced by performing inverse fast Fourier transform (IFFT) on the collection of the dc signals. In the real situation, the ultrasound is transmitted in the form of wide pulse rather than continuously to reduce the power of the transmitting system. Moreover, the transmission of the ultrasound pulse of the same frequency and collection of its induced current signal is repeated multiple times to apply the coherent addition before jumping to the next frequency. The stimulating signal to the ultrasound transducer is $$\begin{alignat}{1} T_{f_k}(t)=\,&A_0 \sin\Big[2\pi \Big(f_0 +\frac{\Delta f}{N}k\Big)t+\varphi _0\Big],\\ &0\le t < T,~~~~ k=0, 1, \ldots, N-1,~~ \tag {2} \end{alignat} $$ where $A_{0}$ is the amplitude of the stimulating signal, $f_{0}$ is the initial frequency, $\Delta f$ is the total frequency bandwidth, $\varphi_{0}$ is the initial phase, $N$ is the number of steps, and $T$ is the stimulating pulse width. Because the amplitude-frequency response of the ultrasound transducer is not flat within the frequency band (1.4–3.4 MHz), the transmitted ultrasound pulses from the transducer do not have the same amplitude although the stimulating signals to the ultrasound transducer have the same amplitude. However, the frequency of the transmitted ultrasound pulse signal still has the same frequency as that of the stimulating signal to the transducer. The induced current signal, as shown in Eq. (1), is the time-domain integral of the ultrasonic pressure signal. The resulting current signal has the same instantaneous frequency as the ultrasound pressure signal because the time-domain integral of a sinusoid results in a sinusoid of the same frequency. However, as the frequency increases between different $T_{f_k}(t)$, the integral of ultrasound pressure also introduces amplitude variation between different $R_{f_k} (t)$. To sum up, we obtain $R_{f_k}(t)$ as $$\begin{align} R_{f_k} (t)=\,&\sum\limits_{n=0}^{N-1}A_n (t)\sin\Big[2\pi \Big(f_0 +\frac{\Delta f}{N}k\Big)\\ &\cdot\Big(t-\frac{z_n}{c}\Big)+\varphi _0\Big],\\ &\frac{z_0}{c}\leqslant t < \frac{z_{N-1}}{c}+T,\\ &~ k=0, 1, \ldots, N-1,~~ \tag {3} \end{align} $$ where $z_{n}$ is the $n$th target range, $z_{0}$ is the 1st target range, and $A_{n}(t)$ accounts for the overall amplitude-frequency response of the transducer, the ultrasound propagating medium and the acousto-electrical transforming system. For the detection of the accumulated phase, there are two demodulation schemes: one is the direct demodulation scheme and the other is the in-phase quadrature (IQ) demodulation scheme. In the direct demodulation scheme, where only the in-phase reference signal is used to demodulate the detected LFEIT signal, the intermediate frequency (IF) signal has the form $$\begin{align} I_{f_k} (t)=\,&\sum\limits_{n=0}^{N-1} A_n (t)\cos\Big[2\pi \Big(f_0 +\frac{\Delta f}{N}k\Big)\frac{z_n}{c}\Big],\\ z_{n}=\,&z_{0}+n\cdot{\it \Delta}_{z},~\frac{z_0}{c}\leqslant t < \frac{z_{N-1}}{c}+T,\\ &~k=0, 1, \ldots, N-1.~~ \tag {4} \end{align} $$ After multiplying $I_{f_k}(t)$ with $e^{j2\pi ({f_0 +\frac{\Delta f}{N}k})\frac{z_0}{c}}$ and applying the restrictions $\frac{{\it \Delta}_z \Delta f}{c}=1$ and $z_0 =\frac{3N+1}{2}{\it \Delta}_z$, $I_{f_k}(t)$ is transformed into $$\begin{align} I'_{f_k}(t)=\,&\sum\limits_{n=0}^{N-1}[A_n (t)e^{-j2\pi f_0 \frac{n{\it \Delta}_z}{c}}e^{-j\frac{2\pi}{N}nk}\\ &+A_{N-1-n} (t)e^{j2\pi f_0\frac{2z_0 +(N-1-n){\it \Delta}_z}{c}}e^{-j\frac{2\pi}{N}nk}],\\ &\frac{z_0}{c}\leqslant t < \frac{z_{N-1}}{c}+T,\\ &~k=0,1, \ldots, N-1.~~ \tag {5} \end{align} $$ From Eq. (5), we can obtain that mathematically $I'_{f_k}(t)$ is the FFT of the summation of $A_n (t)e^{-j2\pi f_0 \frac{n{\it \Delta}_z}{c}}$ and $A_{N-1-n} (t)e^{j2\pi f_0 \frac{2z_0 +(N-1-n){\it \Delta}_z}{c}}$, where $A_n (t)e^{-j2\pi f_0 \frac{n{\it \Delta}_z}{c}}$ is an existence function of the electrical conductivity variation at $z=z_{0}+n\cdot{\it \Delta}_{z}$, while $A_{N-1-n} (t)e^{j2\pi f_0 \frac{2z_0 +(N-1-n){\it \Delta}_z}{c}}$ is an artificial existence function of the electrical conductivity variation at $z=z_{0}+(N-1-n)\cdot{\it \Delta}_{z}$. Therefore, when carrying out IFFT on $I'_{f_k}(t)$, a real target and an artificial target are recovered for one electrical conductivity variation. The restriction of $z_0 =\frac{3N+1}{2}{\it \Delta}_z$ in the transformation from Eq. (4) to Eq. (5) guarantees the mirror for the real and artificial targets locates at $z=\frac{N}{2}{\it \Delta}_z$. To avoid confusing the artificial targets with the real targets, the detection range is limited within $({z_0,z_0 +({\frac{N}{2}-1}){\it \Delta}_z})$. Indeed, the recovered targets locating in the range $({z_0,z_0 +({\frac{N}{2}-1}){\it \Delta}_z})$ are judged as the real targets, while those locating in the range $({z_0 +\frac{N}{2}{\it \Delta}_z,z_0 +({N-1}){\it \Delta}_z})$ are judged as the artificial targets. In the IQ-demodulation scheme, in addition to the in-phase IF component, the quadrature IF signal is also obtained as $$\begin{align} I_{If_k} (t)=\,&\sum\limits_{n=0}^{N-1} A_n (t)\cos\Big[2\pi \Big(f_0 +\frac{\Delta f}{N}k\Big)\frac{z_n}{c}\Big], \\ I_{Qf_k} (t)=\,&\sum\limits_{n=0}^{N-1} -A_n (t)\sin\Big[2\pi \Big(f_0 +\frac{\Delta f}{N}k\Big)\frac{z_n}{c}\Big], \\ z_{n}=\,&z_{0}+n\cdot{\it \Delta}_{z},~\frac{z_0}{c}\leqslant t < \frac{z_{N-1}}{c}+T,\\ &~k=0, 1, \cdots, N-1.~~ \tag {6} \end{align} $$ Therefore, using the in-phase and quadrature IF components the complex form of IF signal is obtained as $$\begin{alignat}{1} \!\!\!\!\!\!\!\!I_{f_k} (t)=\,&I_{If_k} (t)+jI_{Qf_k} (t)\\ =\,&\sum\limits_{n=0}^{N-1} A_n (t)e^{-j2\pi \Big({f_0 +\frac{\Delta f}{N}k}\Big)\frac{z_0 +n{\it \Delta}_z}{c}},\\ &\frac{z_0}{c}\leqslant t < \frac{z_{N-1}}{c}+T,~~k=0, 1,\cdots, N-1.~~ \tag {7} \end{alignat} $$ Reducing the constant phase by multiplying the term $e^{j2\pi ({f_0 +\frac{\Delta f}{N}k})\frac{z_0}{c}}$, $I_{f_k} (t)$ is changed to $$\begin{align} I'_{f_k}(t)=\,&\sum\limits_{n=0}^{N-1} A_n (t)e^{-j2\pi f_0 \frac{n{\it \Delta}_z}{c}}e^{-j\frac{2\pi}{N}nk},\\ &\frac{z_0}{c}\leqslant t < \frac{z_{N-1}}{c}+T~{\rm and}~k=0,1,\ldots,N-1,~~ \tag {8} \end{align} $$ from which it can be obtained that $I'_{f_k}(t)$ is the FFT of $A_n (t)e^{-j2\pi f_0 \frac{n{\it \Delta}_z}{c}}$. Therefore, by IFFT of $I'_{f_k}(t)$, $A_{n}(t)$ containing the location information of the electrical conductivity variations can be reconstructed. Numerical simulations of LFEIT using the step-frequency technique are carried out based on the direct and IQ demodulation schemes. Generation of the LFEIT signal has taken into consideration the amplitude-frequency response of the transducer and the integral in the acousto-electrical transformation. System parameters used are the same as those in Fig. 2. Two electrical conductivity variations 78 mm and 93 mm far away from the transducer exist. Here 64 discrete frequencies with equal step are used and $z_{0}$ is set to 68.93 mm so that $z_{0}$ equals $\frac{3N+1}{2}{\it \Delta}_z$. In Figs. 5(a) and 5(b), the lines in black with circles represent the reconstructed electrical conductivity variations using the direct and the IQ demodulation schemes, respectively. Experiments were also carried out to demonstrate the feasibility of this imaging method. The experimental setups are shown in Fig. 3. The in-phase and quadrature channels of monotonic frequency pulse were generated with pulse width 100 μs and starting phase differing by 90$^{\circ}$, i.e., the in-phase channel was $\sin(\omega t)$ and the quadrature channel was $\cos(\omega t)$. The monotonic frequency started at 1.4 MHz and increased by a step of 31.25 kHz, and the number of frequencies used was 64, thus the frequency bandwidth used was 2 MHz. The in-phase channel monotonic frequency signal was split into two channels, of which one was amplified (HSA4101) to 37.5 Vpp to stimulate the ultrasound transducer, and the other was used as the local reference signal to the mixer. The transducer used is the Olympus C306, which is a flat transducer with the element size of diameter 1.27 cm and the 6 dB bandwidth starting at 1.4 MHz and ending at 3.38 MHz. Two permanent magnets (NdFeB N45, diameter 15 cm and height 3 cm) placed along the same axis and in the same direction were used to generate the static magnetic field with magnetic induction density about 260 mT.
cpl-35-1-014301-fig3.png
Fig. 3. Schematic layout of LFEIT experimental setup using step-frequency technique. The parts in gray were specific to the IQ demodulation scheme.
The LFEIT current signal detected by electrodes was firstly amplified (Olympus 5662) by 34 dB, high-pass and low-pass filtered, then converted to baseband by mixing (ZAD-5+) it with the local reference signal. In the direct demodulation scheme, the detected current signal was only mixed with the in-phase local reference signal. In the IQ demodulation scheme, the detected current signal was first split into two channels, of which one channel was mixed with the in-phase local reference signal and the other channel was mixed with the quadrature local reference signal. The output signals of the mixers were sampled using the Tektronix oscilloscope (DPO2014B). Two frequency components, the dc and the double-frequency component, existed in the output signals of the mixers. Only the dc component was useful and the double-frequency component was filtered using a 5th order Butterworth digital low-pass filter. The dc component contained the accumulated phase information which was related to the distances between the variations of $\sigma(x)$ and the transducer. The different phases were reconstructed by carrying out IFFT on the dc signals. The range information of the electrical conductivity variations was finally obtained by linear operation on the reconstructed phases. The samples used in the experiment were two narrow strips of copper foils (0.1 mm thick) 5 mm in width and 75 mm in length, as shown in Fig. 4. Two ends of each foil were stuck on the acrylic sheet frame. The copper foils were placed in parallel along $e_{z}$ with the plane surface perpendicular to $e_{x}$. Along $e_{y}$, the transducer was placed in the middle between the two copper foils, and along $e_{z}$, the transducer located in the middle between the two ends of the each copper foil. In the $x$ direction, the distances between the transducer and two copper foils were 78 mm and 93 mm, respectively. The transducer transmitted ultrasound right in the $x$ direction.
cpl-35-1-014301-fig4.png
Fig. 4. Schematic of the sample of two copper foils.
Using the method described above, experiments of detection of electrical conductivity variation, the existence of copper foils, were carried out. Here 64 monotonic pulses sequentially were used to stimulate the transducer. The dc signals containing the accumulated phase were used to reconstruct the range information of the electrical conductivity variation. The experiments were carried out only using the IQ demodulation scheme, but verification of the direct demodulation scheme was accomplished using only the in-phase channel signals. In Figs. 5(a) and 5(b), the lines in red with asterisks represented the reconstructed electrical conductivity variation using the direct and IQ demodulation methods respectively. Both numerical simulation and experiments showed that in the direct demodulation method (Fig. 5(a)), in addition to two target electrical conductivity variations 78.21 mm and 93.21 mm far away from the transducer being recognized, two artificial variations respectively 104.6 mm and 89.64 mm far away from the transducer appeared, whereas in the IQ demodulation method (Fig. 5 (b)), only two electrical conductivity variations 78.21 mm and 93.21 mm far away from the transducer were recognized. The reconstructed variations locating at 78.21 mm and 93.21 mm far away from the transducer coincide with the real distances (78 mm and 93 mm), with a minor amount of error.
cpl-35-1-014301-fig5.png
Fig. 5. Reconstructed electrical conductivity variation distribution from LFEIT current signal using step-frequency technique by numerical simulation and experiments. (a) Direct demodulation method only using the in-phase channel signal. (b) IQ demodulation method using both the in-phase and the quadrature channel signals.
Better low-noise amplifying circuit can be explored to detect weak LFEIT current signal in experiments using weakly conductive gel phantoms. Although the method of increasing the number of the step frequencies can also improve the signal-to-noise ratio and increase the detecting range, this method is not preferred as the operating time of the step frequency technique increases linearly with the number of step frequencies. In summary, we have carried out an in-depth study of the application of step-frequency technique in LFEIT. Theory of adaptation of step-frequency technique to LFEIT is formulated. Although the uneven frequency response of the transducer and the integral in the acousto-electrical transformation undermined the integrity of the accumulated phase, it is still viable to recognize the range information of electrical conductivity variations using the step-frequency technique. The direct demodulation scheme and the IQ-demodulation scheme are formulated. Using the same ultrasound stimulation, the direct demodulation method can only detect half the range of the electrical conductivity variation as the IQ-demodulation method due to the mirroring artifacts, but compared with the IQ-demodulation method, the direct demodulation method has the advantage of hardware simplicity. Experimental verification of both methods using copper foil samples has been carried out. The results show that the ranges of copper foil could be reconstructed precisely. Therefore, LFEIT using sinusoidal pulse signal and step-frequency technique suggests an alternative direction for the development of LFEIT. References Hall effect imagingScanning Electric Conductivity Gradients with Ultrasonically-Induced Lorentz ForceLorentz force electrical impedance tomographyThe dielectric properties of biological tissues: I. Literature surveyIn vivo electrical conductivity of hepatic tumoursLorentz force electrical impedance tomography using pulse compression techniqueA hologram matrix radarDetection of nonmetallic buried objects by a step frequency radarStep‐frequency radarUltrawideband Gated Step Frequency Ground-Penetrating RadarStepped-frequency continuous-wave microwave-induced thermoacoustic imagingFrequency-modulated magneto-acoustic detection and imagingSPIE Proceedings
[1] Wen H, Shah J and Balaban R S 1998 IEEE Trans. Biomed. Eng. 45 119
[2] Montalibet A, Jossinet J and Matias A 2001 Ultrason. Imaging 23 117
[3] Grasland-Mongrain P, Mari J M, Chapelon J Y and Lafon C 2013 IRBM 34 357
[4] Gabriel C, Gabriel S and Courhout E 1996 Phys. Med. Biol. 41 2231
[5] Haemmerich D, Staelin S T, Tsai J Z, Tungjitkusolmun S, Mahvi D M and Webster J G 2003 Physiol. Meas. 24 251
[6]Sun Z, Liu G, Xia H and Catheline S 2017 IEEE Trans. Ultrason. Ferroelect. Freq. Control. (accepted)
[7] Sun Z, Liu G and Xia H 2017 Chin. Phys. B 26 124302
[8] Iizuka K, Ogura H, Yen J L, Nguyen V K and Weedmark J R 1976 Proc. IEEE 64 1493
[9] Iizuka K and Freundorfer A P 1983 Proc. IEEE 71 276
[10] Iizuka K, Freundorfer A P, Wu K H, Mori H, Ogura H and Nguyen N K 1984 J. Appl. Phys. 56 2572
[11] Oyan M J, Hamran S K, Hanssen L, Berger T and Plettemeier D 2012 IEEE Trans. Geosci. Remote Sens. 50 212
[12] Nan H and Arbabian A 2014 Appl. Phys. Lett. 104 224104
[13] Aliroteh M S, Scott G and Arbabian A 2014 Electron. Lett. 50 790
[14] Natarajan S, Singh R S, Lee M, Cox B P, Culjat M O, Grundfest W S and Lee H 2010 Proc. SPIE 7629 76290D
[15]Podilchuk C, Bajor M, Stoddart W, Barinov L, Hulbert W, Jairaj A and Mammone R 2013 IEEE Signal Processing in Medicine and Biology Symposium 1