Finite Difference Time Domain (FDTD) Method for Modeling the Effect of Switched Gradients on the Human Body in MRI

Transcription

1 Finite Difference Time Domain (FDTD) Method for Modeling the Effect of Switched Gradients on the Human Body in MRI Huawei Zhao, 1 Stuart Crozier, 2 * and Feng Liu 1 Magnetic Resonance in Medicine 48: (2002) Numerical modeling of the eddy currents induced in the human body by the pulsed field gradients in MRI presents a difficult computational problem. It requires an efficient and accurate computational method for high spatial resolution analyses with a relatively low input frequency. In this article, a new technique is described which allows the finite difference time domain (FDTD) method to be efficiently applied over a very large frequency range, including low frequencies. This is not the case in conventional FDTD-based methods. A method of implementing streamline gradients in FDTD is presented, as well as comparative analyses which show that the correct source injection in the FDTD simulation plays a crucial rule in obtaining accurate solutions. In particular, making use of the derivative of the input source waveform is shown to provide distinct benefits in accuracy over direct source injection. In the method, no alterations to the properties of either the source or the transmission media are required. The method is essentially frequency independent and the source injection method has been verified against examples with analytical solutions. Results are presented showing the spatial distribution of gradient-induced electric fields and eddy currents in a complete body model. Magn Reson Med 48: , Wiley-Liss, Inc. Key words: FDTD method; eddy current; peripheral nerve stimulation; low frequency electromagnetic wave; gradient coil 1 Centre for Magnetic Resonance, The University of Queensland, St. Lucia, Brisbane, Queensland, Australia. 2 School of Information Technology and Electrical Engineering, The University of Queensland, St. Lucia, Brisbane, Queensland, Australia. Grant sponsor: Australian Research Council. *Correspondence to: Stuart Crozier, PhD, School of Information Technology and Electrical Engineering, The University of Queensland, St. Lucia, Brisbane, Queensland 4071, Australia. stuart@itee.uq.edu.au Received 11 December 2001; revised 6 August 2002; accepted 6 August DOI /mrm Published online in Wiley InterScience ( Wiley-Liss, Inc Peripheral nerve stimulation (1 6) presents a fundamental limit to the gradient fields that can be safely used in a clinical MRI environment. The interaction between the human body and electromagnetic fields generated by gradient switches in an MRI examination is both spatially and temporally complex. In this work, we present a variant on the finite difference time domain (FDTD) method that provides both spatial and temporal information on the induced fields in the human body during gradient switching. The FDTD technique is a well-established numerical technique for computationally modeling EM problems. It is applicable for very complicated, inhomogeneous dielectric structures. The technique was first proposed by Yee (7) and later developed by Umashankar and Taflove (8). Recently, there has been increased research in applying the FDTD technique to obtain a high-resolution (HR) (on the order of 10 4 points per wavelength) solution for a diverse range of bioelectromagnetic problems (9,10). However, there are two major difficulties involved in modeling the eddy currents that are induced in the human body due to gradient switching. The first problem is that the switching frequency of gradient coil current is very low relative to those normally modeled using FDTD methods (11,12); the second is the method used to inject the source into the computational domain. Current applications of the conventional FDTD technique can be hampered by its intrinsic time-stepping stability criterion. For example, if the required resolution is on the order of 10 2 m or less and the wavelength is on the order of 10 3 m, a conventional application of the FDTD technique results in a high-definition problem that is notoriously difficult to solve due to the significant computation expense required. Recently, the numerical technique HR-FDTD (13) was developed to overcome such issues. A special time-frequency conversion was used, so that it was only necessary to run the simulation for a fraction of the source period. In terms of the source injection, the EM field source is the current in the gradient coil; a slow changing current source will produce a strong zero-order (i.e., not changing with time) magnetic field response and weak higher orders. The coupling between magnetic and electric fields has only a small effect on the behavior of the magnetic fields in this situation. The FDTD method is normally used to solve problems for strongly coupled fields. The zero-order magnetic field can produce large numerical errors in FDTD simulations. This problem has been overcome in this work, by injecting the time derivative of the source function rather than the source itself. In this article, details of the implementation of a complete, streamline gradient set in an FDTD modeling space are given. A method for the implementation of the current source in the coil is described and the results are verified against examples with analytical solutions. Furthermore, the issue of what frequency should be used to determine the lossy dielectric data of the human body is discussed. The biological eddy currents induced by a gradient coil system in a clinical MRI scanner are highlighted in the final section. FDTD MODELING WITH PULSED CURRENT SOURCES FDTD Method FDTD methods are popular in MRI for modeling RF coil structures and the interaction of RF fields with body

2 1038 Zhao et al. models (11,12). The mathematical model is based on Faraday and Ampere s laws which provide a system of equations that form an independent set of coupled relationships between time-varying electric and magnetic fields. The time-varying EM fields are described in vector form as: F ƒ G I, [1] t where: F D B, G H E, and I J 0. The constitutive relation is F a G where a 0 0. D is the electric displacement, E is the electric field intensity, ε is the permittivity, B is the magnetic flux density, H is the magnetic field intensity, and is the permeability. The electric current density is given by J E. According to the Yee cell (see Fig. 1), Eq. [1] can be written in an FDTD scheme as: H 1/2 xl,m,n H 1/2 xl,m,n î H 1/2 yl,m,n ĵ H 1/2 zl,m,n kˆ î H 1/2 yl,m,n ĵ H 1/2 zl,m,n kˆ Ch l,m,n î ĵ kˆ x y z ; [2] E xl,m,n E yl,m,n E zl,m,n 1 E xl,m,n 1 1 î E yl,m,n ĵ E zl,m,n kˆ Ce ll,m,n E xl,m,n î E yl,m,n ĵ E zl,m,n kˆ Ce 2l,m,n î ĵ kˆ x y z 1/2 H xl,m,n 1/2 H yl,m,n 1/2 H zl,m,n, where: Ch l,m,n 1 / l,m,n; Ce ll,m,n 2ε l,m,n l,m,n t /2ε l,m,n l,m,n t ; Ce 2l,m,n 2 t /2ε l,m,n l,m,n t, where the difference operator is defined as x V l,m,n V l1/ 2,m,n V l1/ 2,m,n / x. This difference equation is fully explicit and is solved using an iterative method, subject to boundary and source conditions. Discrete Space In our FDTD model, the main objective is to investigate the interaction between the switched gradient coil and the human body; the main magnet inner bore is therefore simply treated as a conducting cylinder, with the two open ends truncated by absorbing boundaries, so-called perfectly matched layers (PML) (14). Our body model is discussed in a later section. A staircase mesh was used to approximate the magnet s inner bore (Fig. 2a). For the gradient coil, the wire pattern was mapped into a regular grid (Fig. 2b). The current only flows along the edge of each mesh element. Since the FDTD algorithm is second-order in space, in general the error introduced from the staircase boundary and discreted current rapidly becomes small as the cell size decreases. This assures that an accurate solution will be obtained when the grid size is small enough. [3] where Ce 3(l,m,n) t / (l,m,n).inthefinal analyses, all components of J are considered. This equation represents that the curl of total magnetic field intensity minus the source current density is equated to temporal electric field changes. This is a good approximation in FDTD when J is changing rapidly (high frequency). However, when J is changing very slowly, as in this case, the magnetic field is a half time step behind the electric field in the FDTD scheme. Therefore, the electric field is immediately generated by the current density at the start point of a simulation. The total magnetic field in FDTD simulation involves both zero and higher order fields. In the ideal case, the curl of the zero-order magnetic field should be zero in free space. However, the curl operator could produce a large error when small numerical errors occur in the magnetic field. Consequently, the computed electric field solution could be larger than the true solution. This is illustrated when the frequency of the current source goes to zero; at this time, according to Eq. [4], electric field is still being generated in the system. It follows that it is preferable to avoid introducing the zeroorder magnetic field terms in these simulations. In order to overcome this problem, we first review the nature of the low-frequency current source. For static field Source Injection As the current source is approximated by a path along the edge of the Yee cells, the finite difference form is taken as (first considering J to be Z-directed): 1 E zl,m,n Ce ll,m,n E zl,m,n Ce 2l,m,n H 1/2 yl,m,n 1/2 H yll,m,n x H 1/2 xl,m,n 1/2 H xl,ml,n y 1/2 Ce 3l,m,n J zl,m,n [4] FIG. 1. A typical Yee cell representation.

3 FDTD Method Modeling Switched Gradients 1039 solution for the electric field distribution can be expressed in terms of a Hankel function of the second kind (15) as: E S 1 4 H 0 2 kr dl dt. [9] FIG. 2. Discretization of the space. a: The magnet bore as a perfect conductor. b: Edge approximation of an arc of the gradient coil. problems, there is no coupling between the electric and magnetic fields, except in a medium of finite conductivity, such as the human body, where: ƒ H J E. [5] Thus, in a nonconducting medium the static and fields may be calculated independently of each other. The coupling between the dynamic fields is due to the terms B/t and D/t, and J E in Eq. [1]. This coupling becomes negligible as the time rate of change of the fields diminish. For such fields, it is possible to obtain serial solutions by initially omitting the B/t and B/t terms from Maxwell s equations and then introducing their effects as correction terms. To analyze the behavior of dynamic fields, Maxwell s equations could be written in perturbed form, where the series solutions for E and H have the relations: ƒ E 0 0; ƒ E n B n 1 ; t ƒ H 0 J 0 ; ƒ H n J n D n 1 ; t ƒ D 0 0 ; ƒ D n n ; n 0. ƒ B 0 0; ƒ B n 0; n 0. The relations in Eqs. [6] and [7] show that the time changes of each term of electric field will produce higher-order terms of magnetic field, and vice versa. These relations have to be solved subject to boundary conditions, which is not a simple task. However, if we know the H 0 term then we can calculate the E 1 term and the rest of the terms in the series are only dependent on the E 1 term. This analysis gives us an indication of how to structure an FDTD method to produce solutions for dynamic EM fields. So, instead of using the perturbed forms of Eqs. [6] and [7], we have used an FDTD method to solve for all of the higher-order fields, by using E 1 as the source. From Maxwell s equations, the field at source point can be specified as: [6] [7] Equation [9] is singular at r 0. However, due to the finite size of the grid cell in the FDTD simulation, the field may be inaccurate close to the singularity. A modification of the source term for FDTD models can be achieved by using the effective radius r 0 of wire (16) to match the analytical solution. At the source point, in the discrete form, the electric field can be approximated as: E Sl,m,n 1 4 H 0 2 kr 0 dl l,m,n. [10] dt Where H 0 2 kr 0 is a scale factor. If, for example, a source wire is positioned along the y-edge of the Yee-cell, the FDTD scheme could be written as: 1/2 H zl,m,n 1/2 H zl,m,n C hl,m,n E yll,m,n E ysl,m,n x E xl,m 1,n E xl,m,n, [11] y where E ysl,m,n is the source term, which can be directly specified as a function of J/t. Test Case We now consider the test case used by Keltner et al. (17): a loop coil of radius R coil (26 cm) carrying uniform current I is placed adjacent to a lossy dielectric sphere of radius R body (20 cm), the distance between the coil center and the sphere center is 10 cm (Fig. 3). The frequency of the current is 2.5 khz and the amplitude is 1A. For the sphere, the conductivity is 0.2 S/m and the relative permittivity is 1. A reference calculation of electric and magnetic fields induced inside the sphere was made using the multipole expansion method (MEM). ƒ 2 E ε 2 E t 2 J t 1 ƒ. [8] ε If there is no free charge ( 0), the electric field is only a function of J/t. Consider a line current source; the exact FIG. 3. Schematic arrangement of a homogeneous conductive sphere near a circular current loop.

4 1040 Zhao et al. FIG. 4. Comparison of the numerical analytic results for the magnitude of the electric field in the sphere along the line y 2 cm. Both J and J/t source FDTD schemes were simulated; the comparison with MEM results is illustrated in Fig. 4 along the line y 2 cm. These results demonstrate that, for this type of problem, accurate solutions for dynamic fields can be obtained by the injection of the current-change as a source. If the current source was sinusoidal (J J 0 sin(t)), then the derivative source would be a cosinusoid (J/t J 0 cos(t)). The peak electric field would then be proportional to J 0. Therefore, higher frequency current sources generate stronger electric fields. Since we are mostly concerned with the peak induced electric field, and the spatial dimension is only a small fraction of the wavelength, it is (thankfully) not necessary to run the simulation for a long period of time. A ramp function, E E 0 (1cos(t/t 0 ))/2, can be used to smoothly reach the maximum field at the source point, where the t is the simulation time and the t 0 is the time for the E to reach the peak E 0. The variable t 0 is selected to be large enough such that ct 0 (c is speed of light) is far greater than the space dimension and small enough to make the simulations efficient. The total simulation time should be long enough to let the peak pass through every point in the human body. To illustrate this FIG. 6. The current source driving the gradient coil and a sinusoid for choosing tissue property frequency (only). point, we have run the simulation with three different source functions (Fig. 5a) and they all give the similar eddy current distributions in a uniform cylindrical body excited by a pulsed, streamline z-gradient coil (Fig. 5b). Source Frequency The relative permittivity and conductivity of each cell in the body model are frequency-dependent, so the input source frequency alters the computational body model. In general, we only focus on the average behavior of the input current change in the gradient coil. This may not give the best choice for the input frequency in relation to calculating the material properties of tissue. For a simple trapezoidal current waveform, in the time interval t 0, I AI 0 t and A 1/t 0 (Fig. 6), the generated electric field is constant E(AI 0 ). In the time interval t 1, I I 0 and the electric field is zero. So the question is, what source frequency should be used in the simulation for the purpose of determining the human body s dielectric properties? We are mostly interested in the maximum E field and induced current in the body from a safety viewpoint. Relative permittivity decreases with increasing frequency, while conductivity slightly increases. Given that the cell size in the FIG. 5. Test source function waveforms for a gradient coil of diameter 70 cm, generating 10 mt/m with a 400 sec rise-time: (a) the source function form, each time step represents seconds; (b) the peak eddy current in a homogeneous cylindrical body or diameter 40 cm, conductivity 0.15 S/m and a relative permittivity of 1.0. The total calculation time on a Sun Enterprise 450 was 4 hr.

5 FDTD Method Modeling Switched Gradients 1041 FIG. 7. The FDTD computational domain. body relative to the (dielectric shortened) wavelength will always be very small for gradient coil work, choosing the largest frequency component in the source waveform generates the highest E field strength (worst case) in the human body. This worst case, in general, occurs at the very beginning of the gradient switch, as shown in Fig. 6. The maximum amplitude of electric field is proportional to AI 0. In reality, the gradient switch is not perfect and will have some form of gradual onset. If current rise time t second, measurements of current waveforms taken from our Bruker 2T whole-body instrument (see Fig. 6) allow estimates of the maximum frequency during the rising of the gradient field to be about 25 khz, while the average still remains at 10 khz. We have in the past (13) used Fourier decomposition to reconstitute the trapezoidal waveform altering the body model with each of the harmonic components. In these examples, however, we take a worst-case scenario (25 khz) in determining the value of the body s dielectric properties. It is, of course, important that the form of the input waveform be accurately known, including the influence of pre-emphasis. FIG. 8. The maximum human body eddy current density induced by gradient coil in the MRI scanner. Each longitudinal point represents the maximum eddy current through the cross-section. placed in the center of the MRI scanner model. A trapezoidal current wave form, I AI 0 t, was used as the source with A 10 4 in the gradient coils and the gradient strength is db/dz 40 mt/m per coil. The numerical simulation ran for 5000 iterations with a ramp-up peak at 2000 iterations. The total computational time was about 8 hr using four CPU processors on a SUN Enterprise 450. SIMULATION RESULTS An MRIfdtd computer code with parallel computation was developed. The gradient coil and the conductor boundaries are automatically mapped to the grid system. A frequencydependent tissue model, developed by the United States Air Force Research Laboratory ( hedr), was used to calculate the exact human body dielectric and conductivity data. This real human body has length 187 cm, the relative permittivity are in the range from 1.0 to , and conductivities are in the range from 0.0 to The inner surface of the magnet was treated as a perfect conducting wall. A PML absorbing boundary, which truncates computational domain, was used to surround the human body. Complete details of the FDTD computational domain are illustrated in Fig. 7. In this study a cell size of 10 mm was used, so the entire computational domain is divided into N x N y N z cells. A human body model containing 357,918 cells was FIG. 9. Eddy current density distribution generated by the xyz gradient coil combination in ma/m 2 :(a) cross-sections at the largest induced current level in the chest; (b) cross-sections at the level of the knees.

6 1042 Zhao et al. From these simulations it is clear that the electromagnetic field penetrates deeply into the body and that the induced eddy currents have complex distributions. In future work we plan to complete simulations of different gradient sets, patient positions, and pulse sequences and attempt to correlate these with experimental reports of PNS. CONCLUSION A new method for the analysis of gradient-induced effects in the human body has been presented. The method is a variant on the FDTD approach and essentially allows the interaction of EM fields with tissue at any frequency. It is therefore capable of modeling the effects of gradient and RF fields in MRI scanning. The analysis can accept a full imaging pulse sequence and analyze the induced fields at almost any point in time and space within the body model. The distribution of gradient-induced E-fields and eddy currents in the body is important in predicting the safety of a pulse sequence and the biological consequences of particular gradient coil designs. REFERENCES FIG. 10. The peak amplitude electric field distribution (V/m) for all time points. Figure 8 shows the maximum eddy current density induced along the human body for switched z, y, and xyz gradient coil combinations. As expected, the xyz gradient coil combination produces the largest eddy current density and z-gradient coil produces the smallest. For comparative purposes, we also simulated a sinusoidal gradient waveform often used in EPI, with a principal frequency of 833 Hz. The maximum eddy currents have a similar spatial distribution to those induced by the trapezoidal switch but have much smaller amplitudes. This demonstrates an advantage of using sinusoidal gradient waveforms where practicable. The two-dimensional eddy current density distributions for the trapezoidal pulses are illustrated in Fig. 9 and clearly show the eddy current patterns in the body. The maximum amplitude of the electric field distributions generated by the xyz gradient coils are shown in coronal sections in Fig. 10. These electric fields only exist during the transient state of current flow in the gradient coils. When the current reaches a constant value, these fields subside. The timing of this decay depends on the electrical relaxation behavior of the tissues and will be the subject of future studies. 1. Furse CM, Gandhi OP. Calculation of electric fields and currents induced in a millimeter-resolution human model at 60 Hz using the FDTD method. Bioelectromagnetics 1998;19: Parkinson WC. Electromagnetic fields in biological studies. Ann Biomed Eng 1985;3: Reilly JP. Peripheral nerve stimulation by induced electric currents: exposure to time-varying magnetic fields. Med Biol Eng Comput 1989; 27: Mouchawar GA, Nyenhhuis JA, Geddes LA. Magnetic stimulation of excitable tissue: calculation of induced eddy-currents with a three-dimensional finite element model. IEEE Trans Magn 1993;29: Irnich W, Schmitt F. Magnetostimulation in MRI. Magn Reson Med 1995;33: Bowtell R, Bowley RM. Analytic calculations of the E-fields induced by time-varying magnetic fields generated by clindrical gradient coils. Magn Reson Med 2000;44: Yee S. Numerical solution of initial boundary value problems involving Maxwell s equations in isotropic media. IEEE Trans Antennas Propagat 1966;14: Umashankar K, Taflove A. A novel method to analyze electromagnetic scattering of complex objects. IEEE Trans Electromagn Compat 1982; 24: Holland R. Finite-difference time-domain (FDTD) analysis of magnetic diffusion. IEEE Trans Electromagn Compat 1994;36: Moerloose JD, Dawson TW, Stuchly MA. Application of the finite difference time domain algorithm to qusi-static field analysis. Radio Sci 1997;32: Collins CM, Li S, Smith MB. SAR and B1 field distributions in a heterogeneous human head model within a birdcage coil. Magn Reson Med 1998;40: Ibrahim TS, Lee R, Baertlein BA. Computational analysis of the high pass birdcage resonator: finite difference time domain simulations for high-field MRI. Concepts Magn Reson 2000;18: Zhao H, Crozier S, Liu F. A high definition, finite difference time domain (HD-FDTD) method. Appl Math Model 2002, in press. 14. Berenger JP. A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 1994;114: Dudley DG. Mathematical foundations for electromagnetic theory. IEEE Press Series on Electromagnetic Waves, IEEE, New York; Waldschmidt G, Taflove A. The determination of the effective radius of a filamentary source in the FDTD mesh. IEEE Microwave Guided Wave Lett 2000;10: Keltner JR, Carlson JW, Roos MS, Wong STS, Wong TL, Budinger TF. Electromagnetic fields of surface coil in vivo NMR at high frequencies. Magn Reson Med 1991;22: