Forward Problem in 3D Magnetic Induction Tomography (MIT)

Transcription

1 Forward Problem in 3D Magnetic Induction Tomography (MIT) Manuchehr Soleimani 1, William R B Lionheart 1, Claudia H Riedel 2 and Olaf Dössel 2 1 Department of Mathematics, UMIST, Manchester, UK, m.soleimani@umist.ac.uk 2 Insitut für Biomedizinische Technik, Universität Karlsruhe (TH), Germany Claudia.Riedel@ibt.uni-karlsruhe.de ABSTRACT Simulation of the forward problem in Magnetic Induction Tomography (MIT) is one of the key issues in development of the technology of MIT. We simulate the process of measurement in MIT using an edge based finite element technique. We also present the simulation of this forward problem by a program, MAFIA based on finite integration technique. In both methods the same geometrical arrangements are simulated and results are compared. Keywords Forward Problem, Magnetic Induction Tomography, Edge Finite Element, Finite Integration Technique 1 INTRODUCTION Magnetic Induction Tomography (MIT) is a relatively new imaging modality in which internal electric and magnetic material property distributions are reconstructed from the measured impedance data from sensing and exciting coils. In medical applications the non-invasive measurement of the impedance of biological tissue can yield data of diagnostic relevance. For example a change in impedance can give an indication of the healing process of wounds and of skin irritations. The traditional way is to apply the current directly to the tissue and measure the voltage with electrodes. This leads to stray capacitances between the electrodes as well as between the ground and the patient especially at frequencies above 500 khz (Netz 1998). If the patient has acute pain it is not always possible to touch the skin, so a non-contact method must be used. In some applications (e.g. impedance measurement in the brain) impedance can hardly be measured with surface electrodes (Netz 1998). Such a sensor could be an instrument, which is moved around on the skin of a patient, in order to detect areas with impedances that deviate from normal. The non-contact measurement is based on the idea that a time varying magnetic field induces eddy currents in the conductive tissue. These eddy currents will create a field by themselves and a change in the signal can be detected. This change is expected to be very small, so a high resolution of the measuring system is necessary (Scharfetter 2000). Before attempting to reconstruct the unknown material property, it is necessary to solve the forward problem. A solution of the eddy current forward problem in medical MIT has been published by Merwa (2003) and Hollhaus (2002), which considers the low conductivity case and uses the A r, A r V formulation (Biro, 1999) along with a second order edge finite element for reduced magnetic vector potential (A r ) and a second order nodal finite element for electric scalar potential (V). The electric and magnetic field in the interior will be needed for the image reconstruction. Also for an accurate reconstruction it is necessary to be able to predict the measurement at the sensing coils for an arbitrary inhomogeneous material. The forward problem of MIT is modelled in terms of the electromagnetic fields. The forward models are three-dimensional problems and one cannot constrain electromagnetic field to a plane. 2 ELECTROMAGNETIC LAWS Forward solution, which means simulation of the exciting and sensing procedure where the materials and exciting currents are known. There are several ways to describe the problem. The first are used in geophysics and use a dipole method, later this method is used (Tarjan 1968; Gencer 1998) for a medical application. Griffiths (1999) and Gencer (1998) are using more analytical methods. Also, numerical methods for the forward calculation have been reported by several authors (e.g. Gencer 1999; Scharfetter 2000; Merwa 2003). The objective of the forward solver is the evaluation of the electromagnetic field under known material distributions. In fact we need to evaluate induced voltage on the sensing coils due to induced current, which is the evaluation of the trans-impedance between sensing and exciting coils. 275

2 Therefore the magnetic field of the excitation coil generates a primary solenoidal magnetic field. If conducting material is in the field, eddy currents are produced. These eddy currents themselves create a secondary field, which is approximately in quadrature to the primary field in medical application (Griffiths 1999). From the calculated electromagnetic field we can calculate the induced voltage, which is measured in practice. Using the eddy current approximation and a formulation in terms of electric field, Maxwell s equations can be written 1 µ E + iω ( σ + iωε ) E = iω J e (1) where E is electric field, J e is excitation current density, σ is conductivity,µ is permeability, ω is angular frequency and ε is permittivity. 3 SOLVING THE DISCRETISIED EDDY CURRENT PROBLEM 3.1 Finite Integration Technique (FIT) We use two numerical methods, first a Finite Integration Technique. The simulations are done using MAFIA (Weiland 1996). The Maxwell equations are written in discrete form, the field quantities E and B on different grids. These two grids are dual to each other. This results in a discrete representation of the curl, div and grad operators. This preserves the analytical properties of curl, div, and grad. i.e. div curl = 0 and curl grad = 0 are fulfilled also by their discrete counterparts (Hanhne 1992). Since we deal in with a time harmonic problem, for linear material the equations can be written in frequency space. Materials can be anisotropic with real ε, µ andσ. Therefore the material properties are described by the diagonal matrices D ε, D µ and D σ in the grid and with the matrices D ε, D µ and D σ in the dual grid. Using equation (1) creates a current density j e in a discrete form D 1 A C D S D 1 µ D 1 A CD S ω 2 D ε + 1 iω D σ e = iωj e (2) The discrete curl and div operators are described by the matrices C and S in the grid and by C and S in the dual grid. The geometrical dataset of the grid and the dual grid are described in diagonal matrices D S and D S that determine the length of one mesh segment, D A and D A are also diagonal matrices that describe the surface (Hanhne 1992; Hanhne and Weiland. 1992). 3.2 Finite Element Method (FEM) The second numerical technique is on Finite Element Method. We chose edge finite element over nodal elements, which has well-established advantages for vector fields (Biro 1999). The electromagnetic field in the eddy current problem can be defined either by field or by potentials. There are different possible combinations of these quantities as state variables (Biro 1999). In different regions different approximations of Maxwell s equations can be used. To exclude the wave propagation effects, we neglect the displacement current, which is a low frequency approximation. In the excitation coil region, we assumed there is no eddy current so we can use an electrostatic approximation. With the same assumption, (no eddy current) the sensing coil is considered as a magnetostatic area. In the software we developed, coils with any shape can be simulated and the result of the coil modelling comes explicitly into the eddy current model. We use a formulation in terms of magnetic vector potential. The space of shape function of the first order edge element (one form Withney elements) includes the space of the gradient of the zero form Withney element which is in fact the nodal FEM (Biro 1999), so we can define a modified magnetic vector potential in conductive region * in which E = iωa, in non conductive region A is used. Galerkin s approximation to equation (1) using the A, A* formulation of the eddy current problem is Ω 1 N A dv + ( i ω ( σ + i ωε ) N A ) dv = ( NJ ) e dv. (3) µ Ω e Ω c 276

3 where N is the first order vector based shape function, Ω, Ω e, Ω c are whole region, eddy current region and coil region respectively, and A is the magnetic vector potential. The linear system of equations arising from this edge FEM formulation is solved using a preconditioned BiCSTAB method. 4.1 Simulation of Measuring Values 4 SIMULATION MODEL Using MAFIA the induced voltage that is measured in practise is calculated. The complex measured voltage U meas can be calculated evaluating the line integral for FIT field results and a volume integral for FEM field results where E is the electric field and C is the integration path, j is the current density for an artificially unit current passing the coil and Ω is the coil s volume. 4.2 Simulation of the Excitation Coil U meas = Edl or U meas = E. jdv (4) C Ω In FIT formulation the excitation coil is also simulated with line segments. These line segments are carrying a current with amplitude and a phase. For this formulation we used only one line segment instead of 22. The current of one winding is multiplied by number of turns. The resulting current density is used in equation (5). In FEM formulation we calculate the current density in the coil analytically in simple coil shapes or by finite element solution for a complex shape. For example for a cylindrical coil shape the current density is J = J me ˆ θ, in which J m is the amplitude of the current density and ê θ is the unit vector in tangential direction. The modelling of the current source is important for the accuracy and speed of FEM simulation of the eddy current problem (Ren 1996). 4.3 Boundary Condition In FIT formulation we use open volume boundary condition. For the simulation, 500 iterations are used, which gives us a residual of In FEM software we define far field boundary conditions so we need to mesh area surrounding the coils and conductive object and in the far field we define normal component of the magnetic flux density to be zero. 4.4 Dimension of the Sensor and the Geometrical Arrangement Figure 1 shows the geometrical arrangements for the numerical simulation. The upper coil presents the excitation coil, a coils with 22 windings and an excitation current of 0.2 A. The measurement coil is between the excitation coil and the tissue and represents also a coil with 22 windings. The coils are rectangular coils with length of 10 mm on one side. The sensing coil is 1.5 mm on top of the conductive object. It has 1mm thickness in Z direction. There is 1mm distance between two coils and exciting coil also has 1mm thickness. In figure 1b the geometry of coil is been shown. The volume which is used in calculation is 90 mm x 45 mm x 24 mm, and each of the two conductor blocks are 45 mm x 45 mm x 24 mm. The main regions are vacuum and the conducting tissue. A mesh with equidistant mesh points and a separation of 1 mm is used. The frequency of the alternating current is 1 MHz. The simulations are made with the sensor in the middle to avoid the edge effects. 5.1 Simulation with no Tissue 5 RESULTS Table 1 shows the induced voltage results of both simulations; they are both in the same range. Using an analytical calculation as a crosscheck, we calculate the magnetic field of a round current loop with I = 4.4 A, and R = m by using Biot-Savart s law. From this, at a distance of z = m we have a 277

4 Figure 1: Arrangement of the coil system and the tissue block scale in (m). The excitation and the measurement coils are simulated as rectangular coils. The tissue block is divided into two halves. For the simulation, the tissue was moved under the sensor. dimensions of coil from above. calculated magnetic field B = (T) in the centreline. We use this field to subsequently calculate both the flux Φ and the induced voltage, so that the induced voltage U ind of one winding is -2 given approximately by U = iω Φ = i 2 π f B a = i V, which is 2.22 V for 22 turns coil. ind Induced voltage in the secondary coil Finite Integration Technique i 1.97 V Edge Finite Element Technique i 2.21 V Table 1: Induced voltage in the secondary coil with no tissue under the sensor. The results are calculated with the Finite Integration Technique and the Edge Finite Element Technique. 5.2 Simulation with the Sensor in the Middle Over the Tissue For the following results, the sensor is placed in the middle of the tissue block, as shown in figure 2. So the sensor is right over the area where the conducting parts come together. Induced voltage in the secondary coil Finite Integration Technique Edge Finite Element Technique i i 2.08 Table 2: Induced voltage in the secondary coil with tissue. The results are calculated with the Finite Integration Technique and the Edge Finite Element Technique Figure 2: Edge FEM software Imaginary part of the Magnetic flux density in x-z plane, for a current density 1 (A/m 2 ) from exciting coil and magnetic vector potential for free space in x-y plane. 278

5 Figure 3: Eddy current density FIT shows the x-y plane with the imagery part of the current density in the top layer of the tissue. The maximum arrow presents 8.6 A/m 2 shows the y component of the current density cut in the x z plane in A/m 2. Figure 4: Real part of the magnetic flux density FIT: shows the x z plane. The maximum arrow represents Vs/m 2. shows the magnitude of the real part of the magnetic flux density in 3 Vs/m 2. 6 CONCLUSION The main objective of this study is to compare the numerical results from two forward solutions programs and also better understanding of the numerical behaviour of the eddy current problem. One is the commercial software MAFIA, the other is a home made edge based finite element program. The numerical comparisons show a good agreement. One of the main differences is the simulation of the coil and we suggest that the differences in results are mainly due to this. In numerical simulation of the forward problem of MIT, the modelling of the coil is important in order to have accurate prediction of the induced voltage. Modelling of the coil also has effect on the speed of the forward solver. If we mesh the coil region we must use a dense enough mesh. To compute an accurate forward model it is also important to have a dense enough (typically at least 3 elements) in skin depth region. Too coarse a mesh in skin depth region produces inaccurate sensed voltages. 7 REFERENCES BIRO O., (1999), Edge Element Formulations of Eddy Current Problems, Computer methods in applied mechanics and engineering, 169, pp GENCER N.G. and TEK. N., (1998), Imaging Tissue Conductivity via Contactless Measurements: A Feasibility Study, Turkish Journal of Electrical Engineering ELEKTRIK, 6(3) pp GENCER N.G. and TEK. M.N., (1999), Forward Problem Solution for Electrical Conductivity Imaging via Contactless Measurements, Physiological Measurement, 44, pp

6 GRIFFITHS H., STEWART W.R. and GOUGH. W., (1998), Magnetic Induction Tomography: Measurements with a Single Channel, In Proc: X-ICEBI, Barcelona, pp GRIFFITHS H., STEWART W.R. and GOUGH W., (1999), Magnetic Induction Tomography Electrical Bioimpedance Methods: Applications to Medicine and Biotechnology, 873, pp GRIFFITHS H., (2001), Magnetic Induction Tomography, Meas. Sci. Technol., 12(8) pp HAHNE P. and WEILAND T., (1992a), 3D Eddy Current Computation in the Frequency Domain regarding the Displacement Current, IEEE Transactions on Magnetics, 28(2) pp HAHNE P., (1992b), Zur numerischen Berechnung zeitharmonischer elektromagnetischer Felder, Ph.D. Thises, Techn. Univ. Darmstadt HOLLHAUS K., MAGELE C., BRANDSTÄTTER B., MERWA R., and SCHARFETTER H., (2002), Numerical Simulation of the Forward Problem in Magnetic Induction Tomography of Biological Tissue, In Proc: 10th IGTE Symposium, pp MERWA R., HOLLAUS K., BRANDSTATTER B. and SCHARFETTER H., (2003), Numerical Solution of the General 3D Eddy Current Problem for Magnetic Induction Tomography (Spectroscopy), Physiological. Measurement, 24(2), pp NETZ J., FORNER E. and HAAGEMANN S., (1993), Contactless Impedance Measurement by Magnetic Induction, a Possible Method for Investigation of Brain Impedance, Physiological Measurement, 14(4) pp REN Z., (1996), Influence of the R.H.S on the Convergence Behaviour of the Curl-Curl Equation, IEEE Trans. MAG, 32(3) May. SCHARFETTER H., LACKNER H.K. and ROSELL J., (2000), High Resolution Hardware for Magnetic Induction Tomography (MIT) at Low Frequencies, In Proc: EIT Conference, London, UK. TARJAN P.P. and MCFEE R., (1968), Electrodeless Measurements of the Effective Resistivity of the Human Torso and Head by Magnetic Induction, IEEE Trans. BME, 15(4), pp WEILAND T., (1996), MAFIA version 4, In Proc: Computational Accelerator Physics Conference (CAP), pp