Sensitivity Analysis of 3D Magnetic Induction Tomography (MIT)

Transcription

1 Sensitivity Analysis of 3D Magnetic Induction Tomography (MIT) W R B Lionheart 1, M Soleimani 1, A J Peyton 2 1 Department of Mathematics UMIST, Manchester, UK, bill.lionheart@umist.ac.uk, 2 Department of Engineering University of Lancaster, Lancaster, UK, a.peyton@lancaster.ac.uk ABSTRACT One of the major tools for both design and the image reconstruction of an MIT system are the spatial sensitivity analysis of the object space. This sensitivity analysis effectively maps the sensitivity of a particular excitation / detection coil pair to a small perturbation of the conductivity of the material in the object space. We derive a formula for the sensitivity of the measured voltage in terms of the magnetic vector potential A, which gives to an efficient method of sensitivity calculation using Finite Element Method. Keywords Magnetic Induction Tomography, Sensitivity Analysis 1 INTRODUCTION Magnetic Induction Tomography (MIT) is a technique that attempts to image the conductivity distribution (Griffiths, 2001; Peyton, 1995) within an object space or to image changes in the conductivity distribution. The magnetic field produced by a system of excitation coils generates a secondary eddy current field within the conductive object material, which in turn produces a secondary magnetic field that can be detected by the sensing coils. In this paper we derive a formula for the sensitivity of the induced voltage in a measurement coil to a small perturbation in the conductivity. The formula has the advantage that it involves only the inner product of the electric fields or the magnetic vector potentials when two different coils are excited, and these are convenient computationally. MIT has applications to industrial process monitoring, non-destructive testing, medical diagnosis and geophysics. The calculation of the sensitivity can be used as an aid to the design of system, so the coil configuration can be designed to maximize the difference in induced voltage measured for conductivity changes that need to be detected. The sensitivity to a change in each image voxel gives the Jacobian matrix, which can be used in a regularized Gauss-Newton reconstruction algorithm. 2 MAXWELL S EQUATIONS Assuming time-harmonic fields with angular frequency ω Maxwell s equations are E = iω H, H = 0 (1) ( + iωε ) E + J, E = 0 H = σ s ε (2) Here E and H are the magnetic and electric fields, σ is conductivity, magnetic permeability and ε permittivity. The sources of current are represented by current density J s. The inverse boundary value problem for Maxwell s equations is the recovery of the material parameters σ, ε and from measurements of the tangential components n H and n E of the fields on some surface Γ (with normal n) enclosing the region were the material parameters are unknown. Uniqueness of solution for this inverse boundary value problem was established by (Ola, 1993) provided ω is not a resonant frequency. In this work they take J s = 0 as the sources are assumed to be included in boundary conditions. It worth notice that in sensing coil the measurement induced voltage can be expressed as line integrals of the tangential component of E along the coil, it can also be described as surface integrals of the normal component of the magnetic flux density B. The methodology for establishing the derivative of boundary measurements with respect to a perturbation of a material parameter was established in the fundamental paper (Calderón, 1980) for the static case ( ω = 0 ). The general case was established by (Somersalo, 1992). These results 239

2 require some slight modification for application to MIT. In this case, we are not measuring on an isolated boundary. Typically we have an arrangement of coils on some surface Γ but boundary conditions (such as screening by a conductive or magnetic shield) apply on some surface containing this. We can think of an idealized excitation coil as imposing a predetermined H on Γ, and our idealized measurement as an integration of E around an infinitesimal loop on Γ. This is no worse that the idealization in the low frequency case (Electrical Resistance Tomography ERT) that we can apply arbitrary current patterns to the surface and measure the voltage everywhere. In practice we measure a finite subset of the idealized data, but it is important to know at least that if we collected ideal data then the material parameters are uniquely determined. This question, called uniqueness of solution by mathematicians, is the practical question of sufficiency of data for the engineer. The measurement arrangements of MIT using a system of coils do not fit exactly in to this formalism. There is now barrier to electric and magnetic fields in the surface containing the coils so we must model them by a current source term J s, and impose boundary conditions on some larger enclosing surface. We will address this in the next section. For the moment, our ideal data is the transfer impedance on the surface Γ, where we have complete control of the tangential component of H and knowledge of the transfer impedance of E (or vice versa). There is of course a parallel impedance due to the exterior of Γ, which we will assume is known by calibration and has already been subtracted. It is convenient to recast the data on Γ in an integral of the Poynting vector power-flux, we obtain E H that represents the Γ δ ( E H) n = δ H H + ( δσ + iωδε ) E E + Higher order terms (3) Now taking the electric and magnetic fields from two different excitations from coils 1 and 2, but with the same material perturbations, and applying the above to E = E 1 ± E2 and H = H 1 ± H2 then subtracting we obtain Γ δ ( E H2 ) n = δ H1 H2 + ( δσ + iωδε ) E1 E2 1 + Higher order terms (4) Now taking the magnetic field on Γ to be prescribed and the tangential magnetic field to be measured, the left hand side reduces to Γ δ E 1 H 2. (5) Taking H 2 to be the field due to the excitation of measurement coil 2 with a unit current, this reduces to δv21the change of the induced voltage on the measurement coil 2 when coil 1 is excited. Although one could in principle calculate the sensitivity using a numerical solver for Maxwell s equations by successively making small perturbations to small voxels in the model, this would result in a large number of field solutions, whereas calculation using this formula requires only one E and H solution for each coil. 3 COIL MODEL AND SENSITIVITY There are a number of ways to model the excitation and measurement coils. As in ERT where the conductive electrodes must be modelled, the presence of the coils can affect the fields. Rather than modelling individual turns of copper wire, we will use a simplified model of a coil as a surface, (topologically at least) an open ended cylinder. When used as an excitation coil this surface carries a tangential current J s. This is equivalent to a surface that is perfectly conducting in one direction (angular for a cylinder) and an insulator in another (axial) direction, with each loop fed by a perfect current source. In Figure 1 we see a section of a typical arrangement of excitation and measurement coils for an MIT system. The external screen is modelled as an electrical conductor, which means that the tangential 240

3 component of E vanishes. This could be at a greater distance than shown in the illustration, and where shielding is not possible one would nevertheless need to apply far field boundary conditions to Maxwell s equations. It is important to note that the electromagnetic fields inside the sensor area and between the coils and the shield are coupled so that we can no longer apply the above approximation where measurement is made on a surface, which decouples the problem. Instead we apply the boundary condition n E = 0 on the shield Γ, and include source terms J s for the coils as above. Figure1: MIT array and external screen. excitation coils, detection coils, and a conducting target are shown. Combining (1) and (2) we obtain 1 E + iωξe = iωj s (6) Where ξ is the complex admittivity ξ = σ + iωε. We now consider the case where we excite one coil, suppose that the admittivity is perturbed ξ ξ + δξ with resulting change in the field E E + δe while the current J s is held constant. Our aim is to find the linearized change in the voltage measured on some other coil, so in this derivation we will neglect second and higher order terms. A more detailed derivation along the lines of (Calderon 1980) would prove that this is the Fréchet derivative in suitable normed spaces. Applying (6) to E and E + δe, then subtracting and neglecting higher order terms gives Taking the dot product with E yields 1 δe + iω( δξe + ξδe) = 0. (7) 1 E ( δe) + iωδξe E + iωξe δe = 0 (8) from which we seek to remove the term in δ E (in the interior). We use the identity to give ( E δ E) = E δe + ( E) ( δe) (9) using (6) and subtracting (11) from (9) gives ( δ E E) = δe E + ( δe) ( E) (10) = iωξδ E E iωδe Js + ( δe) ( E) (11) ( E δ E δe E) = E δe + iωξe δe + iωδe (12) eliminating the δ E terms using (8) then integrating over the domain and using Gauss theorem, together with the vanishing of the tangential components of E and δ E on Γ finally gives J s 241

4 ( δ E J s ) dv = δξ( E E) dv (13) which, unsurprisingly has the same right hand side as (4). One can calculate the sensitivity of a voltage measured on coil 2 when coil 1 is excited, following a similar argument to Section 2, ( δ E Js ) dv = δξ( E E ) dv (14) The left hand side here is now the change in voltage induced on our ideal coil provided a unit current is driven in coil 2. It must be emphasised that with non-zero (for example impedance) boundary conditions on the shield Γ the sensitivity would involve boundary terms that are unknown. 4 EDDY CURRENT APPROXIMATION AND NUMERICAL CALCULATION The eddy current approximation, which is valid for sufficiently large σ compared with, ωε is to ignore the displacement current ε E in (2) in the conductive area. In areas (such as the air gap surrounding the coils) the same approximation of ignoring the displacement current results in the magnetostatic approximation H = 0. This does not allow wave propagation effects and is valid provided our system is small compared with the wavelength of electromagnetic waves in air. Our coils are considered, as electro-magnets not radio transmitting antennas. Combining (1) and (2) with the eddy current approximation gives ( ) 1 E + iωσe = iωj s, which is no different from taking ε = 0. The same sensitivity formula (13) holds in this case, although the uniqueness result of (Ola 1993) explicitly assumes ε > 0. Reconstruction of conductivity requires a forward solver so that predicted data can be compared with measured data, and if regularised Gauss -Newton methods are used an efficient scheme for calculation of the Jacobian. Edge Finite Element Method (FEM) has advantages over nodal FEM for vector field computation in eddy current problem (Biro 1999), and it is a powerful tool for simulation of the forward problem in MIT (Hollhaus 2002, Merwa 2003). Two major formulations are known as A,A and A,A-V. For the A,A-V formulation a magnetic vector potential A and scalar potential V are used in the conducting region 1 A + iωξa + iωξ V = J s (15). ξ ( iωa + iω V ) = 0 (16) Here one would normally see V / t but this has been replaced by V for convenience. E in the conductive region is then E = iω A iω V (17) In the A,A formulation by edge FEM, A in conductive region includes the gradient of the electric scalar potential. 1 A + iωξ A = J. (18) In which the electric field in conductive region is ( ) s E = iωa (19) In the Ar, Ar V formulation (Merwa 2003, Hollhuas 2002 ) in which A r is the reduced magnetic vector potential A = A + A (20) s r 242

5 where, A s is the magnetic vector potential caused by source, and equation (17) can be used for calculation of E. The sensitivity formula in equation (14) is verified numerically with the forward solver implemented (Merwa 2003) in A, A V formulation presented by (Hollhuas 2003). In both cases magnetic field can be calculated by r r 1 H = A (21) and the magnetostatic approximation assumed in the non-conducting region. With the A-A formulation and using edge finite element, the sensitivity to a change in the conductivity of the conducting area can be calculated using (14), where the integral becomes the inner product of A fields and the Jacobian can be calculated by performing this integration for a chosen basis for the conductivity perturbation δσ. Using the shape function using edge elements { N e }, the potential A inside each element can be expressed as follows A = { N e }{ A e } (22) where { Ae } are defined along edges. With that the sensitivity term for each element as follow V 2 ij ω i S = = { A e } { N σ k Ii. I j ek e } { N e T j } dv { A e } (23) Equation (23) gives us sensitivity of the induced voltage pairs of coils of element and ek is the volume of element number k and I j i, j where with respect to an and Ii are excitation current for coils. In edge based FEM software we developed for image reconstruction we also can calculate A in all elements by (22) where { N e} is a matrix of shape functions for all elements and { A e} is a vector of the solution of the edge FE solution of the forward problem. We can then use equation (23) simply for region f includes more than one finite element. Then the computation of the Jacobian matrix is matrix vector multiplication for each measurement. The sensitivity calculated in (23) is a complex number S = S r + isi and S r, Si which are real and imaginary parts of the sensitivity term, they are representing the change in V V r, i real and imaginary part of the measurement voltage ( V = V r + ivi ). There are some MIT measurement systems that are measuring phase (φ ) or amplitude of the induced voltage ( V ), the sensitivity term with respect to the phase and the amplitude then is calculated as follow φ Vr Si Vi Sr Sφ = = (24) σ V S amp V Vr Sr + Vi Si = = (25) σ V The sensitivity map will change with the background conductivity (Scharfetter 2002, Scharfetter 2003 ). With a conductive background close to the surface, we have higher eddy currents, and that means those areas have higher sensitivity. Sensitivity also depends on the geometrical configuration of the sensing and exciting coils. For example using a single frequency and fixed shape of the conductive background, for high conductivity the higher eddy current density region is very small and regions very close to the boundary are more detectable, when the conductivity decreases the area of high sensitivity spreads toward the centre, finally when the conductivity goes to zero the more sensitive 243

6 area is no longer effected by the conductive background shape and it is only effected by the geometrical configuration of the sensing and exciting coils. 5 CONCLUSION This paper has presented a derivation of the sensitivity map in MIT. The advantage of this formulation is that it allows efficient computation of the Jacobian matrix in 3D MIT. The shape of the sensitivity map in MIT depends on, each coil s geometry, configuration of sensing and exciting coils, shape of the conductive background, conductivity of the background, and frequency. It is important to understand that the sensitivity is a combination of the electric field created by the magnetic field directly and the electric field created by the eddy current itself. 6 REFERENCES BIRO O, (1999), Edge element formulations of eddy current problems, Computer methods in applied mechanics and engineering, 169, CALDERÓN AP, (1980), On an inverse boundary value problem. Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pp , Soc. Brasil. Mat., Rio de Janeiro. HOLLAUS 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, GRIFFITHS H, (2001), Magnetic induction tomography, Measurement Science and Technology, 12, 8, 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. Volume 24, Number 2, pp OLA P, PÄIVÄRINTA L, and SOMERSALO, E, (1993), An inverse boundary value problem in electrodynamics. Duke Math. J PEYTON A.J., YU ZZ, AL-ZEIBAK S., SAUNDERS NH, AND BORGES A.R., (1995), Electromagnetic imaging using mutual inductance tomography: Potential for process applications, Part. Part. Syst. Charact., vol. 12, SCHARFETTER H, RIU P, POPULO M and ROSELL J, (2002), Sensitivity maps for low-contrastperturbations within conducting background in magnetic induction tomography (MIT) Physiol Meas, 23: SOMERSALO E, ISAACSON D, and CHENEY M, (1992), A linearized inverse boundary value problem for Maxwell's equations. J. Comput. Appl. Math.,42, HOLLAUS K, (2003), Fast Calculation of the sensitivity matrix in magnetic induction tomography by tetrahedral edge finite elements, In Proc. 4th Conference on Biomedical Applications of Electrical Impedance Tomography, Manchester. SCHARFETTER H, HOLLAUS K, MERWA R, (2003), Sensitivity maps for magnetic induction tomography (MIT): Low-contrast perturbations in a background with physiological conductivity, In Proc. 4th Conference on Biomedical Applications of Electrical Impedance Tomography, Manchester. 244