Biomagnetic inverse problems:

Transcription

1 Biomagnetic inverse problems: Magnetic resonance electrical property tomography (MREPT) and magnetoencephalography (MEG) 2018 Aug. 16 The University of Tokyo Takaaki Nara 1

2 Contents Measurement What is reconstructed physically mathematically MREPT Magnetic field inside the body Electrical conductivity and permittivity Coefficients of the timeharmonic Maxwell equation MEG Type of inverse problem Active Passive Magnetic field outside the head Current in the brain Source of the Poisson equation 2

3 I. Magnetic resonance electrical property tomography (MREPT) Joint work with M. Fushimi (The University of Tokyo) 3

4 Background Electrical conductivity and permittivity imaging inside the human body pathological information: breast carcinoma (tumor) exhibits conductivity changes up to a factor of 10. [Katscher, 2009] Small tumor/cancer tissues can be detected 4

5 Magnetic resonance electrical property tomography (MREPT) `Birdcage coil generates the rotating magnetic field with a frequency of e.g. 128MHz MRI scanner What is measured: magnetic field inside the body What should be reconstructed: electric conductivity permittivity 5

6 How to measure the magnetic field using MRI 1) Actively flip the magnetic moment 2) Fourier transform of the xy comp. of magnetic moment is measured. z y The flip angle is affected by the magnetic field. amplitude x phase 8

7 Governing equation: time harmonic Maxwell s equations Faraday s law Ampere s law Silver Muller radiation condition z conductivity : admittivity permittivity MREPT inverse problem: given inside the body, reconstruct. 9

8 Conventional method (1) `standard method in U.Katscher, IEEE Tr. Medical Imaging, By eliminating, a non linear PDE for is obtained: By neglecting, a reconstruction formula is obtained: 10

9 Significant errors occur where J.K.Seo, IEEE Tr. Medical Imaging, Conductivity, S/m true reconstructed 11

10 Conventional methods (2) for inohomogeneous Y.Song and J.K.Seo, SIAM J. Appl. Math, 2013: iterative method for semilinear elliptic PDE for 2D case H.Ammari et al., Inverse Problems, 2015: iterative method for semilinear elliptic PDE for 3D case F.S.Hafalir, et al., IEEE Trans. Med. Imaging, 2014: FEM for a linear PDE for 2 nd order spatial derivatives are required E.Balidemaj, et al., IEEE Trans. Med. Imaging, 2015: integral equation based approach Iterative algorithm E and H without the human body should be computed. 12

11 Objective Derivation of an explicit reconstruction formula for inhomogeneous admittivity from the measured magnetic field 1. Derivation of a direct reconstruction formula for the 2D problem 2. Stability and regularization 3. Extension to the 3D problem 13

12 1. Derivation of a direct reconstruction formula Assumptions: 1) 2) Usual assumptions from the structure of `birdcage coils does not change w.r.t the z axis z First, we consider the 2D problem and derive a direct reconstruction formula. Second, we remove Assumption 2) and propose an iterative algorithm. 14

13 Dbar equation forms of Maxwell s equations Define the compelx derivatives: Let Then, under the two assumptions, Maxwell s equations in are written: Dbar equation of Ampere s law Dbar equation of Faraday s law 15

14 Ampere s law Complex derivative of the measured magnetic field z component of the electric field Faraday s law We use Faraday s equation to represent in terms of 16

15 Generalized Cauchy formula for Dbar equations, : domain bounded by a Jordan contour satisfies a Dbar equation: (inhomogeneous Cauchy Riemann system) Generalized Cauchy formula Cf. (Cauchy Riemann equation) Cauchy formula 17

16 Generalized Cauchy formula for Dbar equations, : domain bounded by a Jordan contour satisfies a Dbar equation: (inhomogeneous Cauchy Riemann system) Generalized Cauchy formula Faraday s law The z comp. of the electric field can be expressed as 18

17 By substituting to Ampere s law, we have Dirichlet boundary value of 19

18 Direct reconstruction formula of the admittivity Given in and on, then, the adtmivity at an arbitrary point is reconstructed by 20

19 How to give the boundary value of Scenario 1: Take as a whole body and measure near the body surface Scenario 2:Take as a local domain inside which a small cancer may exist and give a normal value of on :body body 21

20 Numerical simulations Three cylinders (conductivity = 1S/m, diameters = 10mm, 20mm, 30mm) in a cylinder (conductivity = 0.5 S/m). Forward solution was computed by an FEM software (COMSOL). f=123mhz (corresponding to 2.89T) : 82mm x 82mm 82mm Pixel: 1.4mm x 1.4mm 22

21 Singular integral For computation of and, Gaussian filter + Savitzky Golay Filter were used. (2 nd degree polynomial approximation) 23

22 Reconstruction (noiseless) Proposed Conventional (standard) The three domains were well reconstructed. The large errors were observed at the boundary. 24

23 Reconstruction (1%noise) Proposed Conventional (standard) 25

24 Phantom experiments ROI Abnormal 1S/m Normal 0.5 S/m 3T MRI Phantom 26

25 Proposed Conventional (standard) discontinuous jump of the conductivity robustness against noise 27

26 2. Stability and regularization The relative error of : Measurement noise is enhanced by division by 28

27 Zero point of the electric field: Computed in the phantom exp. Reconstructed conductivity When using the birdcage coil, becomes zero at around the center of body. Reconstruction error becomes large due to division by the small 29

28 Regularization Setting sample positions, the point by point reconstruction formula is expressed as TV regularization can be easily incorporated: 30

29 `Physical regularization :control of the location where Putting an object with high permittivity The zero of Ez can be moved. 31

30 true explicit with regularization 32

31 3. Extension to the 3D problem Assumptions: z 1) 2) Usual assumptions from the structure of `birdcage coils does not change w.r.t the z axis Ampere s law (z compnents) + i Ampere s law (xy compnent) Faraday s law (xy compnents) 33

32 Extension without the assumption The Dbar equation is changed to: Iterative Algorithm: 1) Use the direct reconstruction formula to obtain an initial estimate. 2) Compute with the reconstructed. 3) Go to 2). 34

33 Numerical simulations 64MHz, plane wave input Z axis sampling: 2mm : central difference Gaussian filter: 3x3x3, sd=1pixel Convergence criteria: z=14 mm z=6 mm z 20 mm 35

34 Noise 1% 36

35 Summary of part I For the Magnetic Resonance Electrical Property Tomography (MREPT) inverse problem, the direct reconstruction formula was derived where the admittivity was explicitly reconstructed in terms of the magnetic field. Ill posedness can be physically controlled. TV regularization is easily incorporated. An iterative algorithm for the 3D problem was proposed based on the direct reconstruction formula. 37

36 II. Magnetoencephalography (MEG) inverse problem Joint work with K. Kabashima (The University of Tokyo) K. Watanabe (The University of Tokyo) K. Amano (CiNet, NICT) 38

37 II. Magnetoencephalography (MEG) inverse problem Neural currents in the brain forward problem inverse problem magnetic field Applications analysis of the brain function diagnosis of epilepsy T.Takeda, Univ. Tokyo 39

38 MEG inverse problem Spherically head model Given at outside the head, determine. 40

39 Conventional approaches to MEG inverse problems Parametric approach Equivalent current dipoles Imaging approach Current distribution on gird points The size and shape of a source domain cannot be determined. L2 reg.: over smoothed source L1 reg.: patch is not reconstructed 41

40 L1 norm regularization Reg. parameter was chosen by the generalized Cross Validation The reconstructed source is scattered around the true patch source 42

41 Objective Propose a method that determines patch sources, i.e., domains with explicit boundaries. Difficulty How do we express focal domains on the cortical surface? How do we move them in optimization constrained on the cortical surface? 43

42 Mapping from a sphere to the cortical surface mapping We construct a mapping from a unit sphere to the cortical surface Domains on the cortical surface are mapped from domains on the sphere 44 Optimization of the shapes of the domain on the sphere is much easier.

43 Conformal mapping from a cortical surface to a sphere X.GU, Y.Wang, T.F.Chan, P.M.Tompson, and S T.Yau, Genus zero surface conformal mapping and its application to brain surface mapping, IEEE Trans. Biomedical Eng., 23, :points on the cortical surface :points on the unit sphere Since the genus of is zero, : harmonic mapping conformal mapping : harmonic mapping 45

44 Discrete harmonic mapping 1) Initial points on the unit sphere are generated by the Gauss map: 2) Renew the points on the unit sphere by the Ricci flow: Mesh generated by MRI s.t 46

45 Parametric expression of domains mapping center radius We assume a domain on the cortical surface mapped from the circle on the sphere. The shape is limited, but the spatial extent of the domain is well represented with a few parameters. In order to express more complex domain, the radius of a domain on the sphere is expressed with the truncated Fourier series. 47

46 Parametric expression of domains mapping center radius Cost function: Unknowns: Optimization in the cartesian products Global minimum can be efficiently obtained by a Lipschitzian optimization algorithm. 48

47 Global optimization for parameters in a hyper rectangular space D. R. Jones, C. D. Perttunen, and B. E. Stuckman, Lipschitzian Optimization Without the Lipschitz Constant, J. Optim. Theory Appl., vol. 79, no. 1, pp , Global minimum in a Cartesian product can be efficiently searched. 49

48 Parametric expression of domains mapping center radius Cost function: Unknowns: the center and radius of the circle on the sphere: Optimization in the cartesian products Global minimum can be efficiently obtained by a Lipschitzian optimization algorithm. 50

49 Numerical simulations Sensors: 306ch (Elekta Neuromag) Noise: 10% Gaussian 51

50 True Proposed L1 52

51 radius 3deg MEG for visual stimulus Joint work with K. Amano (CiNet) and K. Watanabe (U. Tokyo) fmri result: Activated domains get larger as the size of stimulation increases. 6deg 12 deg

52 7.2 e e e 1 radius on the 54 sphere [rad]

53 Summary of part II Domains on the cortical surface are expressed as images mapped from the unit sphere. The center and radius of a circle on the unit sphere are obtained by a Lipschitzian optimization algorithm. A focal and connected source domain can be identified by using real visual stimulation / epileptic MEG data. 55