Three-dimensional MR Reconstruction Of Magnetic Susceptibility By The Variational Born Iterative Method Based On The Magnetic Field Volume Integral Equation

Quantitative susceptibility mapping(QSM)becomes more and more important in the clinical research and treatment.It can provide a non-invasive means and is in favor of diagnosis and monitoring of cerebrovascular disease and neurodegenerative disorder.The traditional QSM takes the phase information as the original data and establishes the relationship between the phase information and the magnetic susceptibility distribution based on the quasi-static approximation.Furthermore,the traditional QSM makes weak contrast approximation.However,it is well known that the traditional QSM provides only the "relative" quantification of the magnetic susceptibility rather than the absolute physical quantity.This is due to that the phase and frequency values of MRI are relative and are affected by the accuracy of the data preprocessing.What's more,there is a lack of common and reliable frequency reference in in vivo imaging.In order to provide high quality and high contrast magnetic susceptibility mapping,a three-dimensional MR reconstruction method for magnetic susceptibility based on the magnetic field volume integral equation with the variational Born iterative method(VBIM)is developed.The magnetic susceptibility is reconstructed from the positive rotating magnetic field component H1+ of the RF field acquired by B1 mapping.The implement of the stabilized biconjugate gradient fast Fourier transform(BCGS-FFT)method in the forward problem to solve for the magnetic field and the conjugate gradient fast Fourier transform(CG-FFT)method in the inverse problem to reconstruct the magnetic susceptibility distribution significantly reduces the memory cost and CPU time.The proposed method is based on Maxwell's equations and doesn't make any weak-contrast approximation,so it is more accurate in the reconstruction of high-contrast susceptibility.Besides,the accuracy of the result only depends on the data acquisition process and the method itself because the method doesn't have data preprocessing.Firstly,the methods of solving the electromagnetic(EM)inverse scatteringproblem are studied,which are mainly the volume integral equation(VIE)and the Born approximation(and the extended Born approximation).The Krylov subspace method(BCGS-FFT method and CG-FFT method)which can be used to solve the forward solution is given.And the simulation results of Born approximation method and BCGS-FFT method are provided to verify their effectiveness and accuracy.Secondly,in order to solve the nonlinear relation in the VIE,the methods to solve the nonlinear inverse problem are studied,including the Born iterative method(BIM),the distorted Born iterative method(DBIM)and the variational Born iterative method(VBIM).The similarities and differences between the above three methods are compared.And the corresponding cost functions are given as well.Then,this paper proposes a 3D MR reconstruction method of magnetic susceptibility by the VBIM based on the magnetic field VIE.The detailed formula is given corresponding to the idea of this method.This part mainly includes the weak-form discretization of the magnetic field VIE,the acquisition of H1+ field and the steps of the VBIM.Finally,the validity and accuracy of the method is validated by the 3D human brain model and the high susceptibility model.The result reconstructed using the Born approximation is shown to make clear the effect of the weak contrast approximation.In addition,the results under 40 dB,20 dB,10 dB AWGN are investigated to verify the noise immunity of this method.Moreover,this study is to lay the foundation for reconstructing the relative permittivity,conductivity and permeability simultaneously.By combing the QSM and EPT based on the volume integral equations,reconstructing the relative permittivity,conductivity and permeability simultaneously can be achieved.