A General Three-Dimensional Tensor FDTD-Formulation for Electrically Inhomogeneous Lossy Media Using the Z-Transform

A general three-dimensional tensor finite-difference time-domain (TFDTD) formulation is derived to model electrically inhomogeneous lossy media of arbitrary shapes. The time domain representation of electric losses is achieved using Z-transforms. The regular cubical grid structure is maintained everywhere in the calculation domain by defining a 3-D face-fraction based 3 x 3 permittivity tensor on the interfaces that describes the relationship between the (known) average flux density vector and the (unknown) local electric field vector. For electrically lossy media, this tensor is complex in the frequency domain. However, it can be modified for use with the Z-transform. Only this modified real form is inverted, then transformed from the frequency into the Z-domain, and finally into the time domain. Furthermore, a local interface matrix is used to describe the relationship between the local electric field in the grid node and its counterpart on the other side of the interface. This matrix is complex in the frequency domain for lossy media. By applying the Z-transform, this matrix can also be transformed into the time domain using only real modified matrix elements. The accuracy of the method is confirmed by comparisons with analytical solutions.     

Evaluation of MR-induced hot spots for different temporal SAR modes using a time-dependent finite difference method with explicit temperature gradient treatment

An investigation of magnetic resonance (MR)-induced hot spots in a high-resolution human model is performed, motivated by safety aspects for the use of MR tomographs. The human model is placed in an MR whole body resonator that is driven in a quadrature excitation mode. The MR-induced hot spots are studied by varying the following: (1) the temporal specific absorption rate (SAR) mode ("steady imaging", "intermittent imaging"), (2) the simulation procedure (related to given power levels or to limiting temperatures), and (3) different thermal tissue properties including temperature-independent and temperature-dependent perfusion models. Both electromagnetic and thermodynamic simulations have been performed. For the electromagnetic modeling, a commercial finite-integration theory (FIT) code is applied. For the thermodynamic modeling, a time-domain finite-difference (FD) scheme is formulated that uses an explicit treatment of temperature gradient components. This allows a flux-vector-based implementation of heat transfer boundary conditions on cubical faces. It is shown that this FD scheme significantly reduces the staircase errors at thermal boundaries that are locally sloped or curved with respect to the cubical grid elements.     

Experimental and numerical investigation of feed-point parameters in a 3-D hyperthermia applicator using different FDTD models of feed networks

Experimental and numerical methods were used to determine the coupling of energy in a multichannel three-dimensional hyperthermia applicator (SIGMA-Eye), consisting of 12 short dipole antenna pairs with stubs for impedance matching. The relationship between the amplitudes and phases of the forward waves from the amplifiers, to the resulting amplitudes and phases at the antenna feed-points was determined in terms of interaction matrices. Three measuring methods were used: 1) a differential probe soldered directly at the antenna feed-points; 2) an E-field sensor placed near the feed-points; and 3) measurements were made at the outputs of the amplifier. The measured data were compared with finite-difference time-domain (FDTD) calculations made with three different models. The first model assumes that single antennas are fed independently. The second model simulates antenna pairs connected to the transmission lines. The measured data correlate best with the latter FDTD model, resulting in an improvement of more than 20% and 20 degrees (average difference in amplitudes and phases) when compared with the two simpler FDTD models.     

A thin-rod approximation for the improved modeling of bare and insulated cylindrical antennas using the FDTD method

A general and consistent integral finite-difference time-domain (FDTD) formulation on cubical grids for modeling of cylindrical antennas with or without dielectric coating is derived. No additional grid points or modifications of the integral paths are necessary. Instead, effective material properties are modified in the FDTD grid. Thus, even for insulated antennas, the simple cubical structure is maintained. Special integral factors are defined on cubical elements, which take into account the behavior of fields in all directions in the neighborhood of the antenna. Applying these factors to the gap region and along the antenna's axis allows a correct modeling of the influence of the antenna's thickness. Furthermore, integral factors derived for the antenna's ends improve the modeling of the antenna's length. The accuracy of the method is confirmed by a systematic comparison with analytical and numerical results.     

Investigation of static and quasi-static fields inherent to the pulsed FDTD method

Demonstrates that trailing dc offsets, which can affect E- or H-fields in finite-difference time-domain simulations, are physically correct static solutions of Maxwell's equations instead of being numerically induced artifacts. It is shown that they are present on the grid when sources are used, which generates nondecaying charges. Static solutions are investigated by exciting electric and magnetic dipoles models with suitable waveforms.     

A 3-D tensor FDTD-formulation for treatment of sloped interfaces in electrically inhomogeneous media

A general integral finite-difference time-domain (FDTD) formulation on cubical grids for the modeling of electrically inhomogeneous media of arbitrary shape is derived. The simple cubical structure is maintained and no modifications of the integral paths are necessary. Instead, a three-dimensional tensor relationship between the average electric flux density and electric field is determined beforehand and used during the simulation to account for discontinuities in the neighborhood of sloped interfaces. The accuracy of the method is confirmed by comparisons with analytical solutions.     

A clinical water-coated antenna applicator for MR-controlled deep-body hyperthermia: a comparison of calculated and measured 3-D temperature data sets

A magnetic resonance (MR)-compatible three-dimensional (3-D) hyperthermia applicator was developed and evaluated in the magnetic resonance (MR) tomograph Siemens MAGNETOM Symphony 1.5 T. Radiating elements of this applicator are 12 so-called water coated antenna (WACOA) modules, which are designed as specially shaped and adjustable dipole structures in hermetically closed cassettes that are filled by deionized water. The WACOA modules are arranged in the applicator frame in two transversal antenna subarrays, six antennas per subarray. As a standard load for the applicator an inhomogeneous phantom was fabricated. Details of applicator's realization are presented and a 3-D comparison of calculated and measured temperature data sets is made. A fair agreement is achieved that demonstrates the numerically supported applicator's ability of phase-defined 3-D pattern steering. Further refinement of numerical models and measuring methods is necessary. The applicator's design and the E-field calculations were performed using the finite-difference time-domain (FDTD) method. The calculation and optimization of temperature patterns was obtained using the finite element method (FEM). For MR temperature measurements the proton resonance frequency (PRF) method was used.     

A volume-surface integral equation method for solving Maxwell'sequations in electrically inhomogeneous media using tetrahedral grids

Starting with the solution of Maxwell's equations based on the volume integral equation (VIE) method, the transition to a volume-surface integral equation (VSIE) formulation is described. For the VSIE method, a generalized calculation method is developed to help us directly determine E fields at any interface combination in three-dimensional (3-D) electrically inhomogeneous media. The VSIE implementation described is based on separating the domain of interest into discrete parts using nonuniform tetrahedral grids. Interfaces are described using curved or plane triangles. Applying linear nodal elements, a general 3-D formulation is developed for handling scatter field contributions in the immediate vicinity of grid nodes, and this formulation is applicable to all multiregion junctions. The special case of a smooth interface around a grid node is given naturally by this formulation. Grid nodes are split into pairs of points for E-field calculation, and node normals are assigned to these points. The pairs of points are assigned to the elements adjoining the grid node. For each pair of points, the correct field jumps on the interface are given by a surface integral over the polarization surface charge density     

3-D computation of E fields by the volume-surface integral equation(VSIE) method in comparison with the finite-integration theory (FIT)method [clinical hyperthermia application]

An algorithm has been developed for calculation of 3-D electric (E) fields by the volume-surface integral equation (VSIE) method. Integration over surface elements is performed using elementary analytical formulas, assuming a linear interpolation of surface charges. Grid points at electrical interfaces are split off, taking into account the E field behavior at these contours, specifically at sharp bends and multimedia junctions. Averaging procedures are utilized in order to avoid undefined or infinite values at critical points. The VSIE is solved by iteration using the GMRES (general minimum residuum) solver on a SUN workstation SPARC-IPX or Cray XMP, whereby convergence speed decreases considerably as the heterogeneity of the problem increases. Results for 3-D test cases (plane wave illuminating a layered cylinder) generally agree well with the finite-integration-theory (FIT) method if high E field gradients occur perpendicular to electrical boundaries. The VSIE method predicts slightly higher E fields only in critical regions. On the other hand, the FIT method at present is more efficient with respect to computation time for large domains with high cell numbers (>100000 cells)