Biomedical Engineering Reference
In-Depth Information
As the phases and amplitudes of the individual antennas are not
known in advance (this is the task of the optimization step), the
common approach is to individually compute the fields induced
by each antenna and compute the total field for the entire appli-
cator by linear superposition of the scaled and correctly phase-
shifted fields of each antenna. The remaining challenge is to
take into account the mutual coupling of the antennas and their
interaction with the amplifiers.
Various techniques have been used to solve the Maxwell equa-
tions: the Finite-Difference Time-Domain (FDTD) method [29,
82, 141, 149, 160, 165] and the Finite-Element method (FEM)
[140, 189] (based on the differential formulation of the Maxwell
equations), the Weak Form of the Conjugate Gradient Fast
Fourier Transformation (WF-CGFFT) method [184, 198], and the
Volume Surface Integral Equation (VSIE) method [192] (based
on the Integral formulation of the Maxwell equations). FDTD
and WF-CGFFT mostly use rectangular meshes, while FEM
and VSIE typically apply tetrahedral grids. FDTD on rectilinear
meshes is ideally suited for highly inhomogeneous meshes and
scales well with the mesh size. It offers simple parallelization and
hardware acceleration. Applying transient excitations, it provides
information for multiple frequencies in one run and potentially
reduces convergence issues related to high-Q applicators. FEM on
unstructured meshes is naturally conformal, thus avoiding issues
related to staircasing. It furthermore allows for simple error esti-
mation. FDTD and FEM results have been compared in [158].
In order to correctly capture interface effects and hot spots,
it has been shown that high-resolution modeling is necessary,
much finer than the coarse discretizations of only 10 mm often
applied in HTP [167]. [128] uses graded rectilinear meshes that
allow for sub-millimeter resolution where required (e.g., antenna
feed points) without resulting in excessive numbers of voxels
(volume pixels), resulting in a computational effort in the order
of minutes at 435 MHz using moderately priced personal com-
puters with GPU-based hardware acceleration and optimized
grid generation [122, 128].
Other techniques speed up FDTD simulations by allowing
larger time steps (e.g., using Alternating-Direction Implicit [ADI]
time integration [28, 122, 121]) or by calculating on a coarser
grid and using special post-processing techniques to approxi-
mate high-resolution information (e.g., using quasistatic zoom-
ing [166-169] or a special interpolation technique that handles
interfaces in a physically correct manner [119]). Analytical local
solutions can be used to improve thin antenna handling without
resorting to high-resolution simulations [43, 54, 118, 122].
One of the major sources of error in predicting the SAR pattern
is insufficient accounting for the scattered field on the antenna
elements. The feeding networks with the matching circuits have
to be considered correctly and methods have been developed to
couple simulations with a model of the feeding network [116,
117, 122]. These methods improve the predictive value of simula-
tions, but the feeding network still remains a major source of
uncertainty. Another source of error is the staircasing errors of
FDTD [1, 20]. Conformal subcell techniques increase the accu-
racy and help reduce staircasing errors [12, 122]. Various other
problems have been encountered that compromise the accuracy
of energy deposition predictions. For example, the high-Q cav-
ity formed by an annular phased array hyperthermia applicator
can lead to an extremely long convergence time, mode flipping
can occur before convergence [149], and a whispering gallery
effect has been reported where the water bolus acts in a lens-like
manner, refocusing the energy from one antenna on the opposite
patient side. The last point is a physical effect and has since been
demonstrated experimentally.
7.5 thermal Simulations
A large number of thermal models have been developed (see
reviews in [3, 37, 104, 188]) that essentially differ in their descrip-
tion of the impact of perfusion. Most of the approaches are based
on early work by Pennes [143], which links temperature increase
over time to external heat sources, metabolic heat generation,
heat diffusion, and a homogeneous heat sink term that is pro-
portional to tissue perfusion as well as the difference between
local temperature and perfusing blood temperature. Pennes
interprets this homogeneous heat sink term as being due to rapid
equilibration occurring at the level of the microvasculature.
This assumption has since been questioned as much of the heat
exchange already occurs between larger counter-current vessels.
This has led to the development of more complex heat equations
(e.g., the Weinbaum-Jiji model, see [3]) that often couple a series
of differential equations describing the heat distribution of the
tissue as well as the arterial and the venous blood [3, 68]. Under
certain conditions, the effect of counter-current vessels can be
modeled by replacing the thermal conductivity with an effective
thermal conductivity (e.g., the simplified Weinbaum-Jiji model,
see [3] and a critical analysis of this approach in [187]). The pre-
dictions of the Pennes model can differ significantly from those
obtained using an effective thermal conductivity approach
[95, 188]. Various authors have proposed models interpolat-
ing between the Pennes and the effective thermal conductivity
approaches with weights depending on the location or the local
perfusion [18, 188]. The Pennes term has been reinterpreted to
actually model heat exchange due to bleed-off, and is currently
believed to be valid in the vicinity of large nonequilibrated ves-
sels in addition to tissue where only microvasculature is present.
A detailed multiscale analysis has provided additional justifi-
cation for the Pennes term [48, 49]. Despite its shortcomings,
the Pennes model continues to be the one used most frequently,
which can be partly justified by the large amount of available
experimentally determined tissue parameters for the Pennes
model. Each model has its own range of validity [3]. [197] has
studied vessels with diameters in the order of 0.1 mm and has
shown that the detailed vessel branching pattern has an impor-
tant influence on how the perfusion effect should be modeled.
Others have criticized the Pennes equation for ignoring the
directivity of blood flow. Variants including a flow-field term
have therefore been introduced [3], sometimes based on a porous
medium assumption [68, 99]. Another important fact ignored
by the Pennes equation is the discreteness of blood vessels and
