E. F. Meliado1,2,3, A. Sbrizzi1,2, C.A.T. van den Berg2,4, B. R. Steensma1,2, P.R. Luijten1, and A. J. E. Raaijmakers1,2,5
1Department of Radiology, University Medical Center Utrecht, Utrecht, Netherlands, 2Computational Imaging Group for MR diagnostics & therapy, Center for Image Sciences, University Medical Center Utrecht, Utrecht, Netherlands, 3Tesla Dynamic Coils BV, Zaltbommel, Netherlands, 4Department of Radiotherapy, Division of Imaging & Oncology, University Medical Center Utrecht, Utrecht, Netherlands, 5Biomedical Image Analysis, Dept. Biomedical Engineering, Eindhoven University of Technology, Eindhoven, Netherlands
Synopsis
The introduction of a linear safety factor to address peak
local SAR uncertainties (e.g. inter-subject variation and modeling
inaccuracies) bears one considerable drawback: it often results in over-conservative
scanning constraints. In this work, we present a more efficient and
probabilistic approach to define a variable safety margin based on the
conditional probability density function of the true peak local SAR value given
the estimated peak local SAR value.
Results show that the proposed approach leads to
a reduction of peak local SAR overestimation up to 30%.
PURPOSE
The
assessment of 10g-averaged peak local SAR (pSAR) for MR imaging with
multi-transmit arrays is typically done by off-line numerical simulations using
one or several generic models1. Subsequently, the post-processed simulation
results can be used for on-line pSAR estimation2,3. To compensate for the discrepancies
between the generic simulation models and the subject under examination an additional
safety factor is required to avoid pSAR underestimation. This approach is
suboptimal as illustrated by Figure 1. Here, for 23 simulation models4 with each 250 random phase settings, the true
pSAR (pSART) is plotted versus the estimated pSAR (pSARE,
based on the generic model Duke5). pSARE is often lower
than pSART so a safety factor is applied which transforms pSARE
into a corrected pSARE value (pSARE,C) as indicated by
the red line in Figure 1. pSAR underestimation is now avoided as the scatter
points are below the red line. However, this representation clearly shows that
for large pSARE values, the pSARE,C heavily overestimates
the pSART. In this study we
present the conditional safety margin as an alternative approach for the linear
safety factor based on probability theory.THEORY
The scatter plot in Figure 1 represents samples
from the joint probability distribution of pSARE and pSART,
which is described by the probability density function $$$f_{E,T}(pSAR^E,pSAR^T)$$$.
When we estimate a pSARE value, we would like to know the
probability of pSAR underestimation i.e. $$$P(pSAR^T>pSAR^E)$$$. Thus, we
need to know the conditional probability density function $$$f_{T|E}(pSAR^T|pSAR^E)$$$
which is the probability distribution of pSART for a
given pSARE. This function is calculated in Eq. [1] where $$$f_E(pSAR^E)$$$
is the (marginal) probability density function of pSARE (regardless
of pSART). See Figure 2.
$$f_{T|E}(pSAR^T|pSAR^E)=\frac{f_{E,T}(pSAR^E,pSAR^T)}{f_E(pSAR^E)}$$ [1]
Now
the probability $$$P_{underest}$$$ that pSART is actually larger
than some estimated value pSARE=E1 is given by [2].
$$P_{underest}=P(pSAR^T>E_1|pSAR^E=E_1)=\int_{E_1}^{\infty}
f_{T|E}(pSAR^T|pSAR^E=E_1)dpSAR^T$$ [2]METHODS
The probability density
functions can be estimated from the observed data. Therefore, a representative set
of (pSART,pSARE) samples is generated by means of our database
of 23 subject-specific models4
with an 8-fractionated dipole array6,7 for
prostate imaging at 7T.
For each of the 23 models,
driving the array with uniform input power (8×1W) and random phase settings,
250 pSART values are determined. Three state-of-the-art methods are used
to estimate the pSARE values: 1) using just the generic model “Duke”5; 2) using our model
library to assess the maximum pSAR value over all models; 3) using a recently
published deep learning-based method, where the SAR distribution is determined
by a deep learning network with an acquired B1+-map as an
input8
Conditional Safety Margin
The pairs (pSARE,pSART)
are samples from the joint probability density function $$$f_{E,T}(pSAR^E,pSAR^T)$$$, which can be modelled as a
Gaussian mixture (Figures 2.a,b). All pSARE
values are samples from the marginal probability density function $$$f_E(pSAR^E)$$$
and follow what seems to be a Gamma
distribution (Figure 2.c).
Therefore, using equation [2], the
conditional probability density function can be calculated
for each possible estimated pSARE=E1 value and used to numerically determine
a corrected pSARE,C value with a probability of underestimation equal
to an arbitrary (small) ε (Figure 2.d).
$$$Find$$$ $$$pSAR^{E,C}$$$ $$$such$$$ $$$that$$$
$$P(pSAR^T>pSAR^{E,C}|pSAR^E=E_1)=\int_{pSAR^{E,C}}^{\infty}
f_{T|E}(pSAR^T|pSAR^E=E_1)dpSAR^T=\epsilon$$ [3]
This corrected pSARE,C value
represents our conditional safety margin (CSM).
Performance
Evaluation
To assess the performance of the conditional safety margin, a larger
test set (23×1000=23000 test samples) is generated and the mean overestimation
of the conditional safety margin is evaluated and compared to the mean
overestimation of the corrected pSAR values using the linear safety factor
(LSF)8.RESULTS AND DISCUSSIONS
To
determine the joint and
marginal probability density functions, for each considered
pSAR estimation method, the scatter plot pSART versus pSARE
is fitted with a mixture of five Gaussian distributions, and the pSARE
histogram is fitted with a Gamma
distribution. Subsequently, the conditional safety margin is
evaluated for ε=0.001.
Figure
3.a shows the scatter plot pSART versus pSARE using the
generic model, and the corrected pSARE,C values. The red line is the corrected pSARE,C
by LSF and the green line is the corrected pSARE,C by CSM.
Similar
scatter plots for the model library and the deep learning method are shown in Figures
3.b and 3.c respectively.
Figure
3 clearly shows that the suggested approach is reducing overestimation because
the green line remains closer to the distribution of actually expected pSART
values. The mean overestimation over all models and drive settings for each
method is reported in Figure 3.d.
Results show that the conditional
safety margin allows to reduce the mean overestimation from 108% to 89% using
the generic model (18% reduction), from 104% to 83% using the model library
(20% reduction) and from 57% to 41% with the deep learning method (28%
reduction).
Note that these are
averaged values. For individual cases with high pSARE values, the
reduction in overestimation may reach up to 40%. CONCLUSIONS
An
alternative, probabilistic approach for peak local SAR correction is proposed.
It allows to define a variable safety margin to deal better with peak local SAR
uncertainties. It reduces the average overestimation up to 30% compared to the
more conventional safety factor.Acknowledgements
No acknowledgement found.References
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