Synopsis

Keywords: Safety, Parallel Transmit & Multiband, Safety margins, SAR estimation

Specific absorption rate (SAR) in parallel transmission (pTx) ultra-high field MRI is typically evaluated by performing offline electromagnetic simulations on a generic human model. Conventionally, a safety margin that addresses the issue of intersubject variability is selected by choice of the 99th or higher percentile of the SAR distribution. In this work, we argue that this approach underestimates the risk of exceeding the SAR limit. As an alternative, we propose the use of Extreme Value Theory (EVT) for controlling the probability of extremal events in the form of large SAR deviations.

Purpose

The specific absorption rate (SAR) in parallel transmission (pTx) is typically evaluated by performing offline electromagnetic (EM) simulations on a generic human model. Conventionally, a safety margin that addresses the issue of intersubject variability is selected by choice of the 99^th or higher percentile of the SAR distribution¹. In this work, we argue that more direct control of SAR exceedances can be achieved using Extreme Value Theory (EVT). EVT states that the maxima of random variables are distributed according to the Generalised Extreme Value (GEV) distribution, analogous to the Central Limit Theorem’s statement that sums of random variables are Gaussian distributed. We demonstrate that modelling maximal SAR events provides control of SAR to acceptable risk levels and, as a result, can assist in making more informed selections of safety margins.

Methods

SAR Distribution
It is assumed that the 10g peak local SAR (pSAR_10g) distribution is known, found for example using the method of Le Garrec et al.¹ Chosen anatomical parameters of interest, such as head breadth, head length, linear shifts and electrical properties, are modelled as Gaussian random variables. A set of representative anatomies is generated following the unscented transform sampling scheme², followed by Q-matrix construction using a commercially available EM simulator. Assuming pSAR_10g changes smoothly with respect to each of the parameters, a multivariate polynomial can be fit to the data. With estimates of the polynomial coefficients, pSAR_10g for the whole population can be inferred. In this work, all reported SAR values are normalised by the average model.

Extreme Value Theory for SAR Modelling
Consider pSAR_10g to be an independent and identically distributed (i.i.d.) random variable, $$$X$$$. For blocks of $$$n$$$ scans (for instance, the number of patients scanned in a given period of time), consider within each block the maximum SAR value, $$$M_{n}=max(X_{1}, ... , X_{n})$$$. According to the Fisher-Tippet-Gnedenko theorem, the distribution of $$$\frac{M_{n}-\mu}{\beta}$$$ converges asymptotically to the GEV distribution³,
$$\lim_{n\rightarrow \infty}P \left( \frac{M_{n}-\mu}{\beta} \le z \right) = GEV(z) \tag{1}$$
for some scale $$$\beta > 0$$$ and location $$$\mu \in \mathbb{R}$$$ parameters. The distribution of SAR maxima, $$$M_{n}$$$, is Gumbel distributed (Type I GEV) with the following cumulative and probability density functions:
$$F(z)=e^{-e^{-z}} \tag{2a}$$
$$f(z)=e^{-(z+e^{-z})} \tag{2b}$$
The parameters $$$\mu$$$ and $$$\beta$$$ can be estimated using maximum likelihood.

Let $$$r_{n,k}$$$ be the smallest pSAR_10g value expected to be exceeded at least once in $$$k$$$ blocks. That is, for every block there is a $$$1/k$$$ % probability that the level $$$r_{n,k}$$$ is exceeded. The risk of underestimating pSAR_10g is encapsulated by the ‘return’ level for $$$k$$$ blocks of size $$$n$$$,
$$P \left( M_{n}>r_{n,k} \right) = \frac{1}{k} \tag{3}$$
The total probability that SAR is underestimated at least once within $$$k$$$ blocks is therefore
$$P \left( \text{SAR is underestimated at least once} \right) = 1 - \left( 1-\frac{1}{k} \right) ^{k} \tag{4}$$

Results & Discussion

Conventional SAR control: A SAR cut-off value that corresponds to 1% in the right tail of a given pSAR_10g distribution (Fig. 1) is equivalent to choosing the 99^th percentile as a safety margin. The probability of underestimating pSAR_10g at least once using the conventional approach rapidly approaches 1 for increasing numbers of scans (Fig. 2). Raising the safety margin by controlling 0.1% in the tail of the SAR distribution has the obvious effect of a larger number of scans before with probability 1 the SAR threshold is exceeded. For a given pTx coil, the question becomes one of how to select threshold values that control SAR exceedances to an acceptable level of risk.

SAR control using EVT: Explicit in the EVT framework is the notion of block size, that is, the number of scans over which to control SAR exceedances. Two Gumbel distributions corresponding to $$$n=175$$$ and $$$n=750$$$ scans are depicted in Fig. 3. As expected, the distribution of the maxima shifts towards the higher values for the larger block size. For each Gumbel distribution, a return value curve was computed (Fig. 4). These curves aid risk profile comparison and the selection of acceptable SAR thresholds. For example, by establishing an acceptable probability of an extremal SAR being exceeded within any given block at $$$1/k=1/100=1$$$ %, we can set a safety margin at 1.72 and 1,62 for blocks of $$$n=750$$$ and $$$n=175$$$, respectively. Those margins would indicate, on average, a single SAR underestimation within $$$k=100$$$ blocks.

Conclusion

In this work, we have proposed the use of Extreme Value Theory as an intuitive framework by which to assess the risk of underestimating SAR and the subsequent choice of an appropriate safety margin. It was demonstrated that the conventional approach based on the raw distribution of pSAR_10g values can result in a substantial probability of exceeding the SAR as scan numbers increase. In contrast, EVT provides the mechanism by which to select acceptable risk levels for given numbers of scans.

Acknowledgements

No acknowledgement found.