Background

SAR is used for evaluating induced heating effects in human tissue at a hot spot. SAR, specifically averaged over a mass of 0.1 gram of human tissue (SAR_0.1g), serves as an input parameter in assessing the potential temperature rise (ΔT) caused by RF energy at the hot spot in some studies¹. When direct ΔT calculation cannot be made, SAR_0.1g is used by some researchers as a metric for comparison between the expected ΔT for hot spot of different types or geometry of implants. In our previous work, we presented analytical solutions for estimating ΔT as a result of the SAR distribution exhibiting spherical symmetry with both uniform and Gaussian distributions². Here we aim at offering a comprehensive analysis of ΔT by examining cases where SAR_0.1g remains constant but the spatial distribution of SAR inside and outside the averaging volume is different.

Methods

We consider for convenience the scenario of a sphere of radius R enclosed in an infinite medium. Due to the inherent symmetry, the temperature distribution inside and outside the sphere depends solely on the radial distance r from the center, provided that the SAR distribution itself adheres to spherical symmetry. If the inner sphere is subjected to a uniform SAR (referred to as SAR₀), and there exists no SAR contribution beyond the boundaries of the sphere, we were able to derive analytical solutions for the transient temperature changes at the center of the sphere, where the maximum ΔT occurs¹.
$$$\triangle{T}(t)=\frac{{\rho}SAR_{0}}{2k}\cdot{R}^{2}\cdot(1-\sqrt{\frac{4{\alpha}t}{{\pi}R^{2}}}e^{-\frac{R^{2}}{4{\alpha}t}}-(1-2\frac{{\alpha}t}{R^{2}})\cdot{erf}(\frac{R}{2\sqrt{{\alpha}t}}))$$$ (1)
where $$$\alpha=k/(\rho\cdot{C})$$$ is the diffusivity coefficient, k, ρ and C are thermal conductivity, density and the specific heat of the tissue, respectively.

For $$$R\rightarrow\infty$$$, the solution is
$$$\triangle{T}(t)=\frac{SAR_{0}}{C}\cdot{t} $$$ (2)
This last case with SAR₀ = SAR_0.1g is label as ‘linear’.
For a SAR source with a Gaussian distribution (standard deviation of σ, b is the SAR value at the center of the Gaussian) located in an infinite medium of uniform properties:
$$$SAR(x,y,z)=b\cdot{e}^{-\frac{x^{2}+y^{2}+z^{2}}{2\cdot{\sigma^{2}}}}$$$ (3)
the analytical transient solution in the center, where maximum temperature occurs, is¹
$$$\triangle{T}(t)==\frac{{\rho}\cdot{b}\cdot\sigma^{2}}{k}\cdot(1-\frac{1}{\sqrt{1+\frac{2\cdot{\alpha}\cdot{t}}{\sigma^{2}}}})$$$ (4)

A sphere with a radius R_0.1g = 2.88 mm (SR_0.1g) includes tissue weighing of 0.1g. We calculated the transient ΔT resulting from SAR₀ =1 W/kg for a set of spheres with radii of {2.88,6.2,13,26,39}mm. The second and the third radii correspond to spheres with a tissue mass of 1g, and 10g, respectively. For the Gaussian distribution case, we derived the transient ΔT for a set of σ ranging from 0.65 mm to 26 mm and maintained SAR_0.1g=1W/kg in SR_0.1g. This set of distributions aims at covering the RF power deposition scenarios due to the presence of various types of implants, from a catheter needle to an orthopedic device. All these scenarios therefore differ in the SAR spatial distribution, yet all of them yielding the same average SAR over SR_0.1g. Tissue properties: ρ=1000kg/m³, C=4000J/kg/K, k=0.5 W/m/K. These values remained constant regardless of changes in tissue temperature.

Results and Discussion

For the Gaussian distributions with σ<1.625mm, power deposition mainly occurred within the sub-sphere and b significantly depends on σ (Figure1). Transient results for uniform Gaussian distributions are shown in Figure3 and Figure4, respectively. When σ>2×R_0.1g, power deposition tends to be uniform across SR_0.1g and remains relatively unaffected by σ. On the contrary, ΔT is significantly affected by σ over the entire studied range. When σ>0.45×R_0.1g , smaller σ values correspond to smaller ΔT values and shorter times (τ_0.632) needed for ΔT(t) to reach $$$(1-e^{-1})\cdot\triangle{T}(t\rightarrow\infty)$$$ (Table1).
The more uniform was the SAR profile, the closer b was to SAR_0.1g. For the sharpest profile, i.e., σ=0.65mm, b=22.9>>SAR_0.1g. The ratio ΔT(t=15 min) for the linear case and sphere with R=39mm was 1.017. Thus, the impact of SAR distribution outside sphere with R=39mm (assuming the RF exposure remains within the level of the assessed sphere) is negligible. On the contrary, consider a set of electrodes located on inner surface of a ring with diameter<15mm. In this scenario, a significant power deposition near other electrodes makes Equation4 unsuitable for estimating ΔT. The 'linear' scenario, SAR₀=SAR_0.1g, represents the most critical situation for t>1.5min assuming no other hotspot is closer than 39mm to the hotspot under assessment.

Conclusion

Key results in our study are:
(i) using SAR_0.1g without consideration of the distribution of SAR within and outside the enclosed volume cannot be a reliable tool for ΔT estimation;
(ii) for Gaussian distributions of power deposition:
(a) smaller σ leads to shorter τ0.632
(b) two regimes: for σ>0.45×R_0.1g smaller σ leads to smaller ΔT, but for σ≤0.45×R_0.1g smaller σ leads to higher ΔT at given time while maintaining the same SAR_0.1g.

Acknowledgements

This work has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 885876 (AROMA project).