Quantities of Interest

1. Time-Averaged Power Absorbed (in each Tissue and Total)

One of the key quantities of interest in the benchmark is the time-averaged power absorbed in each tissue

$$\begin{equation} \Pt{}(\omega)=\iiint\limits_{V_\text{tissue}}\overline{P}(\vec{r},\omega)dV\quad (\text{W}). \end{equation}$$

Here, $\omega$ is the angular frequency of interest and the pointwise time-averaged absorbed power density

$$\begin{equation} \overline{P}(\vec{r},\omega)=\frac{\omega}{2\pi}\int\limits_{2\pi/\omega} \sigma(\vec{r},\omega)\vec{E}(\vec{r},t)\cdot\vec{E}(\vec{r},t) dt = \frac{1}{2}\sigma(\vec{r},\omega)\vecT{E}(\vec{r},\omega)\cdot\vecT{E}^*(\vec{r},\omega) \quad (\text{W}) \end{equation}$$

is integrated over the volume of the tissue $V_\text{tissue}$. Integrating over all of the tissues yields another quantity of interest: the total time-averaged power absorbed by the models $\Ptotal$, which is supplied for each problem in the benchmark.

2. Cell-Averaged Time-Averaged Absorbed Power Density

The absorbed power distribution is also important for BioEM applications. Indeed, obtaining an accurate $\Ptotal$ does not guarantee that the distribution is accurate [1]. It is difficult, however, to compare the absorbed power distribution computed by different methods because the benchmark problems should be independent of the mesh used (e.g., meshes could be surface or volume elements, tetrahedra, voxels, or higher-order cells). Nonetheless, it is important to compare the distributions objectively to reference results whenever possible. To do this, we propose enclosing the model volume in a grid comprised of uniform 1×1×1 mm³ cells and reporting the cell-averaged, time-averaged absorbed power density in each cell k,

$$\begin{equation} \Pc{}(\omega)=\frac{1}{V_k}\iiint\limits_{V_k}\overline{P}(\vec{r},\omega)dV\quad (\text{W}). \end{equation}$$

Note that the integral in (3) must be carefully evaluated to ensure that the error from numerical integration is not greater than the error in the computation that yields $\overline{P}(\vec{r},\omega)$, especially given that there will be material discontinuities in the cells due to the curved layers.

3. Co-Polarized Bistatic Radar Cross Section

The co-polarized bistatic radar cross section, $\RCST{}$, is also a quantity of interest. A sample θ-cut (at φ=180°) is supplied for each benchmark problem.