\Omega_{\rm T}\\
&& \hspace{-1.3cm} (\ahat(\r) \!\!+\!\! \atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le (1+\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm T}\\
&&\hspace{-1.3cm} R\in \mathcal{R}.\end{aligned}$$ Both of the alternative formulations described above can be addressed following the steps derived in Appendix \[app A\] by using $\mathcal{S}$-lemma since we are still dealing with quadratic constraints.

In the next section, we illustrate the reference performance of a nominal scenario where the steering vectors are perfectly known. Then we observe how much power can leak to the healthy tissue and cause damages when subject to uncertain steering vectors. Finally, we evaluate the proposed robust scheme in terms of improving the power deposition along our stated design goals.

Numerical Results {#sec:numerical_results}
=================

To illustrate the performance of the proposed robust scheme, we consider a 2D model of the organ at risk. Here, similar to [@08:Guo], we focus on the ultrasonic hyperthermia treatment for breast cancer where a 10-cm-diameter semi-circle is assumed to model breast tissues with a 16-mm-diameter tumor embedded inside. The tumor center is located at $\r_0 = [0 \hspace{.1cm} 34]^T$ mm. Fig. \[fig:2D\_model\] shows this schematic model. We consider a curvilinear array with $M=51$ acoustic transducers and half wavelength element spacing. Acoustic 