C. Thermal simulations

We use the bioheatExact function in the k-Wave toolbox (MATLAB software) to solve the Pennes bioheat equation in three dimensions.43 The Pennes bioheat equation is given by

where ρ is the tissue density, C is the tissue specific heat capacity, T is the tissue temperature, k is the tissue thermal conductivity, ρb is the blood density, Wb is the blood perfusion rate, Cb is the blood specific heat capacity, Tb is the arterial blood temperature, and Q is the power density contributed by radio frequency absorption. Q is calculated separately for injected and non-injected regions. For non-injected regions, Q=12σE2, where σ is the conductivity of the background tissue (in S/m) and E is the applied electric field strength (in V/m). For injected regions, Q=12σE2×(5.18×10−4Fabs), where 5.18×10−4 is the volume fraction of gold in a 10 mg/ml saline suspension and Fabs is the relative absorption ratio of a single gold prolate spheroid compared to the background tissue.

We refer to Ref. 44 for the thermal properties of both breast cancer and healthy breast tissue. We refer to Refs. 45 and 46 for the thermal properties of myocardium. We refer to Ref. 47 for the thermal properties of the blood. For convenience, these values are reproduced in Table II. The domain size for all simulations is 10 × 10 × 10 mm³, and we use a mesh with 0.1 mm step size to discretize the domain.

Parameters used in Pennes bioheat calculations.

The bioheatExact function assumes periodic boundary conditions. Although these boundary conditions prevent heat from leaving the domain, this is a good assumption over the ∼100 ms timescales used in the simulations. Using the thermal diffusivity of water (D= 1.4 × 10⁻⁷ m²/s, water used to approximate tissue), the characteristic diffusion length LD over 100 ms is given by LD=4Dt≈0.2 mm. Thus, the intense heat generated within the injection region in the center of the domain will not have time to reach the boundaries of the simulation over the timescales considered here.