Spatially heterogeneous power-law attenuation with multiple relaxation mechanisms for ultrasound modeling

Objective. The soft tissue attenuation laws have a magnitude and frequency dependence that varies across tissue types and generally follow power laws. An accurate model of ultrasound propagation in the human body thus may require spatially heterogeneous power-law attenuation alpha(x,f) = alpha_0(x) f^(y(x)). However, a spatially heterogeneous representation of frequency-dependent attenuation is technically challenging, so existing methods introduce simplifying assumptions. For example, prior approaches such as Fullwave 2 achieved <5% error for individual tissue types but required manual parameter tuning for each (alpha_0, y) pair, limiting the construction of realistic tissue libraries. Approach. We introduce a calibration framework that uses derivative-free optimization to systematically fit relaxation parameters across diverse tissue combinations spanning alpha_0 = 0.0022-1.0 dB/(MHz^y cm) and y = 0.4-2.0. The Nelder-Mead algorithm minimizes complex-wavenumber mismatch. The attenuation is extended to a convolutional perfectly matched layer, where the same relaxation formulation is used in the boundaries. Main results. The method achieves mean errors below 3% over 1-20 MHz with dispersion error of 1.1 +/- 0.8 m/s across the clinically relevant core region (y = 0.7-1.4). Boundary reflections remain below -50 dB for clinically relevant tissue exponents (y <= 1.5). We validated the method with two-layer muscle/fat/liver models and confirmed per-layer accuracy (<2.5% normalized error). A 3D abdominal simulation using the Visible Human dataset demonstrates stable propagation with voxel-level heterogeneity in both alpha_0(x) and y(x). Significance. The open-source multi-GPU implementation (Fullwave 2.5) provides a practical tool for patient-specific therapy planning, training data generation, estimation of acoustic radiation force, quantitative imaging, and inverse problem applications.