Perceptualizing tool interactions with deformable structures in surgical procedures remains challenging, as unimodal visualization techniques often fail to capture the complexity of these interactions due to constraints such as occlusion and limited depth perception. This paper presents a novel approach to augment tool navigation in mixed reality environments by providing auditory representations of tool-tissue dynamics, particularly for interactions with soft tissue. BioSonix, a physics-informed design framework, utilizes tissue displacements in 3D space to compute excitation forces for a sound model encoding tissue properties such as stiffness and density. Biomechanical simulations were employed to model particle displacements resulting from tool-tissue interactions, establishing a robust foundation for the method. An optimization approach was used to define configurations for capturing diverse interaction scenarios with varying tool trajectories. Experiments were conducted to validate the accuracy of the sound-displacement mappings. Additionally, two user studies were performed: the first involved two clinical professionals (a neuroradiologist and a cardiologist), who confirmed the method's impact and achieved high task accuracy; the second included 22 biomedical experts, who demonstrated high discrimination accuracy in tissue differentiation and targeting tasks. The results revealed a strong correlation between tool-tissue dynamics and their corresponding auditory profiles, highlighting the potential of these sound representations to enhance the intuitive understanding of complex interactions.