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The safe internal transportation of hazardous materials within healthcare facilities is critical to mitigating risks to patients, staff, and visitors. This paper presents a risk-averse path planning framework for autonomously handling hazardous materials in healthcare systems. We model the indoor environment with grid-based obstacle and risk maps, where risk arises from pedestrian flow density and proximity to critical zones. Our novel risk-averse path planning approach integrates risk directly into each transition cost, thereby enabling more robust and secure path selection. We further improve efficiency through (i) a bidirectional variant that cuts search time and (ii) a post-optimization step that minimizes unnecessary heading changes while respecting a risk budget. We evaluated our framework on multiple simulated grid maps and compared it with established methods, measuring path length, average risk, and computational time. The results demonstrate that the proposed framework consistently generates safe and efficient paths while minimizing computational overhead.