We present a Ǫuantum-Inspired Hybrid 3D Convolutional Neural Network (ǪI-3DCNN) for volumetric brain MRI and CT analysis that integrates classical deep learning with a variational quantum circuit (VǪC)–emulating a processing layer. The quantum-inspired layer implements periodic rotation-like nonlinear mappings and entanglement-inspired cross-channel interactions that approximate the mathematical structure of parameterized quantum circuits, enabling expressive, non-Euclidean feature embeddings while remaining fully differentiable and trainable on classical hardware. The proposed architecture is evaluated on the BraTS brain tumor MRI dataset and the RSNA intracranial hemorrhage CT dataset. Across both benchmarks, the ǪI-3DCNN achieves higher area under the ROC curve, sensitivity, and robustness to noise than conventional 3D convolutional networks, Vision Transformer–based volumetric models, and Neural Ordinary Differential Equation (Neural ODE) architectures. These results indicate that quantum-inspired inductive biases can provide practical benefits for medical image analysis without requiring quantum computing hardware.