Radiation therapy, treating over half of all cancer patients, involves using specialized machines to direct high-energy beams at tumors, aiming to damage cancer cells while minimizing harm to nearby healthy tissues. Customizing the shape and intensity of radiation beams for each patient leads to solving large-scale constrained optimization problems within a clinical time-frame of 5-10 minutes. Central to these problems is a large matrix typically sparsified for computational efficiency by ignoring small elements, which can compromise treatment quality and increase unnecessary radiation to healthy tissues, leading to serious radiation-induced side effects. In this work, we demonstrate for the first time that randomized sketch tools can effectively sparsify this matrix without sacrificing treatment quality. We also develop a novel randomized sketch method that outperforms existing techniques in terms of speed and accuracy, and it comes with competitive theoretical guarantees. Beyond developing a novel randomized sketch method, this work emphasizes the potential of harnessing scientific computing tools, crucial in today's big data analysis, to tackle computationally intensive challenges in healthcare. The application of these tools could have a profound impact on the lives of numerous patients.