Radiofrequency ablation is a thermal therapy for moderately-sized cancerous tumors. A target is killed with high temperatures obtained due to the current passed through one or more electrodes (needles) inserted into it. The needles' trajectory must be meticulously planned to prevent interference with dense organs like bone or puncturing of critical structures like veins. By approximating the thermal lesion to an ellipse, we predefine several valid needle trajectories and then solve an integer programming model to identify pairwise valid needle positions, that meet clinical criteria, using a variation of the classic set cover model. To improve the models' tractability and scalability, we use row generation-based decomposition techniques that determines pairwise validity using two different types of cuts. Finally, we analyze target and OAR damage using several thermal damage models. Thus, for the first time we present a full treatment plan that incorporates novel trajectory planning with thermal dose computations. Our method is tested on 12 liver targets: three targets each with four different surgical margins. We show promising results that meet clinical guidelines while obtaining full target coverage.

Gasoline is produced by blending several different components in ratios such that the blended mixture meets the required quality specifications. The blender produces different batches of gasoline by switching operation from one grade of gasoline to another. Blend planning horizon usually spans 10 to 14 days. Blend plan optimization minimizes the total blend costs by solving a multiperiod problem, where demands need to be satisfied in each period and some inventory is carried into the future time periods to meet the demands. Since blend component production is determined by a longer range refinery production plan, inventory carrying costs are not included in the objective function. It is shown that nonlinear programming (NLP) as well as mixed integer nonlinear programming (MINLP) solvers lead to different blend recipes and different blend volume patterns for the same total cost. The new algorithm described in this work systematically searches for multiple optimum solutions; this opens the way for blend planners to select from different blend plans based on additional considerations (e.g., blend more of regular gasoline earlier in the planning horizon thereby creating an opportunity to meet more demand for it in early periods) instead of having to use only one solution that varies with the choice of the solver. Inherent structure of the proposed algorithm makes it well suited for implementation on parallel CPU machines.

Radiofrequency ablation (RFA) offers localized and minimally invasive treatment of small-to-medium sized inoperable tumors. In RFA, tissue is ablated with high temperatures obtained from electrodes (needles) inserted percutaneously or via an open surgery into the target. RFA treatments are generally not planned in a systematic way, and do not account for nearby organs-at-risk (OARs), potentially leading to sub-optimal treatments and inconsistent treatment quality. We therefore develop a mathematical framework to design RFA treatment plans that provide complete ablation while minimizing healthy tissue damage. Borrowing techniques from radiosurgery inverse planning, we design a two-stage approach where we first identify needle positions and orientations, called needle orientation optimization, and then compute the treatment time for optimal thermal dose delivery, called thermal dose optimization. Several different damage models are used to determine both target and OAR damage. We present numerical results on three clinical case studies. Our findings indicate a need for high source voltage for short tip length (conducting portion of the needle) or fewer needles, and low source voltage for long tip length or more needles to achieve full coverage. Further, more needles yields a larger ablation volume and consequently more OAR damage. Finally, the choice of damage model impacts the source voltage, tip length, and needle quantity.