High dose rate (HDR) brachytherapy is a radiation therapy for cancer in the prostate, cervix, breast, head, and neck, including other sites. In HDR brachytherapy, hollow needles are inserted or placed near the cancer site. Radiation is delivered to the patient by a radioactive source which is sequentially threaded through the needles. The dose distribution is controlled by altering the dwell times, the time spent at pre-defined positions on the needles.HDR brachytherapy has a 90\% cancer-free survival rate at 12 years when used for the treatment of prostate cancer, the focus of this dissertation. However, it can have serious negative side effects such as impotence and incontinence, which are caused by excess radiation exposure and needle puncture of healthy organs near the prostate, or organs at risk (OAR). A major goal of the field is to reduce side effects of HDR brachytherapy without compromising its therapeutic effectiveness. Towards this goal, this dissertation seeks to use mathematical optimization techniques to compute radiation dose distributions which meet clinical objectives and needle configurations which induce less trauma in the patient. We develop planning tools that directly optimize the dose distributions towards the RTOG-0321 standard dose objectives set by the Radiation Therapy Oncology Group and needle configurations which avoid puncturing OAR and use fewer needles than common practice. Specifically, this dissertation makes the following contributions.Contributions:1. We developed Inverse Planning by Integer Program (IPIP), the first integer program which directly optimizes dosimetric indices, the standard metrics used to evaluate HDR brachytherapy dose distributions. However, we showed that for anatomy data taken from patients previously treated at the UCSF clinic and the RTOG-0321 dose objectives, CPLEX could not solve IPIP within 30 minutes of computing time using its default parameters.2. We developed a heuristic algorithm, IPIP-H, which uses two linear programs to compute feasible solutions for IPIP. Thus, it is a polynomial-time heuristic algorithm for IPIP. We used IPIP-H to compute dose plans for the same patients as IPIP. We showed that IPIP-H could compute a dose plan for each patient which met all the dose objectives specified by the RTOG-0321 protocol in less than 30 seconds of computing time (avg. 13 seconds). The solutions computed from IPIP-H were always feasible for IPIP and were within 5% of the optimal solution. We compared IPIP-H to Inverse Planning Simulated Annealing (IPSA), a dose planning model which is clinically deployed and has been used worldwide for over a decade. IPSA was not able to compute a dose plan which met all the dose objectives for any of the patients in our data set using its standard class solution. Therefore, IPSA would require iterations of manual fine tuning of its optimization parameters until a feasible dose plan was found. IPIP-H would not require iteration.3. We formulated the problem of positioning HDR brachytherapy needles as a spatial coverage problem: given a large candidate set of needles for insertion, anatomy data, and a user parameter, delta, find the smallest candidate needle subset such that the minimum distance between any point in the prostate and a needle in the chosen set is less than delta. We showed that this problem could be represented as a set cover integer program. 4. We developed Needle Planning by Integer Program (NPIP), an algorithm which generates a set of candidate needles represented by skew-line segments, solves an integer program which chooses a candidate needle subset that covers the prostate according to the user-parameter, delta, and verifies that the final needle configuration meets dose objectives by computing a dose plan for it using IPIP. NPIP uses a candidate needle set which is approximately 10 times larger than considered with Hyrbid Inverse Planning Optimization (HIPO), the only other fully computerized needle planning system for HDR brachytherapy known to us. By construction, NPIP avoids choosing needles which penetrate OAR and needles which collide with each other. We used NPIP to compute needle configurations for patients previously treated at the UCSF clinic and compared the computed needle configurations to those implanted by the physician. NPIP could find needle configurations which met the RTOG-0321 dose objectives and used 10 or fewer needles; the physician used 16 needles. NPIP always computed a needle configuration that avoided puncturing the penile bulb; the average number of punctures made by the physician was 5. NPIP required an average of 5 minutes of computing time, but there was a wide range of run times, up to almost one hour. We also conducted a sensitivity analysis of NPIP-generated needle configurations to placement errors on the order expected from current needle insertion robots, which was about 2 mm. We showed that, although dose objectives could be met with 10 or fewer needles, 16 needles were required to meet dose objectives robustly.5. We designed and implemented the first end-to-end robotic HDR brachytherapy experiment. Our experiment utilized Contributions 1 through 4, and Acubot-RND, a needle insertion robot specialized for needle insertion. We planned and executed NPIP-generated needle configurations in a fully equipped brachytherapy environment on two anatomically-correct gelatin phantoms. There were non-trivial placement errors between the planned needle configuration and the implanted needle configuration. We separated the error into systematic error and random error. We computed the systematic error as the rigid least squares fit between points regularly sampled along the needles in the planned and actual needle configuration. The total RMS error between the planned and actual needle configuration was 3 mm for the first phantom and 5 mm for the second phantom. We computed the random error as the total RMS error between the planned and actual needle configuration after the systematic error was removed. The random error was 1.4 mm for the first phantom and 2.5 mm for the second phantom. Our random errors were close to the placement error of current needle insertion robots which have a more sophisticated calibration device. Although there were discrepancies between the planned and actual needle configuration, we showed that our end-to-end robotic experiment could execute the planned needle configurations with sufficient accuracy to meet the RTOG-0321 dose objectives and avoid puncturing OAR. We compared the needle configurations executed by our robotic workflow with a needle configuration executed by a world-class brachytherapist, who also used 16 needles, met dose objectives and avoided puncturing OAR. Therefore, the needle configurations executed in our experiment are comparable to an expert physician.In summary, this dissertation has developed mathematical methods which improve the planning of HDR brachytherapy dose distributions and needle configurations. Dose distributions can be directly optimized towards the standard RTOG-0321 dosimetric protocol, or other dose objectives based on constraining dosimetric indices, and needle configurations can be computed which meet dose objectives, use fewer needles than standard practice, and avoid puncturing OAR. We have demonstrated the feasibility of using IPIP and NPIP in a clinical environment using a robotic clinical workflow. These planning methods are a significant step towards reducing side effect of brachytherapy. We leave a clinical translation of these tools to determine if, and the extent, side effects are actually reduced
