Optimization of spatiotemporally fractionated radiotherapy treatments with bounds on the achievable benefit

Spatiotemporal fractionation schemes, that is, treatments delivering different dose distributions in different fractions, may lower treatment side effects without compromising tumor control. This is achieved by hypofractionating parts of the tumor while delivering approximately uniformly fractionated doses to the healthy tissue. Optimization of such treatments is based on biologically effective dose (BED), which leads to computationally challenging nonconvex optimization problems. Current optimization methods yield only locally optimal plans, and it has been unclear whether these are close to the global optimum. We present an optimization model to compute rigorous bounds on the normal tissue BED reduction achievable by such plans. The approach is demonstrated on liver tumors, where the primary goal is to reduce mean liver BED without compromising other treatment objectives. First a uniformly fractionated reference plan is computed using convex optimization. Then a nonconvex quadratically constrained quadratic programming model is solved to local optimality to compute a spatiotemporally fractionated plan that minimizes mean liver BED subject to the constraints that the plan is no worse than the reference plan with respect to all other planning goals. Finally, we derive a convex relaxation of the second model in the form of a semidefinite programming problem, which provides a lower bound on the lowest achievable mean liver BED. The method is presented on 5 cases with distinct geometries. The computed spatiotemporal plans achieve 12-35% mean liver BED reduction over the reference plans, which corresponds to 79-97% of the gap between the reference mean liver BEDs and our lower bounds. This indicates that spatiotemporal treatments can achieve substantial reduction in normal tissue BED, and that local optimization provides plans that are close to realizing the maximum potential benefit

Minimizing Metastatic Risk in Radiotherapy Fractionation Schedules

Metastasis is the process by which cells from a primary tumor disperse and form new tumors at distant anatomical locations. The treatment and prevention of metastatic cancer remains an extremely challenging problem. This work introduces a novel biologically motivated objective function to the radiation optimization community that takes into account metastatic risk instead of the status of the primary tumor. In this work, we consider the problem of developing fractionated irradiation schedules that minimize production of metastatic cancer cells while keeping normal tissue damage below an acceptable level. A dynamic programming framework is utilized to determine the optimal fractionation scheme. We evaluated our approach on a breast cancer case using the heart and the lung as organs-at-risk (OAR). For small tumor $\alpha/\beta$ values, hypo-fractionated schedules were optimal, which is consistent with standard models. However, for relatively larger $\alpha/\beta$ values, we found the type of schedule depended on various parameters such as the time when metastatic risk was evaluated, the $\alpha/\beta$ values of the OARs, and the normal tissue sparing factors. Interestingly, in contrast to standard models, hypo-fractionated and semi-hypo-fractionated schedules (large initial doses with doses tapering off with time) were suggested even with large tumor $\alpha$/$\beta$ values. Numerical results indicate potential for significant reduction in metastatic risk.Comment: 12 pages, 3 figures, 2 table

Biologically-based radiation therapy planning and adjustable robust optimization

Radiation therapy is one of the main treatment modalities for various different cancer types. One of the core components of personalized treatment planning is the inclusion of patient-specific biological information in the treatment planning process. Using biological response models, treatment parameters such as the treatment length and dose distribution can be tailored, and mid treatment biomarker information can be used to adapt the treatment during its course. These additional degrees of freedom in treatment planning lead to new mathematical optimization problems. This thesis studies various optimization aspects of biologically-based treatment planning, and focuses on the influence of uncertainty. Adjustable robust optimization is the main technique used to study these problems, and is also studied independently of radiation therapy applications

Applications of Nonlinear Optimization

We apply an interior point algorithm to two nonlinear optimization problems and achieve improved results. We also devise an approximate convex functional alternative for use in one of the problems and estimate its accuracy. The first problem is maximum variance unfolding in machine learning. The traditional method to solve this problem is to convert it to a semi-definite optimization problem by defining a kernel matrix. We obtain better unfolding and higher speeds with the interior point algorithm on the original non-convex problem for data with less than 10,000 points. The second problem is a multi-objective dose optimization for intensity modulated radiotherapy, whose goals are to achieve high radiation dose on tumors while sparing normal tissues. Due to tumor motions and patient set-up errors, a robust optimization against motion uncertainties is required to deliver a clinically acceptable treatment plan. The traditional method, to irradiate an enlargement of the tumor region, is very conservative and leads to possibly high radiation dose on sensitive structures. We use a new robust optimization model within the framework of goal programming that consists of multiple optimization steps based on prescription priorities. One metric is defined for each structure of interest. A final robustness optimization step then minimizes the variance of all the goal metrics with respect to the motion probability space, and pushes the mean values of these metrics toward a desired value as well. We show similar high dose coverage on example tumors with reduced dose on sensitive structures. One clinically important metric for a radiation dose distribution, that describes tumor control probability or normal tissue complication probability, is Dx, the minimum dose value on the hottest x% of a structure. It is not mathematically well-behaved, which impedes its use in optimization. We approximate Dx with a linear function of two generalized equivalent uniform dose metrics, also known as lp norms, requiring that the approximation is concave so that its maximization becomes a convex problem. Results with cross validation on a sampling of radiation therapy plans show that the error of this approximation is less than 1 Gy for the most used range 80 to 95 of x values

Automating Intensity Modulated Radiation Therapy Treatment Planning by using Hierarchical Optimization

The intensity modulated radiation therapy (IMRT) optimizes the beam’s intensity to deliver the prescribed dose to the target while minimizing the radiation exposure to normal structures. The IMRT optimization is a complex optimization problem because of the multiple conflicting objectives in it. Due to the complexity of the optimization, the IMRT treatment planning is still a trial and error process. Hierarchical optimization was proposed to automate the treatment planning process, but its potential has not been demonstrated in a clinical setting. Moreover, hierarchical optimization is slower than the traditional optimization. The dissertation studied a sampling algorithm to reduce the hierarchical optimization time, customized an open source optimization solver to solve the nonlinear optimization formulation and demonstrated the potential of hierarchical optimization to automate the treatment planning process in a clinical setting. We generated the treatment plans of 31 prostate patients by hierarchical optimization using the same criteria as used by planners to prepare the treatment plans at Memorial Sloan Kettering Cancer Center. We found that hierarchical optimization produced the same or better treatment plans than that produced by a planner using the Eclipse treatment planning system. Therefore, the dissertation demonstrated that hierarchical optimization could automate the treatment planning process and shift the paradigm of the treatment planning from manual trial and error to an ideal automated process

Biologically effective dose (BED) treatment planning for Gamma Knife Radiosurgery

Gamma Knife (GK) radiosurgery treats brain lesions through multiple targeted radiation exposures of varying duration and spatial distribution. Clinical radiosurgery treatment planning only considers the total amount of delivered radiation. A biologically effective dose (BED) model allows quantifying the damage induced in a tissue due to radiation exposure while accounting for cellular repair. With this thesis work, we explore the potential and feasibility of using the more complex BED formulation to generate biologically-aware treatment plans. To this end, we quantify the impact of changes in the temporal domain of treatment delivery (i.e. beam-off periods, order of delivery), which need to be considered at the treatment planning stage to avoid undesirable BED variations. The delivery sequence alone can incur variations in marginal BED by up to 14%. Consideration of treatment delivery timing and sequence creates a nonconvex nonlinear treatment planning problem that is too computationally expensive to solve in a time-sensitive clinical setting. We develop multiple optimisation techniques to identify the most suitable one for a clinical workflow. While a convex underestimator approach provides slightly improved solutions, it requires several orders of magnitude more computational resources than local optimisation approaches that reach similar performance in terms of plan quality. In consultation with our clinical collaborators, we devise a BED treatment planning workflow that further reduces the possible planning times by combining pre-computation of candidate solutions with interactive exploration and refinement of the final treatment plans. To evaluate this workflow, we develop a prototype treatment planning framework. We show that BED optimisation removes the time dependence and further increases plan quality. The results of the proof-of-concept workflow demonstrate the feasibility of a future clinical application of BED planning in GK radiosurgery

INTEGRATED BEAM ORIENTATION OPTIMIZATION FOR ROBUST INTENSITY-MODULATED PROTON THERAPY

Methods The unified framework is formulated to include a dose fidelity term, a heterogeneity-weighted group sparsity term, and a sensitivity regularization term. The dose fidelity term encourages less physical dose deviation from ideal distribution. The L2,1/2-norm group sparsity is used to reduce the number of active beams from the initial 1162 evenly distributed non-coplanar candidate beams, to between 2 and 4. A heterogeneity index, which evaluates the lateral tissue heterogeneity of a beam, is used to weigh the group sparsity term. With this index, beams more resilient to setup uncertainties are encouraged. There is a symbiotic relationship between the heterogeneity index and the sensitivity regularization; the integrated optimization framework further improves beam robustness against both range and setup uncertainties. This Sensitivity regularization and Heterogeneity weighting based BOO and FMO framework (SHBOO-FMO) was tested on two skull-base tumor (SBT) patients and two bilateral head-and-neck (H&N) patients. The conventional CTV-based optimized plans (Conv) with SHBOO-FMO beams (SHBOO-Conv) and manual beams (MAN-Conv) were compared to investigate the beam robustness of the proposed method. The dosimetry and robustness of SHBOO-FMO plan were compared against the manual beam plan with CTV-based voxel-wise worst-case scenario approach (MAN-WC).Results With SHBOO-FMO method, the beams with superior range robustness over manual beams were selected while the setup robustness was maintained or improved. On average, the lowest [D95%, V95%, V100%] of CTV were increased from [93.8%, 91.0%, 70.6%] in MAN-Conv plans, to [98.6%, 98.6%, 96.1%] in SHBOO-Conv plans with range uncertainties. With setup uncertainties, the average lowest [D98%, D95%, V95%, V100%] of CTV were increased from [92.0%, 94.8%, 94.3%, 78.9%] in MAN-Conv plans, to [93.5%, 96.6%, 97.0%, 91.9%] in SHBOO-Conv plans. Compared with the MAN-WC plans, the final SHBOO-FMO plans achieved comparable plan robustness and better OAR sparing, with an average reduction of [Dmean, Dmax] of [6.3, 6.6] GyRBE for the SBT cases and [1.9, 5.1] GyRBE for the H&N cases from the MAN-WC plans. Conclusions A novel robust optimization method was developed for IMPT. It integrates robust BOO and robust FMO into a unified framework, and the resulting optimization problem can be solved efficiently. Compared with the current clinical practice, where beam angles are manually selected and fluence map is optimized by worst-case method, the planning efficiency is improved, and it generates plans with superior dosimetry and good robustness