We investigate quasi-Newton methods for minimizing a strongly convex quadratic function which is subject to errors in the evaluation of the gradients. In particular, we focus on computing search directions for quasi-Newton methods that all give identical behavior in exact arithmetic, generating minimizers of Krylov subspaces of increasing dimensions, thereby having finite termination. The BFGS quasi-Newton method may be seen as an ideal method in exact arithmetic and is empirically known to behave very well on a quadratic problem subject to small errors. We investigate large-error scenarios, in which the expected behavior is not so clear. We consider memoryless methods that are less expensive than the BFGS method, in that they generate low-rank quasi-Newton matrices that differ from the identity by a symmetric matrix of rank two. In addition, a more advanced model for generating the search directions is proposed, based on solving a chance-constrained optimization problem. Our numerical results indicate that for large errors, such a low-rank memoryless quasi-Newton method may perform better than a BFGS method. In addition, the results indicate a potential edge by including the chance-constrained model in the memoryless quasi-Newton method.

BACKGROUND: The steep dose gradients obtained with pencil beam scanning allow for precise targeting of the tumor but come at the cost of high sensitivity to uncertainties. Robust optimization is commonly applied to mitigate uncertainties in density and patient setup, while its application to motion management, called 4D-robust optimization (4DRO), is typically accompanied by other techniques, including gating, breath-hold, and re-scanning. In particular, current commercial implementations of 4DRO do not model the interplay effect between the delivery time structure and the patient's motion.

PURPOSE: Interplay-robust optimization (IPRO) has previously been proposed to explicitly model the interplay-affected dose during treatment planning. It has been demonstrated that IPRO can mitigate the interplay effect given the uncertainty in the patient's breathing frequency. In this study, we investigate and evaluate IPRO in the context where the motion uncertainty is extended to also include variations in breathing amplitude.

METHODS: The compared optimization methods are applied and evaluated on a set of lung patients. We model the patients' motion using synthetic 4D computed tomography (s4DCT), each created by deforming a reference CT based on a motion pattern obtained with 4D magnetic resonance imaging. Each (s4DCT) contains multiple breathing cycles, partitioned into two sets for scenario generation: one for optimization and one for evaluation. Distinct patient motion scenarios are then created by randomly concatenating breathing cycles varying in period and amplitude. In addition, a method considering a single breathing cycle for generating optimization scenarios (IPRO-1C) is developed to investigate to which extent robustness can be achieved with limited information. Both IPRO and IPRO-1C were investigated with 9, 25, and 49 scenarios.

RESULTS: For all patient cases, IPRO and IPRO-1C increased the target coverage in terms of the near-worst-case (5th percentile) CTV D98, compared to 4DRO. After normalization of plan doses to equal target coverage, IPRO with 49 scenarios resulted in the greatest decreases in OAR dose, with near-worst-case (95th percentile) improvements averaging 4.2 %. IPRO-1C with 9 scenarios, with comparable computational demands as 4DRO, decreased OAR dose by 1.7 %.

CONCLUSIONS: The use of IPRO could lead to more efficient mitigation of the interplay effect, even when based on the information from a single breathing cycle. This can potentially decrease the need for real-time motion management techniques that prolong treatment times and decrease patient comfort.

Objective. Online adaptive radiation therapy requires fast and automated contouring of daily scans for treatment plan re-optimization. However, automated contouring is imperfect and introduces contour uncertainties. This work aims at developing and comparing robust optimization strategies accounting for such uncertainties. Approach. A deep-learning method was used to predict the uncertainty of deformable image registration, and to generate a finite set of daily contour samples. Ten optimization strategies were compared: two baseline methods, five methods that convert contour samples into voxel-wise probabilities, and three methods accounting explicitly for contour samples as scenarios in robust optimization. Target coverage and organ-at-risk (OAR) sparing were evaluated robustly for simplified proton therapy plans for five head-and-neck cancer patients. Results. We found that explicitly including target contour uncertainty in robust optimization provides robust target coverage with better OAR sparing than the baseline methods, without increasing the optimization time. Although OAR doses first increased when increasing target robustness, this effect could be prevented by additionally including robustness to OAR contour uncertainty. Compared to the probability-based methods, the scenario-based methods spared the OARs more, but increased integral dose and required more computation time. Significance. This work proposed efficient and beneficial strategies to mitigate contour uncertainty in treatment plan optimization. This facilitates the adoption of automatic contouring in online adaptive radiation therapy and, more generally, enables mitigation also of other sources of contour uncertainty in treatment planning.

The focus in this work is on interior-point methods for inequality-constrained quadratic programs, and particularly on the system of nonlinear equations to be solved for each value of the barrier parameter. Newton iterations give high quality solutions, but we are interested in modified Newton systems that are computationally less expensive at the expense of lower quality solutions. We propose a structured modified Newton approach where each modified Jacobian is composed of a previous Jacobian, plus one low-rank update matrix per succeeding iteration. Each update matrix is, for a given rank, chosen such that the distance to the Jacobian at the current iterate is minimized, in both 2-norm and Frobenius norm. The approach is structured in the sense that it preserves the nonzero pattern of the Jacobian. The choice of update matrix is supported by results in an ideal theoretical setting. We also produce numerical results with a basic interior-point implementation to investigate the practical performance within and beyond the theoretical framework. In order to improve performance beyond the theoretical framework, we also motivate and construct two heuristics to be added to the method.

Objective. Delineating and planning with respect to regions suspected to contain microscopic tumor cells is an inherently uncertain task in radiotherapy. The recently proposed clinical target distribution (CTD) is an alternative to the conventional clinical target volume (CTV), with initial promise. Previously, using theCTDin planning has primarily been evaluated in comparison to a conventionally defined CTV. Wepropose to compare theCTDapproach against CTVmargins of various sizes, dependent on the threshold at which the tumor infiltration probability is considered relevant. Approach. First, a theoretical framework is presented, concerned with optimizing the trade-off between the probability of sufficient target coverage and the penalties associated with high dose. From this framework we derive conventional CTV-based planning and contrast it with theCTDapproach. The approaches are contextualized further by comparison with established methods for managing geometric uncertainties. Second, for both one- and three-dimensional phantoms, we compare a set of CTDplans created by varying the target objective function weight against a set of plans created by varying both the target weight and the CTVmargin size. Main results. The results show that CTD-based planning gives slightly inefficient trade-offs between the evaluation criteria for a case in which near-minimum target dose is the highest priority. However, in a case when sparing a proximal organ at risk is critical, theCTDis better at maintaining sufficiently high dose toward the center of the target. Significance. Weconclude that CTD-based planning is a computationally efficient method for planning with respect to delineation uncertainties, but that the inevitable effects on the dose distribution should not be disregarded.

The focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework. 

The main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give a class of limited-memory quasi-Newton Hessian approximations which generate search directions parallel to those of the BFGS method, or equivalently, to those of the method of preconditioned conjugate gradients. In the setting of reduced Hessians, the class provides a dynamical framework for the construction of limited-memory quasi-Newton methods. These methods attain finite termination on quadratic optimization problems in exact arithmetic. We show performance of the methods within this framework in finite precision arithmetic by numerical simulations on sequences of related systems of linear equations, which originate from the CUTEst test collection. In addition, we give a compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components.

We show that given a feasible primal–dual pair of linear programs in canonical form, there exists a sequence of pivots, whose length is bounded by the minimum dimension of the constraint matrix, leading from the origin to the optimum. The sequence of pivots give a sequence of square and nonsingular submatrices of the constraint matrix. Solving two linear equations involving such a submatrix give primal–dual optimal solutions to the corresponding linear program in canonical form.

Mathematical modelling can provide precious tools for bioprocess simulation, prediction, control and optimization of mammalian cell-based cultures. In this paper we present a novel method to generate kinetic models of such cultures, rendering complex metabolic networks in a poly-pathway kinetic model. The model is based on subsets of elementary flux modes (EFMs) to generate macro-reactions. Thanks to our column generation-based optimization algorithm, the experimental data are used to identify the EFMs, which are relevant to the data. Here the systematic enumeration of all the EFMs is eliminated and a network including a large number of reactions can be considered. In particular, the poly-pathway model can simulate multiple metabolic behaviors in response to changes in the culture conditions. We apply the method to a network of 126 metabolic reactions describing cultures of antibody-producing Chinese hamster ovary cells, and generate a poly-pathway model that simulates multiple experimental conditions obtained in response to variations in amino acid availability. A good fit between simulated and experimental data is obtained, rendering the variations in the growth, product, and metabolite uptake/secretion rates. The intracellular reaction fluxes simulated by the model are explored, linking variations in metabolic behavior to adaptations of the intracellular metabolism.