arXiv subjects

Thomas Bortfeld

Publications and source records attributed to Thomas Bortfeld.

18 recordsLinked to original sources

arXiv2026

Bi-level Multi-criteria Optimization for Risk-informed Radiotherapy

In radiation therapy (RT) treatment planning, multi-criteria optimization (MCO) supports efficient plan selection but is usually solved for population-based dosimetric criteria and ignores patient-specific biological risk, potentially compromising outcomes in high-risk patients. We propose risk-guided MCO, a one-shot method that embeds a clinical risk model into conventional MCO, enabling interactive navigation between dosimetric and biological endpoints. The proposed algorithm uses a special order relation to fuse the classical MCO sandwiching algorithm with bi-level optimization, restricting the Pareto set to plans that achieve improvement in the secondary risk objective for user-defined, acceptable loss in primary clinical objectives. Thus, risk-guided MCO generates risk-optimized counterparts of clinical plans in a single run rather than by sequential or lexicographic planning. To assess the performance, we retrospectively analyzed 19 lung cancer patients treated with RT. The endpoint was the risk of grade 2+ radiation pneumonitis (RP), modeled using bootstrapped stepwise logistics regression with interaction terms, including baseline lung function, smoking history, and dosimetric factors. The risk-guided plans yielded a mean reduction of 8.0% in total lung V20 and 9.5% in right lung V5, translating into an average RP risk reduction of 7.7% (range=0.3%-20.1%), with small changes in target coverage (mean -1.2 D98[%] for CTV) and modest increase in heart dose (mean +1.74 Gy). This study presents the first proof-of-concept for integrating biological risk models directly within multi-criteria RT planning, enabling an interactive balance between established population-wide dose protocols and individualized outcome prediction. Our results demonstrate that the risk-informed MCO can reduce the risk of RP while maintaining target coverage.

physics.med-ph↗

arXiv2025

Ising energy model for the stochastic prediction of tumor islets

A major challenge in diagnosing and treating cancer is the infiltrative growth of tumors into surrounding tissues. This microscopic spread of the disease is invisible on most diagnostic imaging modalities and can often only be detected histologically in biopsies. The purpose of this paper is to develop a physically based model of tumor spread that captures the histologically observed behavior in terms of seeding small tumor islets in prostate cancer. The model is based on three elementary events: a tumor cell can move, duplicate, or die. The propensity of each event is given by an Ising-like Hamiltonian that captures correlations between neighboring cells. The model parameters were fitted to clinical data obtained from surgical specimens taken from 23 prostate cancer patients. The results demonstrate that this straightforward physical model effectively describes the distribution of the size and the number of tumor islets in prostate cancer. The simulated tumor islets exhibit a regular, approximately spherical shape, correctly mimicking the shapes observed in histology. This is due to the Ising interaction term between neighboring cells acting as a surface tension that gives rise to regularly shaped islets. The model addresses the important clinical need of calculating the probability of tumor involvement in specific sub-volumes of the prostate, which is required for radiation treatment planning and other applications.

physics.med-ph↗

arXiv2025

A K-adaptability Approach to Proton Radiation Therapy Robust Treatment Planning

Uncertainties such as setup and range errors can significantly compromise proton therapy. A discrete uncertainty set is often constructed to represent different uncertainty scenarios. A min-max robust optimization approach is then utilized to optimize the worst-case performance of a radiation therapy plan against the uncertainty set. However, the min-max approach can be too conservative as a single plan has to account for the entire uncertainty set. K-adaptability is a novel approach to robust optimization which covers the uncertainty set with multiple (K) solutions, reducing the conservativeness. Solving K-adaptability to optimality is known to be computationally intractable. To that end, we developed a novel and efficient K-adaptability heuristic that iteratively clusters the scenarios based on plan-scenario performance for the proton radiation therapy planning problem. Compared to the conventional robust solution, the developed K-adaptability heuristic increased the worst-case CTV Dmin dose up to 4.52 Gy on average across five head and neck patients. The developed heuristic also demonstrated its superiority in objective value and time-efficiency compared to the competing methods we tested.

math.OC↗

arXiv2023

Using constraint structure and an improved object detection network to detect the 12^{th} Vertebra from CT images with a limited field of view for image-guided radiotherapy

Image guidance has been widely used in radiation therapy. Correctly identifying the bounding box of the anatomical landmarks from limited field of views is the key to success. In image-guided radiation therapy (IGRT), the detection of those landmarks like the 12th vertebra (T12) still requires tedious manual inspections and annotations; and superior-inferior misalignment to the wrong vertebral body is still relatively common. It is necessary to develop an automated approach to detect those landmarks from images. The challenges of training a model to identify T12 vertebrae automatically mainly are high shape similarity between T12 and neighboring vertebrae, limited annotated data, and class imbalance. This study proposed a novel 3D detection network, requiring only a small amount of training data. Our approach has the following innovations, including 1) the introduction of an auxiliary network to build constraint feature map for improving the model's generalization, especially when the constraint structure is easier to be detected than the main one; 2) an improved detection head and target functions for accurate bounding box detection; and 3) an improved loss functions to address the high class imbalance. Our proposed network was trained, validated and tested on anotated CT images from 55 patients and demonstrated accurate distinguish T12 vertebra from its neighboring vertebrae of high shape similarity. Our proposed algorithm yielded the bounding box center and size errors of 3.98\pm2.04mm and 16.83\pm8.34mm, respectively. Our approach significantly outperformed state-of-the-arts Retina-Net3D in average precision (AP) at IoU thresholds of 0.35 and 0.5, with AP increasing from 0 to 95.4 and 0 to 64.7, respectively. In summary, our approach has a great potential to be integrated into the clinical workflow to improve the safety of IGRT.

cs.CV↗

arXiv2022

Modeling the propagation of tumor fronts with shortest path and diffusion models -- implications for the definition of the clinical target volume

Objective: The overarching objective is to make the definition of the clinical target volume (CTV) in radiation oncology less subjective and more scientifically based. The specific objective of this study is to investigate similarities and differences between two methods that model tumor spread beyond the visible gross tumor volume (GTV): 1. The shortest path model, which is the standard method of adding a geometric GTV-CTV margin, and 2. The reaction-diffusion model. Approach: These two models to capture the invisible tumor "fire front" are defined and compared in mathematical terms. The models are applied to example cases that represent tumor spread in non-uniform and anisotropic media with anatomical barriers. Main Results: The two seemingly disparate models bring forth traveling waves that can be associated with the front of tumor growth outward from the GTV. The shape of the fronts is similar for both models. Differences are seen in cases where the diffusive flow is reduced due to anatomical barriers, and in complex spatially non-uniform cases. The diffusion model generally leads to smoother fronts. The smoothness can be controlled with a parameter defined by the ratio of the diffusion coefficient and the proliferation rate. Significance: Defining the CTV has been described as the weakest link of the radiotherapy chain. There are many similarities in the mathematical description and the behavior of the common geometric GTV-CTV expansion method, and the definition of the CTV tumor front via the reaction-diffusion model. Its mechanistic basis and the controllable smoothness make the diffusion model an attractive alternative to the standard GTV-CTV margin model.

physics.med-ph↗

arXiv2022

Does the greater power of pencil beam scanning reduce the need for a proton gantry? A study of head-and-neck and brain tumors

Proton therapy systems without a gantry can be more compact and less expensive in terms of capital cost, and therefore more available to a larger patient population. Would the advances in pencil beam scanning and robotics make gantry-less treatment possible? In this study, we explore if high-quality treatment plans can be obtained without a gantry. We recently showed that proton treatments with the patient in an upright position may be feasible with a new soft robotic immobilization device and imaging which enables multiple possible patient orientations during a treatment. In this study, we evaluate if this new treatment geometry could enable high quality treatment plans without a gantry. We created pencil beam scanning (PBS) treatment plans for seven patients with head-and-neck or brain tumors. Each patient was planned with two scenarios: one with a gantry with the patient in supine position and the other with a gantry-less fixed horizontal beam-line with the patient sitting upright. For the treatment plans, dose-volume-histograms (DVHs), target homogeneity index (HI), mean dose, D_2 and D_98 are reported. A robustness analysis of one plan was performed with +/-2.5 mm setup errors and +/-3.5% range uncertainties with nine scenarios. Most of the PBS-gantry-less plans had similar target HI and OAR mean dose as compared to PBS-gantry plans, and similar robustness with respect to range uncertainties and setup errors. Pencil beam scanning provides sufficient power to deliver high quality treatment plans without requiring a gantry for head-and-neck or brain tumors. In combination with the development of the new positioning and immobilization methods required to support this treatment geometry, this work suggests the feasibility of further development of a compact proton therapy system with a fixed horizontal beam-line to treat patients in sitting and reclined positions.

physics.med-ph↗

arXiv2022

Feasibility study of clinical target volume definition for soft-tissue sarcoma using muscle fiber orientations derived from diffusion tensor imaging

Objective: Soft-tissue sarcoma spreads preferentially along muscle fibers. We explore the utility of deriving muscle fiber orientations from diffusion tensor MRI (DT-MRI) for defining the boundary of the clinical target volume in muscle tissue. Approach: We recruited eight healthy volunteers to acquire MR images of the left and right thigh. The imaging session consisted of (a) two MRI spin-echo-based scans, T1- and T2-weighted; (b) a diffusion weighted (DW) spin-echo-based scan using an echo planar acquisition with fat suppression. The thigh muscles were auto-segmented using CNN. DT-MRI data was used as a geometry encoding input to solve the anisotropic Eikonal equation with Hamiltonian Fast-Marching method. The isosurfaces of the solution modeled the CTV boundary. Main results: The auto-segmented muscles of the thigh agreed with manually delineated with the Dice score ranging from 0.8 to 0.94 for different muscles. Anisotropy of the isosurfaces was compared across muscles with different anatomical orientations within a thigh, between muscles in left and right thighs of each subject, and between different subjects. Analysis showed a high degree of consistency across all comparisons. The distance from the GTV to the isosurface and the eigenvalues ratio are two controlling parameters for the extent and shape of the CTV. Significance: Our feasibility study with healthy volunteers shows the promise of using muscle fiber orientations derived from diffusion weighted MRI data for automated generation of anisotropic CTV boundary in soft tissue sarcoma. Our contribution is significant as it is expected to lead to the improvements in the treatment outcomes of soft-tissue sarcoma patients undergoing radiotherapy and decrease amputation rate for a subset of patients. We expect such improvements to have a strong positive impact for the cancer centers with small volume of sarcoma patients.

physics.med-ph↗

arXiv2021

A full-scale clinical prototype for proton range verification using prompt gamma-ray spectroscopy

We present a full-scale clinical prototype system for in vivo range verification of proton pencil-beams using the prompt gamma-ray spectroscopy method. The detection system consists of eight LaBr3 scintillators and a tungsten collimator, mounted on a rotating frame. Custom electronics and calibration algorithms have been developed for the measurement of energy- and time-resolved gamma-ray spectra during proton irradiation at a clinical dose rate. Using experimentally determined nuclear reaction cross sections and a GPU-accelerated Monte Carlo simulation, a detailed model of the expected gamma-ray emissions is created for each individual pencil-beam. The absolute range of the proton pencil-beams is determined by minimizing the discrepancy between the measurement and this model, leaving the absolute range of the beam and the elemental concentrations of the irradiated matter as free parameters. The system was characterized in a clinical-like situation by irradiating different phantoms with a scanning pencil-beam. A dose of 0.9 Gy was delivered to a 5x10x10 cm$^3$ target with a beam current of 2 nA incident on the phantom. Different range shifters and materials were used to test the robustness of the verification method and to calculate the accuracy of the detected range. The absolute proton range was determined for each spot of the distal energy layer with a mean statistical precision of 1.1 mm at a 95% confidence level and a mean systematic deviation of 0.5 mm, when aggregating pencil-beam spots within a cylindrical region of 10 mm radius and 10 mm depth. Small range errors that we introduced were successfully detected and even large differences in the elemental composition do not affect the range verification accuracy. These results show that our system is suitable for range verification during patient treatments in our upcoming clinical study.

physics.med-ph↗

arXiv2021

Adjustable robust treatment-length optimization in radiation therapy

Traditionally, optimization of radiation therapy (RT) treatment plans has been done before the initiation of RT course, using population-wide estimates for patients' response to therapy. However, recent technological advancements have enabled monitoring individual patient response during the RT course, in the form of biomarkers. Although biomarker data remains subject to substantial uncertainties, information extracted from this data may allow the RT plan to be adapted in a biologically informative way. We present a mathematical framework that optimally adapts the treatment-length of an RT plan based on the acquired mid-treatment biomarker information, while accounting for the inexact nature of this information. We formulate the adaptive treatment-length optimization problem as a 2-stage problem, wherein the information about the model parameters gathered during the first stage influences the decisions in the second stage. Using Adjustable Robust Optimization (ARO) techniques we derive explicit optimal decision rules for the stage-2 decisions and solve the optimization problem. The problem allows for multiple worst-case optimal solutions. To discriminate between these, we introduce the concept of Pareto Adjustable Robustly Optimal (PARO) solutions. In numerical experiments using lung cancer patient data, the ARO method is benchmarked against several other static and adaptive methods. In case of exact biomarker information, there is sufficient space to adapt, and numerical results show that taking into account both robustness and adaptability is not necessary. In case of inexact biomarker information, accounting for adaptability and inexactness of biomarker information is particularly beneficial when robustness (w.r.t. organ-at-risk (OAR) constraint violations) is of high importance. If minor OAR violations are allowed, a nominal folding horizon approach is a good performing alternative.

physics.med-ph↗

arXiv2018

The clinical target distribution: a probabilistic alternative to the clinical target volume

Definition of the clinical target volume (CTV) is one of the weakest links in the radiation therapy chain. In particular, inability to account for uncertainties is a severe limitation in the traditional CTV delineation approach. Here, we introduce and test a new concept for tumor target definition, the clinical target distribution (CTD). The CTD is a continuous distribution of the probability of voxels to be tumorous. We describe an approach to incorporate the CTD in treatment plan optimization algorithms, and implement it in a commercial treatment planning system. We test the approach in two synthetic and two clinical cases, a sarcoma and a glioblastoma case. The CTD is straightforward to implement in treatment planning and comes with several advantages. It allows one to find the most suitable tradeoff between target coverage and sparing of surrounding healthy organs at the treatment planning stage, without having to modify or redraw a CTV. Owing to the variable probabilities afforded by the CTD, a more flexible and more clinically meaningful sparing of critical structure becomes possible. Finally, the CTD is expected to reduce the inter-user variability of defining the traditional CTV.

physics.med-ph↗

arXiv2015

Optimization of Radiation Therapy Fractionation Schedules in the Presence of Tumor Repopulation

We analyze the effect of tumor repopulation on optimal dose delivery in radiation therapy. We are primarily motivated by accelerated tumor repopulation towards the end of radiation treatment, which is believed to play a role in treatment failure for some tumor sites. A dynamic programming framework is developed to determine an optimal fractionation scheme based on a model of cell kill due to radiation and tumor growth in between treatment days. We find that faster tumor growth suggests shorter overall treatment duration. In addition, the presence of accelerated repopulation suggests larger dose fractions later in the treatment to compensate for the increased tumor proliferation. We prove that the optimal dose fractions are increasing over time. Numerical simulations indicate potential for improvement in treatment effectiveness.

physics.med-ph↗

arXiv2014

A Mathematical Programming Approach to the Fractionation Problem in Chemoradiotherapy

In concurrent chemoradiotherapy, chemotherapeutic agents are administered during the course of radiotherapy to enhance the primary tumor control. However, that often comes at the expense of increased risk of normal-tissue complications. The additional biological damage is mainly attributed to two mechanisms of action, which are the independent cytotoxic activity of chemotherapeutic agents and their interactive cooperation with radiation. The goal of this study is to develop a mathematical framework to obtain drug and radiation administration schedules that maximize the therapeutic gain for concurrent chemoradiotherapy. In particular, we analyze the impact of incorporating these two mechanisms into the radiation fractionation problem. Considering each mechanism individually, we first derive closed-form expressions for the optimal radiation fractionation regimen and the corresponding drug administration schedule. We next study the case in which both mechanisms are simultaneously present and develop a dynamic programming framework to determine optimal treatment regimens. Results show that those chemotherapeutic agents that interact with radiation may change optimal radiation fractionation regimens. Moreover, administration of chemotherapeutic agents possessing both mechanisms may give rise to optimal non-stationary radiation fractionation schemes.

physics.med-ph↗

arXiv2014

Exploiting tumor shrinkage through temporal optimization of radiotherapy

In multi-stage radiotherapy, a patient is treated in several stages separated by weeks or months. This regimen has been motivated mostly by radiobiological considerations, but also provides an approach to reduce normal tissue dose by exploiting tumor shrinkage. The paper considers the optimal design of multi-stage treatments, motivated by the clinical management of large liver tumors for which normal liver dose constraints prohibit the administration of an ablative radiation dose in a single treatment. We introduce a dynamic tumor model that incorporates three factors: radiation induced cell kill, tumor shrinkage, and tumor cell repopulation. The design of multi-stage radiotherapy is formulated as a mathematical optimization problem in which the total dose to the liver is minimized, subject to delivering the prescribed dose to the tumor. Based on the model, we gain insight into the optimal administration of radiation over time, i.e. the optimal treatment gaps and dose levels. We analyze treatments consisting of two stages in detail. The analysis confirms the intuition that the second stage should be delivered just before the tumor size reaches a minimum and repopulation overcompensates shrinking. Furthermore, it was found that, for a large range of model parameters, approximately one third of the dose should be delivered in the first stage. The projected benefit of multi-stage treatments depends on model assumptions. However, the model predicts large liver dose reductions by more than a factor of two for plausible model parameters. The analysis of the tumor model suggests that substantial reduction in normal tissue dose can be achieved by exploiting tumor shrinkage via an optimal design of multi-stage treatments. This suggests taking a fresh look at multi-stage radiotherapy for selected disease sites where substantial tumor regression translates into reduced target volumes.

physics.med-ph↗

arXiv2011

Including robustness in multi-criteria optimization for intensity-modulated proton therapy

We present a method to include robustness into a multi-criteria optimization (MCO) framework for intensity-modulated proton therapy (IMPT). The approach allows one to simultaneously explore the trade-off between different objectives as well as the trade-off between robustness and nominal plan quality. In MCO, a database of plans each emphasizing different treatment planning objectives, is pre-computed to approximate the Pareto surface. An IMPT treatment plan that strikes the best balance between the different objectives can be selected by navigating on the Pareto surface. In our approach, robustness is integrated into MCO by adding robustified objectives and constraints to the MCO problem. Uncertainties of the robust problem are modeled by pre-calculated dose-influence matrices for a nominal scenario and a number of pre-defined error scenarios. A robustified objective represents the worst objective function value that can be realized for any of the error scenarios. The optimization method is based on a linear projection solver and is capable of handling large problem sizes resulting from a fine dose grid resolution, many scenarios, and a large number of proton pencil beams. A base-of-skull case is used to demonstrate the robust optimization method. It is demonstrated that the robust optimizationmethod reduces the sensitivity of the treatment plan to setup and range errors to a degree that is not achieved by a safety margin approach. A chordoma case is analyzed in more detail to demonstrate the involved tradeoffs between target underdose and brainstem sparing as well as robustness and nominal plan quality. The latter illustrates the advantage of MCO in the context of robust planning. For all cases examined, the robust optimization for each Pareto optimal plan takes less than 5 min on a standard computer, making a computationally friendly interface possible to the planner.

physics.med-ph↗

arXiv2011

Multicriteria VMAT optimization

Purpose: To make the planning of volumetric modulated arc therapy (VMAT) faster and to explore the tradeoffs between planning objectives and delivery efficiency. Methods: A convex multicriteria dose optimization problem is solved for an angular grid of 180 equi-spaced beams. This allows the planner to navigate the ideal dose distribution Pareto surface and select a plan of desired target coverage versus organ at risk sparing. The selected plan is then made VMAT deliverable by a fluence map merging and sequencing algorithm, which combines neighboring fluence maps based on a similarity score and then delivers the merged maps together, simplifying delivery. Successive merges are made as long as the dose distribution quality is maintained. The complete algorithm is called VMERGE. Results: VMERGE is applied to three cases: a prostate, a pancreas, and a brain. In each case, the selected Pareto-optimal plan is matched almost exactly with the VMAT merging routine, resulting in a high quality plan delivered with a single arc in less than five minutes on average. VMERGE offers significant improvements over existing VMAT algorithms. The first is the multicriteria planning aspect, which greatly speeds up planning time and allows the user to select the plan which represents the most desirable compromise between target coverage and organ at risk sparing. The second is the user-chosen epsilon-optimality guarantee of the final VMAT plan. Finally, the user can explore the tradeoff between delivery time and plan quality, which is a fundamental aspect of VMAT that cannot be easily investigated with current commercial planning systems.

physics.med-ph↗

arXiv2011

A Dynamic Programming Approach to Adaptive Fractionation

We conduct a theoretical study of various solution methods for the adaptive fractionation problem. The two messages of this paper are: (i) dynamic programming (DP) is a useful framework for adaptive radiation therapy, particularly adaptive fractionation, because it allows us to assess how close to optimal different methods are, and (ii) heuristic methods proposed in this paper are near-optimal, and therefore, can be used to evaluate the best possible benefit of using an adaptive fraction size. The essence of adaptive fractionation is to increase the fraction size when the tumor and organ-at-risk (OAR) are far apart (a "favorable" anatomy) and to decrease the fraction size when they are close together. Given that a fixed prescribed dose must be delivered to the tumor over the course of the treatment, such an approach results in a lower cumulative dose to the OAR when compared to that resulting from standard fractionation. We first establish a benchmark by using the DP algorithm to solve the problem exactly. In this case, we characterize the structure of an optimal policy, which provides guidance for our choice of heuristics. We develop two intuitive, numerically near-optimal heuristic policies, which could be used for more complex, high-dimensional problems. Furthermore, one of the heuristics requires only a statistic of the motion probability distribution, making it a reasonable method for use in a realistic setting. Numerically, we find that the amount of decrease in dose to the OAR can vary significantly (5 - 85%) depending on the amount of motion in the anatomy, the number of fractions, and the range of fraction sizes allowed. In general, the decrease in dose to the OAR is more pronounced when: (i) we have a high probability of large tumor-OAR distances, (ii) we use many fractions (as in a hyper-fractionated setting), and (iii) we allow large daily fraction size deviations.

physics.med-ph↗

arXiv2009

On the tradeoff between treatment time and plan quality in rotational arc radiation delivery

A new delivery option for cancer centers equipped with linear accelerators fitted with multi-leaf collimators (MLC) -- i.e. centers which can perform intensity modulated radiation therapy (IMRT) -- is rotational delivery. In rotational delivery, the beam is on while the gantry is rotating and the MLC leaves are moving, thus treating the patient more efficiently (regarding time) than in IMRT. A consideration that should be examined when evaluating this new type of delivery method is the tradeoff between treatment plan quality and delivery time: what do we sacrifice in terms of plan quality by guaranteeing a 2 minute delivery, for example? In this paper we examine this question by studying a simplified 2D phantom where leaf and gantry motion are directly included in the optimization model. We formulate the model as a linear mixed integer program. Because of the difficulty in solving to optimality, we employ additional models which allow us to trap the true optimal solution between upper and lower bounds. The lower bound solutions reveal which beam directions are most useful for treatment (i.e. where the gantry should slow down), and this information is used to produce deliverable solutions close to the lower bound solutions. For the phantom cases studied, we show that when time is not an issue, IMRT solutions are optimal, but when allowable treatment time is constrained sufficiently, rotational delivery is the preferred choice.

physics.med-ph↗

arXiv2009

The number of beams in IMRT - theoretical investigations and implications for single-arc IMRT

The first purpose of this paper is to shed some new light on the old question of selecting the number of beams in intensity-modulated radiation therapy (IMRT). The second purpose is to illuminate the related issue of discrete static beam angles vs. rotational techniques, which has recently re-surfaced due to the advancement of volumetric arc therapy (VMAT). A specific objective is to find analytical expressions that allow one to address the points raised above. To make the problem mathematically tractable, it is assumed that the depth dose is flat and that the lateral dose profile can be approximated by polynomials, specifically Chebyshev polynomials of the first kind, of finite degree. The application of methods known from image reconstruction then allows one to answer the first question above as follows: The required number of beams is determined by the maximum degree of the polynomials used in the approximation of the beam profiles, which is a measure of the dose variability. There is nothing to be gained by using more beams. In realistic cases, in which the variability of the lateral dose profile is restricted in several ways, the required number of beams is of the order of 10 to 20. The consequence of delivering the beams with a `leaf sweep' technique during continuous rotation of the gantry, as in VMAT, is also derived in analytical form. The main effect is that the beams fan out, but the effect near the axis of rotation is small. This result can serve as a theoretical justification of VMAT. Overall the analytical derivations in this paper, albeit based on strong simplifications, provide new insights into, and a deeper understanding of, the beam angle problem in IMRT.

physics.med-ph↗