The transformation model is an essential component of any deformable image registration approach. It provides a representation of physical deformations between images, thereby defining the range and realism of registrations that can be found. Two types of transformation models have emerged as popular choices: B-spline models and mesh models. Although both models have been investigated in detail, a direct comparison has not yet been made, since the models are optimized using very different optimization methods in practice. B-spline models are predominantly optimized using gradient-descent methods, while mesh models are typically optimized using finite-element method solvers or evolutionary algorithms. Multi-objective optimization methods, which aim to find a diverse set of high-quality trade-off registrations, are increasingly acknowledged to be important in deformable image registration. Since these methods search for a diverse set of registrations, they can provide a more complete picture of the capabilities of different transformation models, making them suitable for a comparison of models. In this work, we conduct the first direct comparison between B-spline and mesh transformation models, by optimizing both models with the same state-of-the-art multi-objective optimization method, the Multi-Objective Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (MO-RV-GOMEA). The combination with B-spline transformation models, moreover, is novel. We experimentally compare both models on two different registration problems that are both based on pelvic CT scans of cervical cancer patients, featuring large deformations. Our results, on three cervical cancer patients, indicate that the choice of transformation model can have a profound impact on the diversity and quality of achieved registration outcomes.

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In recent years, the use of Artificial Intelligence (AI) has become prevalent in a large number of societally relevant, real-world problems, e.g., in the domains of engineering and health care. The field of Evolutionary Computation (EC) can be considered to be a sub-field of AI, concerning optimization using Evolutionary Algorithms (EAs), which are population-based (meta-)heuristics that employ the Darwinian principles of evolution, i.e., variation and selection. Such EAs are historically mainly considered for the optimization of difficult, non-linear problems in a Black-Box Optimization (BBO) setting, because EAs can effectively optimize such problems even when very little is known about the optimization problem and its structure. This is in contrast to optimization methods that are specifically designed for certain problems of which the definition and structure are known, i.e., a White-Box Optimization (WBO) setting.@en

Often, real-world problems are of the gray-box type. It has been shown that the Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (RV-GOMEA) is in principle capable of exploiting such a Gray-Box Optimization (GBO) setting using linkage models that capture dependencies between problem variables, resulting in excellent performance and scalability on both benchmark and real-world problems that allow for partial evaluations. However, linkage models proposed for RV-GOMEA so far cannot explicitly capture overlapping dependencies. Consequently, performance degrades if such dependencies exist. In this paper, we therefore introduce various ways of using conditional linkage models in RV-GOMEA. Their use is compared to that of non-conditional models, and to VkD-CMA, whose performance is among the state of the art, on various benchmark problems with overlapping dependencies. We find that RV-GOMEA with conditional linkage models achieves the best scalability on most problems, with conditional models leading to similar or better performance than non-conditional models. We conclude that the introduction of conditional linkage models to RV-GOMEA is an important contribution, as it expands the set of problems for which optimization in a GBO setting results in substantially improved performance and scalability. In future work, conditional linkage models may prove to benefit the optimization of real-world problems.

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The recently introduced real-valued gene-pool optimal mixing evolutionary algorthm (RV-GOMEA) has been shown to be among the state of the art for solving gray-box optimization problems where partial evaluations can be leveraged. A core strength is its ability to effectively exploit the linkage structure of a problem, which often is unknown a priori and has to be learned online. Previously published work on RV-GOMEA, however, demonstrated excellent scalability when the linkage structure is prespecified appropriately. A mutual information-based metric to learn linkage structure online, as commonly adopted in EDA's and the original discrete version of the gene-pool optimal mixing evolutionary algorithm, did not lead to similarly excellent results, especially in a black-box setting. In this article, the strengths of RV-GOMEA are combined with a new fitness-based linkage learning approach that is inspired by differential grouping that reduces its computational overhead by an order of magnitude for problems with fewer interactions. The resulting new version of RV-GOMEA achieves scalability similar to when a predefined linkage model is used, outperforming also, for the first time, the EDA AMaLGaM upon which it is partially based in a black-box setting where partial evaluations cannot be leveraged.11This article is extended from the MSc thesis of Chantal Olieman, available at https://repository.tudelft.nl/ [24].

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Purpose: The purpose of this study is to improve upon a recently introduced bi-objective treatment planning method for prostate high-dose-rate (HDR) brachytherapy (BT), both in terms of resulting plan quality and runtime requirements, to the extent that its execution time is clinically acceptable. Methods: Bi-objective treatment planning is done using a state-of-the-art multiobjective evolutionary algorithm, which produces a large number of potential treatment plans with different trade-offs between coverage of the target volumes and sparing organs at risk. A graphics processing unit (GPU) is used for large-scale parallelization of dose calculations and the calculation of the dose-volume (DV) indices of potential treatment plans. Moreover, the objectives of the previously used bi-objective optimization model are modified to produce better results. Results: We applied the GPU-accelerated bi-objective treatment planning method to a set of 18 patients, resulting in a set containing a few hundred potential treatment plans with different trade-offs for each of these patients. Due to accelerations introduced in this article, results previously achieved after 1 hour are now achieved within 30 seconds of optimization. We found plans satisfying the clinical protocol for 15 of 18 patients, whereas this was the case for only 4 of 18 clinical plans. Higher quality treatment plans are obtained when the accuracy of DV index calculation is increased using more dose calculation points, requiring still no more than 3 minutes of optimization for 100 000 points. Conclusions: Large sets of high-quality treatment plans that trade-off coverage and sparing are now achievable within 30 seconds, due to the GPU-acceleration of a previously introduced bi-objective treatment planning method for prostate HDR brachytherapy. Higher quality plans can be achieved when optimizing for 3 minutes, which we still consider to be clinically acceptable. This allows for more insightful treatment plan selection in a clinical setting.

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The importance and potential of Gray-Box Optimization (GBO) with evolutionary algorithms is becoming increasingly clear lately both for benchmark and real-world problems. We consider the GBO setting where partial evaluations are possible, meaning that sub-functions of the evaluation function are known and can be exploited to improve optimization efficiency. In this paper, we show that the efficiency of GBO can be greatly improved through large-scale parallelism, exploiting the fact that each evaluation function requires the calculation of a number of independent sub-functions. This is especially interesting for real-world problems where often the majority of the computational effort is spent on the evaluation function. Moreover, we show how the best parallelization technique largely depends on factors including the number of sub-functions and their required computation time, revealing that for different parts of the optimization the best parallelization technique should be selected based on these factors. As an illustration, we show how large-scale parallelization can be applied to optimization of high-dose-rate brachytherapy treatment plans for prostate cancer. We find that use of a modern Graphics Processing Unit (GPU) was the most efficient parallelization technique in all realistic scenarios, leading to substantial speed-ups up to a factor of 73.@en

We address the problemof high-dose-rate brachytherapy treatment planning for prostate cancer. The problem involves determining a treatment plan consisting of the so-called dwell times that a radiation source resides at different positions inside the patient such that the prostate volume and the seminal vesicles are covered by the prescribed radiation dose level as much as possiblewhile the organs at risk, e.g., bladder, rectum, and urethra, are irradiated as little as possible. The problem is highly constrained, following clinical requirements for radiation dose distributionwhile the planning process for treatment planners to design a clinically-Acceptable treatment plan is strictly time-limited. In this paper, we propose that the problem can be formulated as a bi-objective optimization problem that intuitively describes trade-offs between target volumes to be radiated and organs to be spared. We solve this problem with the recently-introduced Multi-Objective Real-Valued Genepool Optimal Mixing Evolutionary Algorithm (MO-RV-GOMEA), which is a promising MOEA that is able to effectively exploit dependencies between problem variables to tackle complicated problems in the continuous domain. MO-RV-GOMEA also has the capability to perform partial evaluations if problem structures allow local variations in existing solutions to be efficiently computed, substantially accelerating the overall optimization performance. Experiments on real medical data and comparison with state-of-Theart MOEAs confirm our claims.@en

The recently introduced Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) has been shown to be among the state-of-the-art for solving discrete optimization problems. Key to the success of GOMEA is its ability to efficiently exploit the linkage structure of a problem. Here, we introduce the Real-Valued GOMEA (RV-GOMEA), which incorporates several aspects of the real-valued EDA known as AMaLGaM into GOMEA in order to make GOMEA well-suited for real-valued optimization. The key strength of GOMEA to competently exploit linkage structure is effectively preserved in RV-GOMEA, enabling excellent performance on problems that exhibit a linkage structure that is to some degree decomposable. Moreover, the main variation operator of GOMEA enables substantial improvements in performance if the problem allows for partial evaluations, which may be very well possible in many real-world applications. Comparisons of performance with state-of-the-art algorithms such as CMA-ES and AMaLGaM on a set of well-known benchmark problems show that RV-GOMEA achieves comparable, excellent scalability in case of black-box optimization. Moreover, RV-GOMEA achieves unprecedented scalability on problems that allow for partial evaluations, reaching near-optimal solutions for problems with up to millions of real-valued variables within one hour on a normal desktop computer.@en

A multi-objective optimization approach is often followed by an a posteriori decision-making process, during which the most appropriate solution of the Pareto set is selected by a professional in the field. Conventional visualization methods do not correct for Pareto fronts with irregularly-spaced solutions. However, achieving a uniform spread of solutions can make the decision-making process more intuitive when decision tools such as sliders, which represent the preference for each objective, are used. We propose a method that maps an m-dimensional Pareto front to an (m - 1)-simplex and spreads out points to achieve a more uniform distribution of these points in the simplex while maintaining the local neighborhood structure of the solutions as much as possible. This set of points can then more intuitively be navigated due to the more uniform distribution. We test our approach on a set of non-uniformly spaced 3D Pareto fronts of a real-world problem: deformable image registration of medical images. The results of these experiments are visualized as points in a triangle, showing that we indeed achieve a representation of the Pareto front with a near-uniform distribution of points where these are still positioned as expected, i.e., according to their quality in each of the objectives of interest.@en

The recently introduced Multi-Objective Gene-pool Optimal Mixing Evolutionary Algorithm (MO-GOMEA) exhibits excellent scalability in solving a wide range of challenging discrete multi-objective optimization problems. In this paper, we address scalability issues in solving multi-objective optimization problems with continuous variables by introducing the Multi-Objective Real-Valued GOMEA (MO-RV-GOMEA), which combines MO-GOMEA with aspects of the multi-objective estimation-of-distribution algorithm known as MAMaLGaM. MO-RV-GOMEA exploits linkage structure in optimization problems by performing distribution estimation, adaptation, and sampling as well as solution mixing based on an explicitly-defined linkage model. Such a linkage model can be defined a priori when some problem-specific knowledge is available, or it can be learned from the population. The scalability of MO-RV-GOMEA using different linkage models is compared to the state-of-the-art multi-objective evolutionary algorithms NSGA-II and MAMaLGaM on a wide range of benchmark problems. MO-RV-GOMEA is found to retain the excellent scalability of MO-GOMEA through the successful exploitation of linkage structure, scaling substantially better than NSGA-II and MAMaLGaM. This scalability is even further improved when partial evaluations are possible, achieving strongly sub-linear scalability in terms of the number of evaluations.@en

Taking a multi-objective optimization approach to deformable image registration has recently gained attention, because such an approach removes the requirement of manually tuning the weights of all the involved objectives. Especially for problems that require large complex deformations, this is a non-trivial task. From the resulting Pareto set of solutions one can then much more insightfully select a registration outcome that is most suitable for the problem at hand. To serve as an internal optimization engine, currently used multi-objective algorithms are competent, but rather inefficient. In this paper we largely improve upon this by introducing a multi-objective real-valued adaptation of the recently introduced Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) for discrete optimization. In this work, GOMEA is tailored specifically to the problem of deformable image registration to obtain substantially improved efficiency. This improvement is achieved by exploiting a key strength of GOMEA: iteratively improving small parts of solutions, allowing to faster exploit the impact of such updates on the objectives at hand through partial evaluations. We performed experiments on three registration problems. In particular, an artificial problem containing a disappearing structure, a pair of pre- and post-operative breast CT scans, and a pair of breast MRI scans acquired in prone and supine position were considered. Results show that compared to the previously used evolutionary algorithm, GOMEA obtains a speed-up of up to a factor of ~1600 on the tested registration problems while achieving registration outcomes of similar quality.@en