Inverse treatment planning by physically constrained minimization of a biological objective function

Conic formulation of fluence map optimization problems

The convexity of objectives and constraints in fluence map optimization (FMO) for radiation therapy has been extensively studied. Next to convexity, there is another important characteristic of optimization functions and problems, which has thus far not been considered in FMO literature: conic representation. Optimization problems that are conically representable using quadratic, exponential and power cones are solvable with advanced primal-dual interior-point algorithms. These algorithms guarantee an optimal solution in polynomial time and have good performance in practice. In this paper, we construct conic representations for most FMO objectives and constraints. This paper is the first that shows that FMO problems containing multiple biological evaluation criteria can be solved in polynomial time. For fractionation-corrected functions for which no exact conic reformulation is found, we provide an accurate approximation that is conically representable. We present numerical results on the TROTS data set, which demonstrate very stable numerical performance for solving FMO problems in conic form. With ongoing research in the optimization community, improvements in speed can be expected, which makes conic optimization a promising alternative for solving FMO problems.

A cooperative game solution approach for intensity modulated radiation therapy design: Nash Social Welfare optimization

We study the fluency map optimization problem in Intensity Modulated Radiation Therapy (IMRT) from a cooperative game theory point of view. We consider the cancerous and healthy organs in a patient's body as players of a game, where cancerous organs seek to eliminate the cancerous cells and healthy organs seek to receive no harm. The goal is to balance the trade-offs between the utility of players by forming a grand coalition between them. We do so by proposing a methodology that solves a few convex optimization problems in order to transform the fluency map optimization problem into a bargaining game. To solve the bargaining game, we employ the concept of Nash Social Welfare (NSW) optimization due to the desirable efficiency and fairness properties of its outcomes. The proposed NSW optimization is convex and can be solved by powerful commercial solvers such as CPLEX. An additional advantage of the proposed approach is that it has a new control lever for the fluency map optimization, the so-called negotiation powers, which enables practitioners to put more emphasis on an organ by changing its negotiation power. To show the efficacy of our proposed methodology, we apply it to the TG-119 case and a liver case. We compare our proposed approach with a state-of-the-art approach through creating Dose Volume Histograms.

Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization

Inverse planning is a method of radiotherapy treatment planning where the care team begins with the desired dose distribution satisfying prescribed clinical objectives, and then determines the treatment parameters that will achieve it. The variety in symmetry, form, and characteristics of the objective functions describing clinical criteria requires a flexible optimization approach in order to obtain optimized treatment plans. Therefore, we introduce and discuss a nonlinear optimization formulation called the split common coincidence point problem (SCCPP). We show that the SCCPP is a suitable formulation for the inverse planning optimization problem with the flexibility of accommodating several biological and/or physical clinical objectives. Also, we propose an iterative algorithm for approximating the solution of the SCCPP, and using Bregman techniques, we establish that the proposed algorithm converges to a solution of the SCCPP and to an extremum of the inverse planning optimization problem. We end with a note on useful insights on implementing the algorithm in a clinical setting.

Prescription Value-Based Automatic Optimization of Importance Factors in Inverse Planning

Objective:

An automatic method for the optimization of importance factors was proposed to improve the efficiency of inverse planning.

Methods:

The automatic method consists of 3 steps: (1) First, the importance factors are automatically and iteratively adjusted based on our proposed penalty strategies. (2) Then, plan evaluation is performed to determine whether the obtained plan is acceptable. (3) If not, a higher penalty is assigned to the unsatisfied objective by multiplying it by a compensation coefficient. The optimization processes are performed alternately until an acceptable plan is obtained or the maximum iteration N_max of step (3) is reached.

Results:

Tested on 2 kinds of clinical cases and compared with manual method, the results showed that the quality of the proposed automatic plan was comparable to, or even better than, the manual plan in terms of the dose–volume histogram and dose distributions.

Conclusions:

The proposed algorithm has potential to significantly improve the efficiency of the existing manual adjustment methods for importance factors and contributes to the development of fully automated planning. Especially, the more the subobjective functions, the more obvious the advantage of our algorithm.

An integrated strategy of biological and physical constraints in biological optimization for cervical carcinoma

Background

For cervical carcinoma cases, this study aimed to evaluate the quality of intensity-modulated radiation therapy (IMRT) plans optimized by biological constraints. Furthermore, a new integrated strategy in biological planning module was proposed and verified.

Methods

Twenty patients of advanced stage cervical carcinoma were enrolled in this study. For each patient, dose volume optimization (DVO), biological model optimization (BMO) and integrated strategy optimization (ISO) plans were created using same treatment parameters. Different biological models were also used for organ at risk (OAR) in BMO plans, which include the LKB and Poisson models. Next, BMO plans were compared with their corresponding DVO plans, in order to evaluate BMO plan quality. ISO plans were also compared with DVO and BMO plans, in order to verify the performance of the integrated strategy.

Results

BMO plans produced slightly inhomogeneity and less coverage of planning target volume (PTV) (V95=96.79, HI = 0.10: p < 0.01). However, the tumor control probability (TCP) value, both from DVO and BMO plans, were comparable. For the OARs, BMO plans produced lower normal tissue complication probability (NTCP) of rectum (NTCP = 0.11) and bladder (NTCP = 0.14) than in the corresponding DVO plans (NTCP = 0.19 and 0.18 for rectum and bladder; p < 0.01 for rectum and p = 0.03 for bladder). V95, D98, CI and HI values that were produced by ISO plans (V95 = 98.31, D98 = 54.18Gy, CI = 0.76, HI = 0.09) were greatly better than BMO plans (V95 = 96.79, D98 = 53.42Gy, CI = 0.71, HI = 0.10) with significant differences. Furthermore, ISO plans produced lower NTCP values of rectum (NTCP = 0.14) and bladder (NTCP = 0.16) than DVO plans (NTCP = 0.19 and 0.18 for rectum and bladder, respectively) with significant differences.

Conclusions

BMO plans produced lower NTCP values of OARs compared to DVO plans for cervical carcinoma cases, and resulted in slightly less target coverage and homogeneity. The integrated strategy, proposed in this study, could improve the coverage, conformity and homogeneity of PTV greater than the BMO plans, as well as reduce the NTCP values of OARs greater than the DVO plans.

Multimodality functional imaging in radiation therapy planning: relationships between dynamic contrast-enhanced MRI, diffusion-weighted MRI, and 18F-FDG PET

OBJECTIVES

Biologically guided radiotherapy needs an understanding of how different functional imaging techniques interact and link together. We analyse three functional imaging techniques that can be useful tools for achieving this objective.

MATERIALS AND METHODS

The three different imaging modalities from one selected patient are ADC maps, DCE-MRI, and 18F-FDG PET/CT, because they are widely used and give a great amount of complementary information. We show the relationship between these three datasets and evaluate them as markers for tumour response or hypoxia marker. Thus, vascularization measured using DCE-MRI parameters can determine tumour hypoxia, and ADC maps can be used for evaluating tumour response.

RESULTS

ADC and DCE-MRI include information from 18F-FDG, as glucose metabolism is associated with hypoxia and tumour cell density, although 18F-FDG includes more information about the malignancy of the tumour. The main disadvantage of ADC maps is the distortion, and we used only low distorted regions, and extracellular volume calculated from DCE-MRI can be considered equivalent to ADC in well-vascularized areas.

CONCLUSION

A dataset for achieving the biologically guided radiotherapy must include a tumour density study and a hypoxia marker. This information can be achieved using only MRI data or only PET/CT studies or mixing both datasets.

The use of biologically related model (Eclipse) for the intensity-modulated radiation therapy planning of nasopharyngeal carcinomas

Purpose

Intensity-modulated radiation therapy (IMRT) is the most common treatment technique for nasopharyngeal carcinoma (NPC). Physical quantities such as dose/dose-volume parameters are used conventionally for IMRT optimization. The use of biological related models has been proposed and can be a new trend. This work was to assess the performance of the biologically based IMRT optimization model installed in a popular commercial treatment planning system (Eclipse) as compared to its dose/dose volume optimization model when employed in the clinical environment for NPC cases.

Methods

Ten patients of early stage NPC and ten of advanced stage NPC were selected for this study. IMRT plans optimized using biological related approach (BBTP) were compared to their corresponding plans optimized using the dose/dose volume based approach (DVTP). Plan evaluation was performed using both biological indices and physical dose indices such as tumor control probability (TCP), normal tissue complication probability (NTCP), target coverage, conformity, dose homogeneity and doses to organs at risk. The comparison results of the more complex advanced stage cases were reported separately from those of the simpler early stage cases.

Results

The target coverage and conformity were comparable between the two approaches, with BBTP plans producing more hot spots. For the primary targets, BBTP plans produced comparable TCP for the early stage cases and higher TCP for the advanced stage cases. BBTP plans reduced the volume of parotid glands receiving doses of above 40 Gy compared to DVTP plans. The NTCP of parotid glands produced by BBTP were 8.0±5.8 and 7.9±8.7 for early and advanced stage cases, respectively, while those of DVTP were 21.3±8.3 and 24.4±12.8, respectively. There were no significant differences in the NTCP values between the two approaches for the serial organs.

Conclusions

Our results showed that the BBTP approach could be a potential alternative approach to the DVTP approach for NPC.

Adaptive IMRT using a multiobjective evolutionary algorithm integrated with a diffusion-invasion model of glioblastoma

We demonstrate a patient-specific method of adaptive IMRT treatment for glioblastoma using a multiobjective evolutionary algorithm (MOEA). The MOEA generates spatially optimized dose distributions using an iterative dialogue between the MOEA and a mathematical model of tumor cell proliferation, diffusion and response. Dose distributions optimized on a weekly basis using biological metrics have the potential to substantially improve and individualize treatment outcomes. Optimized dose distributions were generated using three different decision criteria for the tumor and compared with plans utilizing standard dose of 1.8 Gy/fraction to the CTV (T2-visible MRI region plus a 2.5 cm margin). The sets of optimal dose distributions generated using the MOEA approach the Pareto Front (the set of IMRT plans that delineate optimal tradeoffs amongst the clinical goals of tumor control and normal tissue sparing). MOEA optimized doses demonstrated superior performance as judged by three biological metrics according to simulated results. The predicted number of reproductively viable cells 12 weeks after treatment was found to be the best target objective for use in the MOEA.

Optimization of equivalent uniform dose using the L-curve criterion

Optimization of equivalent uniform dose (EUD) in inverse planning for intensity-modulated radiation therapy (IMRT) prevents variation in radiobiological effect between different radiotherapy treatment plans, which is due to variation in the pattern of dose nonuniformity. For instance, the survival fraction of clonogens would be consistent with the prescription when the optimized EUD is equal to the prescribed EUD. One of the problems in the practical implementation of this approach is that the spatial dose distribution in EUD-based inverse planning would be underdetermined because an unlimited number of nonuniform dose distributions can be computed for a prescribed value of EUD. Together with ill-posedness of the underlying integral equation, this may significantly increase the dose nonuniformity. To optimize EUD and keep dose nonuniformity within reasonable limits, we implemented into an EUD-based objective function an additional criterion which ensures the smoothness of beam intensity functions. This approach is similar to the variational regularization technique which was previously studied for the dose-based least-squares optimization. We show that the variational regularization together with the L-curve criterion for the regularization parameter can significantly reduce dose nonuniformity in EUD-based inverse planning.

A theoretical approach to the problem of dose-volume constraint estimation and their impact on the dose-volume histogram selection

This paper outlines a theoretical approach to the problem of estimating and choosing dose-volume constraints. Following this approach, a method of choosing dose-volume constraints based on biological criteria is proposed. This method is called "reverse normal tissue complication probability (NTCP) mapping into dose-volume space" and may be used as a general guidance to the problem of dose-volume constraint estimation. Dose-volume histograms (DVHs) are randomly simulated, and those resulting in clinically acceptable levels of complication, such as NTCP of 5 +/- 0.5%, are selected and averaged producing a mean DVH that is proven to result in the same level of NTCP. The points from the averaged DVH are proposed to serve as physical dose-volume constraints. The population-based critical volume and Lyman NTCP models with parameter sets taken from literature sources were used for the NTCP estimation. The impact of the prescribed value of the maximum dose to the organ, D(max), on the averaged DVH and the dose-volume constraint points is investigated. Constraint points for 16 organs are calculated. The impact of the number of constraints to be fulfilled based on the likelihood that a DVH satisfying them will result in an acceptable NTCP is also investigated. It is theoretically proven that the radiation treatment optimization based on physical objective functions can sufficiently well restrict the dose to the organs at risk, resulting in sufficiently low NTCP values through the employment of several appropriate dose-volume constraints. At the same time, the pure physical approach to optimization is self-restrictive due to the preassignment of acceptable NTCP levels thus excluding possible better solutions to the problem.

Optimization of intensity-modulated radiation therapy with biological objectives

IMRT treatment planning via biological objectives gives rise to constrained nonlinear optimization problems. We consider formulations with nonlinear objectives based on the equivalent uniform dose (EUD), with bound constraints on the beamlet weights, and describe fast, flexible variants of the two-metric gradient-projection approach for solving them efficiently and in a mathematically sound manner. We conclude that an approach that calculates the Newton component of the step iteratively, by means of the conjugate-gradient algorithm and an implicit representation of the Hessian matrix, is most effective. We also present an efficient heuristic for obtaining an approximate solution with a smoother distribution of beamlet weights. The effectiveness of the methods is verified by testing on a medium-scale clinical case.

Dose–volume based ranking of incident beam direction and its utility in facilitating IMRT beam placement

PURPOSE

Beam orientation optimization in intensity-modulated radiation therapy (IMRT) is computationally intensive, and various single beam ranking techniques have been proposed to reduce the search space. Up to this point, none of the existing ranking techniques considers the clinically important dose-volume effects of the involved structures, which may lead to clinically irrelevant angular ranking. The purpose of this work is to develop a clinically sensible angular ranking model with incorporation of dose-volume effects and to show its utility for IMRT beam placement.

METHODS AND MATERIALS

The general consideration in constructing this angular ranking function is that a beamlet/beam is preferable if it can deliver a higher dose to the target without exceeding the tolerance of the sensitive structures located on the path of the beamlet/beam. In the previously proposed dose-based approach, the beamlets are treated independently and, to compute the maximally deliverable dose to the target volume, the intensity of each beamlet is pushed to its maximum intensity without considering the values of other beamlets. When volumetric structures are involved, the complication arises from the fact that there are numerous dose distributions corresponding to the same dose-volume tolerance. In this situation, the beamlets are not independent and an optimization algorithm is required to find the intensity profile that delivers the maximum target dose while satisfying the volumetric constraints. In this study, the behavior of a volumetric organ was modeled by using the equivalent uniform dose (EUD). A constrained sequential quadratic programming algorithm (CFSQP) was used to find the beam profile that delivers the maximum dose to the target volume without violating the EUD constraint or constraints. To assess the utility of the proposed technique, we planned a head-and-neck and abdominal case with and without the guidance of the angular ranking information. The qualities of the two types of IMRT plans were compared quantitatively.

RESULTS

An effective angular ranking model with consideration of volumetric effect has been developed. It is shown that the previously reported dose-based angular ranking represents a special case of the general formalism proposed here. Application of the technique to a abdominal and a head-and-neck IMRT case indicated that the proposed technique is capable of producing clinically sensible angular ranking. In both cases, we found that the IMRT plans obtained under the guidance of EUD-based angular ranking were improved in comparison with that obtained using the conventional uniformly spaced beams.

CONCLUSIONS

The EUD-based function is a general approach for angular ranking and allows us to identify the potentially good and bad angles for clinically complicated cases. The ranking can be used either as a guidance to facilitate the manual beam placement or as prior information to speed up the computer search for the optimal beam configuration. Thus the proposed technique should have positive clinical impact in facilitating the IMRT planning process.

A unifying framework for multi-criteria fluence map optimization models

Models for finding treatment plans for intensity modulated radiation therapy are usually based on a number of structure-based treatment plan evaluation criteria, which are often conflicting. Rather than formulating a model that a priori quantifies the trade-offs between these criteria, we consider a multi-criteria optimization approach that aims at finding the so-called undominated treatment plans. We present a unifying framework for studying multi-criteria optimization problems for treatment planning that establishes conditions under which treatment plan evaluation criteria can be transformed into convex criteria while preserving the set of undominated treatment plans. Such transformations are identified for many of the criteria that have been proposed to date, establishing equivalences between these criteria. In addition, it is shown that the use of a nonconvex criterion can often be avoided by transformation to an equivalent convex criterion. In particular, we show that models employing criteria such as tumour control probability, normal tissue complication probability, probability of uncomplicated tumour control, as well as sigmoidal transformations of (generalized) equivalent uniform dose are equivalent to models formulated in terms of separable voxel-based criteria that penalize dose in individual voxels.