Treatment Optimization Using Physical and Radiobiological Objective Functions

Bi-criteria Pareto optimization to balance irradiation time and dosimetric objectives in proton arc therapy

Objective. Proton arc therapy (PAT) is a new delivery technique that exploits the continuous rotation of the gantry to distribute the therapeutic dose over many angular windows instead of using a few static fields, as in conventional (intensity-modulated) proton therapy. Although coming along with many potential clinical and dosimetric benefits, PAT has also raised a new optimization challenge. In addition to the dosimetric goals, the beam delivery time (BDT) needs to be considered in the objective function. Considering this bi-objective formulation, the task of finding a good compromise with appropriate weighting factors can turn out to be cumbersome.Approach. We have computed Pareto-optimal plans for three disease sites: a brain, a lung, and a liver, following a method of iteratively choosing weight vectors to approximate the Pareto front with few points. Mixed-integer programming (MIP) was selected to state the bi-criteria PAT problem and to find Pareto optimal points with a suited solver.Main results. The trade-offs between plan quality and beam irradiation time (staticBDT) are investigated by inspecting three plans from the Pareto front. The latter are carefully picked to demonstrate significant differences in dose distribution and delivery time depending on their location on the frontier. The results were benchmarked against IMPT and SPArc plans showing the strength of degrees of freedom coming along with MIP optimization.Significance. This paper presents for the first time the application of bi-criteria optimization to the PAT problem, which eventually permits the planners to select the best treatment strategy according to the patient conditions and clinical resources available.

Data for TROTS - The Radiotherapy Optimisation Test Set

The Radiotherapy Optimisation Test Set (TROTS) is an extensive set of problems originating from radiotherapy (radiation therapy) treatment planning. This dataset is created for 2 purposes: (1) to supply a large-scale dense dataset to measure performance and quality of mathematical solvers, and (2) to supply a dataset to investigate the multi-criteria optimisation and decision-making nature of the radiotherapy problem. The dataset contains 120 problems (patients), divided over 6 different treatment protocols/tumour types. Each problem contains numerical data, a configuration for the optimisation problem, and data required to visualise and interpret the results. The data is stored as HDF5 compatible Matlab files, and includes scripts to work with the dataset.

Employing the therapeutic operating characteristic (TOC) graph for individualised dose prescription

Background

In current practice, patients scheduled for radiotherapy are treated according to ‘rigid’ protocols with predefined dose prescriptions that do not consider risk-taking preferences of individuals. The therapeutic operating characteristic (TOC) graph is applied as a decision-aid to assess the trade-off between treatment benefit and morbidity to facilitate dose prescription customisation.

Methods

Historical dose-response data from prostate cancer patient cohorts treated with 3D-conformal radiotherapy is used to construct TOC graphs. Next, intensity-modulated (IMRT) plans are generated by optimisation based on dosimetric criteria and dose-response relationships. TOC graphs are constructed for dose-scaling of the optimised IMRT plan and individualised dose prescription. The area under the TOC curve (AUC) is estimated to measure the therapeutic power of these plans.

Results

On a continuous scale, the TOC graph directly visualises treatment benefit and morbidity risk of physicians’ or patients’ choices for dose (de-)escalation. The trade-off between these probabilities facilitates the selection of an individualised dose prescription. TOC graphs show broader therapeutic window and higher AUCs with increasing target dose heterogeneity.

Conclusions

The TOC graph gives patients and physicians access to a decision-aid and read-out of the trade-off between treatment benefit and morbidity risks for individualised dose prescription customisation over a continuous range of dose levels.

From analytic inversion to contemporary IMRT optimization: radiation therapy planning revisited from a mathematical perspective

In this paper we look at the development of radiation therapy treatment planning from a mathematical point of view. Historically, planning for Intensity-Modulated Radiation Therapy (IMRT) has been considered as an inverse problem. We discuss first the two fundamental approaches that have been investigated to solve this inverse problem: Continuous analytic inversion techniques on one hand, and fully-discretized algebraic methods on the other hand. In the second part of the paper, we review another fundamental question which has been subject to debate from the beginning of IMRT until the present day: The rotation therapy approach versus fixed angle IMRT. This builds a bridge from historic work on IMRT planning to contemporary research in the context of Intensity-Modulated Arc Therapy (IMAT).

Toolkit for determination of dose-response relations, validation of radiobiological parameters and treatment plan optimization based on radiobiological measures

Accurately determined dose-response relations of the different tumors and normal tissues should be estimated and used in the clinic. The aim of this study is to demonstrate developed tools that are necessary for determining the dose-response parameters of tumors and normal tissues, for clinically verifying already published parameter sets using local patient materials and for making use of all this information in the optimization and comparison of different treatment plans and radiation techniques. One of the software modules (the Parameter Determination Module) is designed to determine the dose-response parameters of tumors and normal tissues. This is accomplished by performing a maximum likelihood fitting to calculate the best estimates and confidence intervals of the parameters used by different radiobiological models. Another module of this software (the Parameter Validation Module) concerns the validation and compatibility of external or reported dose-response parameters describing tumor control and normal tissue complications. This is accomplished by associating the expected response rates, which are calculated using different models and published parameter sets, with the clinical follow-up records of the local patient population. Finally, the last module of the software (the Radiobiological Plan Evaluation Module) is used for estimating and optimizing the effectiveness a treatment plan in terms of complication-free tumor control, P(+). The use of the Parameter Determination Module is demonstrated by deriving the dose-response relation of proximal esophagus from head and neck cancer radiotherapy. The application of the Parameter Validation Module is illustrated by verifying the clinical compatibility of those dose-response parameters with the examined treatment methodologies. The Radiobiological Plan Evaluation Module is demonstrated by evaluating and optimizing the effectiveness of head and neck cancer treatment plans. The results of the radiobiological evaluation are compared against dosimetric criteria. The presented toolkit appears to be very convenient and efficient for clinical implementation of radiobiological modeling. It can also be used for the development of a clinical data and health information database for assisting the performance of epidemiological studies and the collaboration between different institutions within research and clinical frameworks.

Variation in radiation sensitivity and repair kinetics in different parts of the spinal cord

BACKGROUND

The spinal cord, known for its strongly serial character and high sensitivity to radiation even when a small segment is irradiated, is one of the most critical organs at risk to be spared during radiation therapy. To compare the sensitivity of different parts of the spinal cord, data for radiation myelopathy have been used.

MATERIAL AND METHODS

In the present study, the relative seriality model was fitted to two different datasets of clinical radiation myelitis concerning cervical spinal cord after treating 248 patients for head and neck cancer and thoracic spinal cord after treating 43 patients with lung carcinoma. The maximum likelihood method was applied to fit the clinical data. The model parameters and their 68% confidence intervals were calculated for each dataset. The alpha/beta ratio for the thoracic cord was also was also found to be 0.9 (0-3.0) Gy.

RESULTS

The dose-response curve for the more sensitive cervical myelopathy is well described by the parameters D(50)=55.9 (54.8-57.1) Gy, gamma=6.9 (5.0-9.2), s=0.13 (0.07-0.24), whereas the thoracic myelopathy is described by the parameters D(50)=75.5 (70.5-80.8) Gy, gamma=1.1 (0.6-1.6), s=36 (3.3-infinity).

DISCUSSION AND CONCLUSIONS

Large differences in radiation response between the cervical and thoracic region of spinal cord are thus observed: cervical myelopathy seems to be characterized by medium seriality, while thoracic spinal cord is characterized by a highly serial dose-response. The much steeper dose-response curve for cervical spinal cord myelopathy can be interpreted as a higher number of functional subunits consistent with a higher amount of white matter close to the brain.

A novel approach to multi-criteria inverse planning for IMRT

Treatment plan optimization is a multi-criteria process. Optimizing solely on one objective or on a sum of a priori weighted objectives may result in inferior treatment plans. Manually adjusting weights or constraints in a trial and error procedure is time consuming. In this paper we introduce a novel multi-criteria optimization approach to automatically optimize treatment constraints (dose-volume and maximum-dose). The algorithm tries to meet these constraints as well as possible, but in the case of conflicts it relaxes lower priority constraints so that higher priority constraints can be met. Afterwards, all constraints are tightened, starting with the highest priority constraints. Applied constraint priority lists can be used as class solutions for patients with similar tumour types. The presented algorithm does iteratively apply an underlying algorithm for beam profile optimization, based on a quadratic objective function with voxel-dependent importance factors. These voxel-dependent importance factors are automatically adjusted to reduce dose-volume and maximum-dose constraint violations.

Design of a fast multileaf collimator for radiobiological optimized IMRT with scanned beams of photons, electrons, and light ions

Intensity modulated radiation therapy is rapidly becoming the treatment of choice for most tumors with respect to minimizing damage to the normal tissues and maximizing tumor control. Today, intensity modulated beams are most commonly delivered using segmental multileaf collimation, although an increasing number of radiation therapy departments are employing dynamic multileaf collimation. The irradiation time using dynamic multileaf collimation depends strongly on the nature of the desired dose distribution, and it is difficult to reduce this time to less than the sum of the irradiation times for all individual peak heights using dynamic leaf collimation [Svensson et al., Phys. Med. Biol. 39, 37-61 (1994)]. Therefore, the intensity modulation will considerably increase the total treatment time. A more cost-effective procedure for rapid intensity modulation is using narrow scanned photon, electron, and light ion beams in combination with fast multileaf collimator penumbra trimming. With this approach, the irradiation time is largely independent of the complexity of the desired intensity distribution and, in the case of photon beams, may even be shorter than with uniform beams. The intensity modulation is achieved primarily by scanning of a narrow elementary photon pencil beam generated by directing a narrow well focused high energy electron beam onto a thin bremsstrahlung target. In the present study, the design of a fast low-weight multileaf collimator that is capable of further sharpening the penumbra at the edge of the elementary scanned beam has been simulated, in order to minimize the dose or radiation response of healthy tissues. In the case of photon beams, such a multileaf collimator can be placed relatively close to the bremsstrahlung target to minimize its size. It can also be flat and thin, i.e., only 15-25 mm thick in the direction of the beam with edges made of tungsten or preferably osmium to optimize the sharpening of the penumbra. The low height of the collimator will minimize edge scatter from glancing incidence. The major portions of the collimator leafs can then be made of steel or even aluminum, so that the total weight of the multileaf collimator will be as low as 10 kg, which may even allow high-speed collimation in real time in synchrony with organ movements. To demonstrate the efficiency of this collimator design in combination with pencil beam scanning, optimal radiobiological treatments of an advanced cervix cancer were simulated. Different geometrical collimator designs were tested for bremsstrahlung, electron, and light ion beams. With a 10 mm half-width elementary scanned photon beam and a steel collimator with tungsten edges, it was possible to make as effective treatments as obtained with intensity modulated beams of full resolution, i.e., here 5 mm resolution in the fluence map. In combination with narrow pencil beam scanning, such a collimator may provide ideal delivery of photons, electrons, or light ions for radiation therapy synchronized to breathing and other organ motions. These high-energy photon and light ion beams may allow three-dimensional in vivo verification of delivery and thereby clinical implementation of the BioArt approach using Biologically Optimized three-dimensional in vivo predictive Assay based adaptive Radiation Therapy [Brahme, Acta Oncol. 42, 123-126 (2003)].

Derivative-free generation and interpolation of convex Pareto optimal IMRT plans

In inverse treatment planning for intensity-modulated radiation therapy (IMRT), beamlet intensity levels in fluence maps of high-energy photon beams are optimized. Treatment plan evaluation criteria are used as objective functions to steer the optimization process. Fluence map optimization can be considered a multi-objective optimization problem, for which a set of Pareto optimal solutions exists: the Pareto efficient frontier (PEF). In this paper, a constrained optimization method is pursued to iteratively estimate the PEF up to some predefined error. We use the property that the PEF is convex for a convex optimization problem to construct piecewise-linear upper and lower bounds to approximate the PEF from a small initial set of Pareto optimal plans. A derivative-free Sandwich algorithm is presented in which these bounds are used with three strategies to determine the location of the next Pareto optimal solution such that the uncertainty in the estimated PEF is maximally reduced. We show that an intelligent initial solution for a new Pareto optimal plan can be obtained by interpolation of fluence maps from neighbouring Pareto optimal plans. The method has been applied to a simplified clinical test case using two convex objective functions to map the trade-off between tumour dose heterogeneity and critical organ sparing. All three strategies produce representative estimates of the PEF. The new algorithm is particularly suitable for dynamic generation of Pareto optimal plans in interactive treatment planning.

Fast, multiple optimizations of quadratic dose objective functions in IMRT

Inverse treatment planning for intensity-modulated radiotherapy may include time consuming, multiple minimizations of an objective function. In this paper, methods are presented to speed up the process of (repeated) minimization of the well-known quadratic dose objective function, extended with a smoothing term that ensures generation of clinically acceptable beam profiles. In between two subsequent optimizations, the voxel-dependent importance factors of the quadratic terms will generally be adjusted, based on an intermediate plan evaluation. The objective function has been written in matrix-vector format, facilitating the use of a recently published, fast quadratic minimization algorithm, instead of commonly applied gradient-based methods. This format also reduces the calculation time in between subsequent minimizations, related to adjustment of the voxel-dependent importance factors. Sparse matrices are used to limit the required amount of computer memory. For three patients, comparisons have been made with a gradient method. Mean speed improvements of up to a factor of 37 have been achieved.

Statistical methods for clinical verification of dose–response parameters related to esophageal stricture and AVM obliteration from radiotherapy

The purpose of this work is to provide some statistical methods for evaluating the predictive strength of radiobiological models and the validity of dose-response parameters for tumour control and normal tissue complications. This is accomplished by associating the expected complication rates, which are calculated using different models, with the clinical follow-up records. These methods are applied to 77 patients who received radiation treatment for head and neck cancer and 85 patients who were treated for arteriovenous malformation (AVM). The three-dimensional dose distribution delivered to esophagus and AVM nidus and the clinical follow-up results were available for each patient. Dose-response parameters derived by a maximum likelihood fitting were used as a reference to evaluate their compatibility with the examined treatment methodologies. The impact of the parameter uncertainties on the dose-response curves is demonstrated. The clinical utilization of the radiobiological parameters is illustrated. The radiobiological models (relative seriality and linear Poisson) and the reference parameters are validated to prove their suitability in reproducing the treatment outcome pattern of the patient material studied (through the probability of finding a worse fit, area under the ROC curve and chi2 test). The analysis was carried out for the upper 5 cm of the esophagus (proximal esophagus) where all the strictures are formed, and the total volume of AVM. The estimated confidence intervals of the dose-response curves appear to have a significant supporting role on their clinical implementation and use.

An integrated two-level hierarchical system for decision making in radiation therapy based on fuzzy cognitive maps

The radiation therapy decision-making is a complex process that has to take into consideration a variety of interrelated functions. Many fuzzy factors that must be considered in the calculation of the appropriate dose increase the complexity of the decision-making problem. A novel approach introduces fuzzy cognitive maps (FCMs) as the computational modeling method, which tackles the complexity and allows the analysis and simulation of the clinical radiation procedure. Specifically this approach is used to determine the success of radiation therapy process estimating the final dose delivered to the target volume, based on the soft computing technique of FCMs. Furthermore a two-level integrated hierarchical structure is proposed to supervise and evaluate the radiotherapy process prior to treatment execution. The supervisor determines the treatment variables of cancer therapy and the acceptance level of final radiation dose to the target volume. Two clinical case studies are used to test the proposed methodology and evaluate the simulation results. The usefulness of this two-level hierarchical structure discussed and future research directions are suggested for the clinical use of this methodology.

[Optimization criteria in intensity-modulated radiotherapy]

The present paper provides an overview on the inverse treatment planning for the assessment of intensity-modulated fields. The problem is to find the optimal dose distribution for given attributes of the irradiated tissue. The attributes of the optimal dose distribution are delineated by an objective function. In practice, models are used that evaluate the physical dose distribution, either directly or through their radiobiological effects. In the simplest case, the squared deviation of the achieved dose distribution is minimized to the prescribed dose distribution. For organs structured in parallel, it is common to introduce dose-volume constraints. Another approach is to optimize a value for the probability of complication-free tumor control. The complication probability for normal tissue, in turn, is a rather complex function. However, using the relative seriality, a simple model can be devised with a certain approximation. Other models of "effective dose" are also presented.

[Design of an inverse planning system for radiotherapy using linear optimization]

Our approach shows that inverse planning for intensity-modulated beams in radiotherapy can be solved efficiently by the mathematical method of linear optimization. The completeness property of this method guarantees that calculated treatment plans fulfill the dose constraints given by the oncologist. Techniques developed by our group can also avoid the possible infeasibility caused by a physically impossible dose distribution. In contrast to other optimization methods, the simplex algorithm used for linear optimization allows a very high optimization speed, i.e., very short planning cycles. These preconditions were integrated in the development of the software system MIPART ("Munich Inverse Planning And Radiotherapy Treatment"). The object-oriented software architecture of MIPART achieves a maximum of extensibility and flexibility. Our clinical tests show that MIPART can be easily integrated in the clinical routine in spite of the complexity of data in the field of intensity-modulated radiotherapy. Moreover, qualitatively better treatment plans can be generated than in conventional treatment planning, especially in complicated cases.

[IMCO(++)--a Monte Carlo based IMRT system]

The application of intensity modulated radiotherapy (IMRT) to dose escalation in the target volume sets particular demands in terms of accuracy of dose calculation. Dose calculation errors due to approximations are compensated by the optimization algorithm, a procedure that ultimately leads to incorrect fluence modulation. Such inaccuracies affect particularly the dose distribution in areas with secondary electron disequilibrium. In case tissues heterogeneity predominates, conventional dose calculation methods (such as Pencil Beam) can produce relative errors up to more than 10%. The accuracy can be significantly improved by the application of a Monte-Carlo (MC) algorithm. This paper describes a MC-based inverse treatment planning algorithm (IMCO++), based on a non-iterative approach with a feedback-controlling process. The convergence behavior of IMCO++ was investigated and the used MC dose-calculation codes MMms and XVMC were compared by means of a heterogeneous phantom. IMCO++ plans were optimized in various phantoms. All plans showed conformity in terms of dose distribution of the target volume and dose reduction in risk organs (according to the requirements of the target parameter), as well as a very fast convergence of the algorithm (in less than 10 optimization steps).

3D conformal intensity-modulated radiotherapy planning: interactive optimization by constrained matrix inversion

BACKGROUND AND PURPOSE

This paper presents a method for interactive optimization of 3D conformal intensity-modulated radiotherapy plans employing a quadratic objective that also contains dose limitations in the organs at risk. This objective function is minimized by constrained matrix inversion (CMI) that follows the same approach as the gradient technique using matrix notation.

MATERIALS AND METHODS

Sherouse's GRATIS radiotherapy design system is used to determine the outlines of the target volume and the organs at risk and to input beam segments which are given by the beam segmentation technique. This technique defines the beam incidences and the beam segmentation. The weights of the segments are then calculated using a quadratic objective function and CMI. The objective function to be minimized consists of two components based on the planning target volume (PTV) and the organ at risk (OAR) with an importance factor w associated with the OAR.

RESULTS

Optimization is tested for concave targets in the head and neck region wrapping around the spinal cord. For a predefined w-value, segment weights are optimized within a few seconds on a DEC Alpha 3000. In practice, 5-10 w-values have to be tested, making optimization a less than 5 min procedure. This optimization procedure predicts the possibility of target dose escalation for a tumour in the lower neck to 120-150 Gy without exceeding the spinal cord tolerance, whereas human planners could not increase the dose above 65-80 Gy.

CONCLUSIONS

Treatment plans optimized using a quadratic objective function and the CMI algorithm are superior to those which are generated by human planners. The optimization algorithm is very fast and allows interactive use. Quadratic optimization by CMI is routinely used by clinicians at the Division of Radiotherapy, U.Z.-Gent.