Simultaneous optimization of dose distributions and fractionation schemes in particle radiotherapy

Spatiotemporal fractionation schemes for stereotactic radiosurgery of multiple brain metastases

BACKGROUND

Stereotactic radiosurgery (SRS) is an established treatment for patients with brain metastases (BMs). However, damage to the healthy brain may limit the tumor dose for patients with multiple lesions.

PURPOSE

In this study, we investigate the potential of spatiotemporal fractionation schemes to reduce the biological dose received by the healthy brain in SRS of multiple BMs, and also demonstrate a novel concept of spatiotemporal fractionation for polymetastatic cancer patients that faces less hurdles for clinical implementation.

METHODS

Spatiotemporal fractionation (STF) schemes aim at partial hypofractionation in the metastases along with more uniform fractionation in the healthy brain. This is achieved by delivering distinct dose distributions in different fractions, which are designed based on their cumulative biologically effective dose (BED α / β ${\rm{BED}}_{{{\alpha}}/{{\beta}}}$ ) such that each fraction contributes with high doses to complementary parts of the target volume, while similar dose baths are delivered to the normal tissue. For patients with multiple brain metastases, a novel constrained approach to spatiotemporal fractionation (cSTF) is proposed, which is more robust against setup and biological uncertainties. The approach aims at irradiating entire metastases with possibly different doses, but spatially similar dose distributions in every fraction, where the optimal dose contribution of every fraction to each metastasis is determined using a new planning objective to be added to the BED-based treatment plan optimization problem. The benefits of spatiotemporal fractionation schemes are evaluated for three patients, each with >25 BMs.

RESULTS

For the same tumor BED10 and the same brain volume exposed to high doses in all plans, the mean brain BED2 can be reduced compared to uniformly fractionated plans by 9%-12% with the cSTF plans and by 13%-19% with the STF plans. In contrast to the STF plans, the cSTF plans avoid partial irradiation of the individual metastases and are less sensitive to misalignments of the fractional dose distributions when setup errors occur.

CONCLUSION

Spatiotemporal fractionation schemes represent an approach to lower the biological dose to the healthy brain in SRS-based treatments of multiple BMs. Although cSTF cannot achieve the full BED reduction of STF, it improves on uniform fractionation and is more robust against both setup errors and biological uncertainties related to partial tumor irradiation.

MOQUI: an open-source GPU-based Monte Carlo code for proton dose calculation with efficient data structure

Objective.Monte Carlo (MC) codes are increasingly used for accurate radiotherapy dose calculation. In proton therapy, the accuracy of the dose calculation algorithm is expected to have a more significant impact than in photon therapy due to the depth-dose characteristics of proton beams. However, MC simulations come at a considerable computational cost to achieve statistically sufficient accuracy. There have been efforts to improve computational efficiency while maintaining sufficient accuracy. Among those, parallelizing particle transportation using graphic processing units (GPU) achieved significant improvements. Contrary to the central processing unit, a GPU has limited memory capacity and is not expandable. It is therefore challenging to score quantities with large dimensions requiring extensive memory. The objective of this study is to develop an open-source GPU-based MC package capable of scoring those quantities.Approach.We employed a hash-table, one of the key-value pair data structures, to efficiently utilize the limited memory of the GPU and score the quantities requiring a large amount of memory. With the hash table, only voxels interacting with particles will occupy memory, and we can search the data efficiently to determine their address. The hash-table was integrated with a novel GPU-based MC code, moqui.Main results.The developed code was validated against an MC code widely used in proton therapy, TOPAS, with homogeneous and heterogeneous phantoms. We also compared the dose calculation results of clinical treatment plans. The developed code agreed with TOPAS within 2%, except for the fall-off and regions, and the gamma pass rates of the results were >99% for all cases with a 2 mm/2% criteria.Significance.We can score dose-influence matrix and dose-rate on a GPU for a 3-field H&N case with 10 GB of memory using moqui, which would require more than 100 GB of memory with the conventionally used array data structure.

Spatiotemporal optimisation of prostate intensity modulated proton therapy (IMPT) treatments

Objective. In intensity modulated particle therapy (IMPT), the adoption of spatially and temporally heterogeneous dose distributions allows to decouple the fractionation scheme from the patient anatomy, so that an hypofractionated schedule can be selectively created inside the tumour, while simultaneously exploiting the fractionation effect in the healthy tissues. In this paper, the authors show the reproducibility of the method on a set of prostate patients, quantifying the dependencies of the achievable benefit with respect to conventional and hypofractionated schemes and the sensitivity of the method to setup errors and range uncertainty. Approach. On a cohort of 9 patients, non-uniform IMPT plans were optimised and compared to conventional and hypofractionated schedules. For each patient, the comparison of the three strategies has been based on the output of the cost function used to optimise the treatments. The analysis has been repeated considering different α/β ratios for the tumour, namely 1.5, 3 and 4.5 Gy. For a single patient, setup errors and beam range uncertainty have been analysed: the plans, for each optimisation strategy, have been iteratively forward planned 500 times with randomly varying the patient position in each fraction, and 200 times for systematic range shift. Main results. An average 10% benefit has been shown for the lowest α/β ratio considered for the tumour, where the non-uniform schedule generally converges to hypofractionation; the benefit decreases to 5%–7% for higher α/β ratios, for which the non-uniform schedule always showed better outcomes with respect to the other fractionation schedules. An increased sensitivity to uncertainty, especially for setup errors, has been shown, which can be associated to the spatial non-uniformity of the dose distributions peculiar of the spatiotemporal plans. Significance. This work represents the first investigation of spatiotemporal fractionation for prostate cancer and the beginning of further investigations before clinical implementation can be considered.

Imputing radiobiological parameters of the linear-quadratic dose-response model from a radiotherapy fractionation plan

The objective in cancer radiotherapy is to maximize tumor-kill while limiting toxic effects of radiation dose on nearby organs-at-risk (OAR). Given a fixed number of treatment sessions, planners thus face the problem of finding a dosing sequence that achieves this goal. This is called the fractionation problem, and has received steady attention over a long history in the clinical literature. Mathematical formulations of the resulting optimization problem utilize the linear-quadratic (LQ) framework to characterize radiation dose-response of tumors and OAR. This yields a nonconvex quadratically constrained quadratic program. The optimal dosing plan in this forward problem crucially depends on the parameters of the LQ model. Unfortunately, these parameters are difficult to estimate via in vitro or in vivo studies, and as such, their values are unknown to treatment planners. The clinical literature is thus replete with debates about what parameter values will make specific dosing plans effective. This paper formulates this as an inverse optimization problem. The LQ dose-response parameters appear in the objective function, the left hand side, and the right hand side of the forward problem, and none of the existing generic methods can provide an exact solution of the inverse problem. This paper exploits the structure of the problem and identifies all possible parameter values that render the given dosing plan optimal, in closed-form. This closed-form formula is applied to dosing-plans from three clinical studies published within the last two years.

Fraction-variant beam orientation optimization for intensity-modulated proton therapy

PURPOSE

To achieve a superior balance between dosimetry and the delivery efficiency of intensity-modulated proton therapy (IMPT) using as few beams as possible in a single fraction, we optimally vary beams in different fractions.

METHODS

In the optimization, 400~800 feasible noncoplanar beams were included in the candidate pool. For each beam, the doses of all scanning spots covering the target volume and a margin were calculated. The fraction-variant beam orientation optimization (FVBOO) problem was formulated to include three terms: two quadratic dose fidelity terms to penalize the deviation of planning target volume fractional dose and organs at risk (OAR) cumulative doses from prescription, respectively; an L2,1/2-norm group sparsity term to control the number of active beams per fraction to between 1 and 4. The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) was applied to solve this problem. FVBOO was tested on a patient with base-of-skull (BOS) tumor of 5 fractions (5f) and 30 fractions (30f) with an average number of active beams per fraction varying between 4 and 1. In addition, one bilateral head-and-neck (H&N) patient, and one esophageal cancer (ESG) patient of 30f were tested with about three active beams per fraction. The results were compared with IMPT plans that use fixed beams in each fraction. The fixed beams were selected using the group sparsity term with a fraction-invariant BOO (FIBOO) constraint.

RESULTS

Varying beams were chosen in either the 5f or 30f FVBOO plans. While similar number of beams per fraction was selected as the FIBOO plan, the FVBOO plans were able to spare the OARs better, with an average reduction of [Dmean, Dmax] from the FIBOO plans by [0.85, 2.08] Relative Biological Effective Gy (GyRBE) in the 5f plan and [1.87, 4.06] GyRBE in the 30f plans. While reducing the number of beams per fraction in the BOS patient, a three-beam/fraction 5f FVBOO plan performs comparably as the four-beam FIBOO plan and a two-beam/fraction 30f FVBOO plan still provides superior dosimetry.

CONCLUSION

Fraction-variant beam orientation optimization allows the utilization of a larger beam solution space for superior dose distribution in IMPT while maintaining a practical number of beams in each fraction.

Optimal modality selection in external beam radiotherapy

The goal in external beam radiotherapy (EBRT) for cancer is to maximize damage to the tumour while limiting toxic effects on the organs-at-risk. EBRT can be delivered via different modalities such as photons, protons and neutrons. The choice of an optimal modality depends on the anatomy of the irradiated area and the relative physical and biological properties of the modalities under consideration. There is no single universally dominant modality. We present the first-ever mathematical formulation of the optimal modality selection problem. We show that this problem can be tackled by solving the Karush-Kuhn-Tucker conditions of optimality, which reduce to an analytically tractable quartic equation. We perform numerical experiments to gain insights into the effect of biological and physical properties on the choice of an optimal modality or combination of modalities.

A feasibility study of spatiotemporally integrated radiotherapy using the LQ model

This paper investigates the feasibility of spatiotemporally modulated radiotherapy (STMRT)-integrated model with explicit constraints on the tumor dose heterogeneity. In particular, we demonstrate the effect of the tumor dose heterogeneity on the tumor biologically effective dose (BED) achievable and optimal fractionation. We propose an STMRT model that simultaneously optimizes the dose distributions and fractionation schedule for each individual case with the maximum and minimum constraints on the tumor BED to explicitly control the level of tumor dose heterogeneity. Sixteen thoracic phantom cases were planned using (1) STMRT and (2) standard fractionation (60 Gy in 30 fractions fixed) IMRT. Constraints on the organs-at-risk (OAR) BED were identical for both plans. BEDs were calculated using the [Formula: see text] ratio of 10 Gy for the tumor and 3 Gy for all OARs. The maximum tumor BED for STMRT plans was constrained to be less than 100%-150% of the maximum tumor BED resulted from the standard fractionation plans. The mean tumor BED from STMRT plans was up to 110.7%, 128.3%, 135.0% and 148.0% of that from the standard fractionation plans when the maximum tumor BED was constrained to be less than 100%, 120%, 130% and 150% of the maximum BED achieved using the standard plans. The optimal number of fractions varied widely for different phantom geometries for the same radiobiological parameter values. The increase in the tumor BED and the range of optimal fractionation was larger with a larger tumor dose heterogeneity allowed. The results have shown the feasibility of personalizing fractionation schedule using an STMRT integrated model to deliver a maximum feasible BED to the tumor for a fixed OAR BED. The potential increase in the tumor BED was positively correlated to the tumor dose heterogeneity allowed.

Selection of external beam radiotherapy approaches for precise and accurate cancer treatment

Physically precise external-beam radiotherapy (EBRT) technologies may not translate to the best outcome in individual patients. On the other hand, clinical considerations alone are often insufficient to guide the selection of a specific EBRT approach in patients. We examine the ways in which to compare different EBRT approaches based on physical, biological and clinical considerations, and how they can be enhanced with the addition of biophysical models and machine-learning strategies. The process of selecting an EBRT modality is expected to improve in tandem with knowledge-based treatment planning.

Fraction-variant beam orientation optimization for non-coplanar IMRT

Conventional beam orientation optimization (BOO) algorithms for IMRT assume that the same set of beam angles is used for all treatment fractions. In this paper we present a BOO formulation based on group sparsity that simultaneously optimizes non-coplanar beam angles for all fractions, yielding a fraction-variant (FV) treatment plan. Beam angles are selected by solving a multi-fraction fluence map optimization problem involving 500-700 candidate beams per fraction, with an additional group sparsity term that encourages most candidate beams to be inactive. The optimization problem is solved using the fast iterative shrinkage-thresholding algorithm. Our FV BOO algorithm is used to create five-fraction treatment plans for digital phantom, prostate, and lung cases as well as a 30-fraction plan for a head and neck case. A homogeneous PTV dose coverage is maintained in all fractions. The treatment plans are compared with fraction-invariant plans that use a fixed set of beam angles for all fractions. The FV plans reduced OAR mean dose and D 2 values on average by 3.3% and 3.8% of the prescription dose, respectively. Notably, mean OAR dose was reduced by 14.3% of prescription dose (rectum), 11.6% (penile bulb), 10.7% (seminal vesicle), 5.5% (right femur), 3.5% (bladder), 4.0% (normal left lung), 15.5% (cochleas), and 5.2% (chiasm). D 2 was reduced by 14.9% of prescription dose (right femur), 8.2% (penile bulb), 12.7% (proximal bronchus), 4.1% (normal left lung), 15.2% (cochleas), 10.1% (orbits), 9.1% (chiasm), 8.7% (brainstem), and 7.1% (parotids). Meanwhile, PTV homogeneity defined as D 95/D 5 improved from .92 to .95 (digital phantom), from .95 to .98 (prostate case), and from .94 to .97 (lung case), and remained constant for the head and neck case. Moreover, the FV plans are dosimetrically similar to conventional plans that use twice as many beams per fraction. Thus, FV BOO offers the potential to reduce delivery time for non-coplanar IMRT.

Optimization of spatiotemporally fractionated radiotherapy treatments with bounds on the achievable benefit

Spatiotemporal fractionation schemes, that is, treatments delivering different dose distributions in different fractions, can potentially lower treatment side effects without compromising tumor control. This can be achieved by hypofractionating parts of the tumor while delivering approximately uniformly fractionated doses to the surrounding tissue. Plan optimization for such treatments is based on biologically effective dose (BED); however, this leads to computationally challenging nonconvex optimization problems. Optimization methods that are in current use yield only locally optimal solutions, and it has hitherto been unclear whether these plans are close to the global optimum. We present an optimization framework to compute rigorous bounds on the maximum achievable normal tissue BED reduction for spatiotemporal plans. The approach is demonstrated on liver tumors, where the primary goal is to reduce mean liver BED without compromising any other treatment objective. The BED-based treatment plan optimization problems are formulated as quadratically constrained quadratic programming (QCQP) problems. First, a conventional, uniformly fractionated reference plan is computed using convex optimization. Then, a second, nonconvex, QCQP model is solved to local optimality to compute a spatiotemporally fractionated plan that minimizes mean liver BED, subject to the constraints that the plan is no worse than the reference plan with respect to all other planning goals. Finally, we derive a convex relaxation of the second model in the form of a semidefinite programming problem, which provides a rigorous lower bound on the lowest achievable mean liver BED. The method is presented on five cases with distinct geometries. The computed spatiotemporal plans achieve 12-35% mean liver BED reduction over the optimal uniformly fractionated plans. This reduction corresponds to 79-97% of the gap between the mean liver BED of the uniform reference plans and our lower bounds on the lowest achievable mean liver BED. The results indicate that spatiotemporal treatments can achieve substantial reductions in normal tissue dose and BED, and that local optimization techniques provide high-quality plans that are close to realizing the maximum potential normal tissue dose reduction.

Spatiotemporal fractionation schemes for liver stereotactic body radiotherapy

BACKGROUND AND PURPOSE

Dose prescription in stereotactic body radiotherapy (SBRT) for liver tumors is often limited by the mean liver dose. We explore the concept of spatiotemporal fractionation as an approach to facilitate further dose escalation in liver SBRT.

MATERIALS AND METHODS

Spatiotemporal fractionation schemes aim at partial hypofractionation in the tumor along with near-uniform fractionation in normal tissues. This is achieved by delivering distinct dose distributions in different fractions, which are designed such that each fraction delivers a high single fraction dose to complementary parts of the tumor while creating a similar dose bath in the surrounding noninvolved liver. Thereby, higher biologically effective doses (BED) can be delivered to the tumor without increasing the mean BED in the liver. Planning of such treatments is performed by simultaneously optimizing multiple dose distributions based on their cumulative BED. We study this concept for five liver cancer patients with different tumor geometries.

RESULTS

Spatiotemporal fractionation presents a method of increasing the ratio of prescribed tumor BED to mean BED in the noninvolved liver by approximately 10-20%, compared to conventional SBRT using identical fractions.

CONCLUSIONS

Spatiotemporal fractionation may reduce the risk of liver toxicity or facilitate dose escalation in liver SBRT in circumstances where the mean dose to the non-involved liver is the prescription-limiting factor.

Optimization of the fractionated irradiation scheme considering physical doses to tumor and organ at risk based on dose-volume histograms

PURPOSE

Radiotherapy of solid tumors has been performed with various fractionation regimens such as multi- and hypofractionations. However, the ability to optimize the fractionation regimen considering the physical dose distribution remains insufficient. This study aims to optimize the fractionation regimen, in which the authors propose a graphical method for selecting the optimal number of fractions (n) and dose per fraction (d) based on dose-volume histograms for tumor and normal tissues of organs around the tumor.

METHODS

Modified linear-quadratic models were employed to estimate the radiation effects on the tumor and an organ at risk (OAR), where the repopulation of the tumor cells and the linearity of the dose-response curve in the high dose range of the surviving fraction were considered. The minimization problem for the damage effect on the OAR was solved under the constraint that the radiation effect on the tumor is fixed by a graphical method. Here, the damage effect on the OAR was estimated based on the dose-volume histogram.

RESULTS

It was found that the optimization of fractionation scheme incorporating the dose-volume histogram is possible by employing appropriate cell surviving models. The graphical method considering the repopulation of tumor cells and a rectilinear response in the high dose range enables them to derive the optimal number of fractions and dose per fraction. For example, in the treatment of prostate cancer, the optimal fractionation was suggested to lie in the range of 8-32 fractions with a daily dose of 2.2-6.3 Gy.

CONCLUSIONS

It is possible to optimize the number of fractions and dose per fraction based on the physical dose distribution (i.e., dose-volume histogram) by the graphical method considering the effects on tumor and OARs around the tumor. This method may stipulate a new guideline to optimize the fractionation regimen for physics-guided fractionation.

The emergence of nonuniform spatiotemporal fractionation schemes within the standard BED model

PURPOSE

Nonuniform spatiotemporal radiotherapy fractionation schemes, i.e., delivering distinct dose distributions in different fractions can potentially improve the therapeutic ratio. This is possible if the dose distributions are designed such that similar doses are delivered to normal tissues (exploit the fractionation effect) while hypofractionating subregions of the tumor. In this paper, the authors develop methodology for treatment planning with nonuniform fractions and demonstrate this concept in the context of intensity-modulated proton therapy (IMPT).

METHODS

Treatment planning is performed by simultaneously optimizing (possibly distinct) IMPT dose distributions for multiple fractions. This is achieved using objective and constraint functions evaluated for the cumulative biologically equivalent dose (BED) delivered at the end of treatment. BED based treatment planning formulations lead to nonconvex optimization problems, such that local gradient based algorithms require adequate starting positions to find good local optima. To that end, the authors develop a combinatorial algorithm to initialize the pencil beam intensities.

RESULTS

The concept of nonuniform spatiotemporal fractionation schemes is demonstrated for a spinal metastasis patient treated in two fractions using stereotactic body radiation therapy. The patient is treated with posterior oblique beams with the kidneys being located in the entrance region of the beam. It is shown that a nonuniform fractionation scheme that hypofractionates the central part of the tumor allows for a skin and kidney BED reduction of approximately 10%-20%.

CONCLUSIONS

Nonuniform spatiotemporal fractionation schemes represent a novel approach to exploit fractionation effects that deserves further exploration for selected disease sites.

Spatiotemporal Fractionation Schemes for Irradiating Large Cerebral Arteriovenous Malformations

PURPOSE

To optimally exploit fractionation effects in the context of radiosurgery treatments of large cerebral arteriovenous malformations (AVMs). In current practice, fractionated treatments divide the dose evenly into several fractions, which generally leads to low obliteration rates. In this work, we investigate the potential benefit of delivering distinct dose distributions in different fractions.

METHODS AND MATERIALS

Five patients with large cerebral AVMs were reviewed and replanned for intensity modulated arc therapy delivered with conventional photon beams. Treatment plans allowing for different dose distributions in all fractions were obtained by performing treatment plan optimization based on the cumulative biologically effective dose delivered at the end of treatment.

RESULTS

We show that distinct treatment plans can be designed for different fractions, such that high single-fraction doses are delivered to complementary parts of the AVM. All plans create a similar dose bath in the surrounding normal brain and thereby exploit the fractionation effect. This partial hypofractionation in the AVM along with fractionation in normal brain achieves a net improvement of the therapeutic ratio. We show that a biological dose reduction of approximately 10% in the healthy brain can be achieved compared with reference treatment schedules that deliver the same dose distribution in all fractions.

CONCLUSIONS

Boosting complementary parts of the target volume in different fractions may provide a therapeutic advantage in fractionated radiosurgery treatments of large cerebral AVMs. The strategy allows for a mean dose reduction in normal brain that may be valuable for a patient population with an otherwise normal life expectancy.