Fraction‐variant beam orientation optimization for spatiotemporal fractionation schemes

Abstract

Background

Spatiotemporal fractionation schemes aim at partial hypofractionation in the tumor along with more uniform fractionation in the normal tissue. This is achieved by delivering distinct dose distributions in different fractions, where each fraction contributes with high doses to complementary parts of target volume, while similar dose baths are delivered to the normal tissue.

Purpose

In this study, we extend spatiotemporal fractionation schemes by allowing for different non‐coplanar beam orientations in each fraction. As different regions of the target volume are treated in different fractions, spatiotemporal fractionation schemes may benefit from selecting different beam orientations in each fraction.

Methods

Spatiotemporally fractionated (STF) treatments are generated using a novel biologically effective dose (BED)‐based direct aperture optimization (DAO) algorithm, which allows for the simultaneous optimization of multiple‐dose distributions to be delivered in the different fractions along with their corresponding sets of multileaf collimated apertures and monitor unit (MU) weights. Each set of apertures specifies a series of control points along a fraction‐specific non‐coplanar dynamic trajectory, which consists of a 360° gantry arc with dynamic bi‐directional couch rotation. The gantry‐couch path is automatically determined during the treatment plan optimization. The proposed DAO approach is demonstrated for stereotactic body radiotherapy (SBRT) of a patient with liver metastases (3 × 12 Gy, patient 1) and stereotactic radiotherapy (SRT) of a patient with a large arteriovenous malformation (AVM) (4 × 7 Gy, patient 2).

Results

For both patients, STF treatments deliver highly non‐uniform dose distributions in distinct fractions. Depending on which region of the target volume is irradiated in each fraction, very different non‐coplanar dynamic trajectories are selected. For a similar target coverage, the STF treatment obtained using fraction‐specific non‐coplanar dynamic trajectories reduces the mean liver BED₃ in patient 1 by 13.7% (32.9 Gy vs. 38.1 Gy) and the mean brain BED₂ in patient 2 by 18.4% (5.15 Gy vs. 6.31 Gy), respectively, compared to a STF treatment obtained using coplanar VMAT arcs in every fraction.

Conclusions

The dosimetric quality of STF treatments can be considerably enhanced by selecting different non‐coplanar beam orientations in different fractions, which are beneficial to treat specific parts of the target volume.

1. INTRODUCTION

Spatiotemporal fractionation schemes have been suggested as a treatment approach to improve the ratio of biological dose in the tumor and the normal tissue. ¹ , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ , ⁹ In contrast to conventional fractionation approaches, in which the same dose distribution is delivered in each fraction, spatiotemporally fractionated (STF) treatments vary the dose distribution between different fractions, aiming to better exploit the fractionation effect. The goal of spatiotemporal fractionation is to approximate an ideal treatment that would deliver the dose to the tumor in a single fraction and at the same time deliver a conventionally fractionated dose to the surrounding normal tissue. STF treatments partly accomplish this by delivering high single‐fraction doses to complementary parts of the tumor in every fraction, while distributing the normal tissue dose more uniformly over all fractions. Consequently, a lower cumulative physical dose is needed to achieve the same level of tumor control compared to conventional fractionation schemes, which translates into a net reduction of the biological dose in the normal tissue.

Prior in‐silico studies have demonstrated the potential of STF treatments to improve the therapeutic ratio by 10%–15% compared to conventional uniformly fractionated treatments, for several treatment sites. ³ , ⁴ , ⁵ , ⁸ Further research investigated robust optimization approaches to mitigate the sensitivity of spatiotemporal fractionation schemes to both setup and motion errors, ⁶ , ⁸ as well as to biological uncertainties related to partial tumor irradiation and the use of varying doses per fraction. ⁹ , ¹⁰ However, in all previous studies, it was assumed that the same pre‐defined set of beam orientations is used in every fraction.

In this study, we aim to further extend spatiotemporal fractionation by allowing different beam orientations to be used in different fractions. As different regions of the target volume are treated in different fractions, spatiotemporal fractionation schemes may benefit from selecting specific beam orientations in each fraction. A direct aperture optimization (DAO) algorithm is proposed that simultaneously optimizes multiple dose distributions to be delivered in different fractions, together with corresponding fraction‐specific non‐coplanar dynamic trajectories. Non‐coplanar dynamic trajectories consist of a treatment technique combining dynamic gantry and couch rotation and represent an efficient way of delivering non‐coplanar radiotherapy treatments using conventional C‐arm linear accelerators. ¹¹ , ¹² , ¹³ , ¹⁴ , ¹⁵ , ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ The proposed approach is demonstrated in‐silico for stereotactic body radiotherapy (SBRT) of a patient with large liver metastases and stereotactic radiotherapy (SRT) of a patient with a large arteriovenous malformation (AVM). Robustness of the treatments against both setup errors and biological uncertainties is not explicitly considered in this study.

2. MATERIALS AND METHODS

2.1. Optimization algorithm

STF treatments are generated by simultaneously optimizing multiple dose distributions to be delivered in the different fractions, together with their corresponding sets of multileaf collimated apertures and monitor unit (MU) weights. Each set of apertures specifies a series of control points along a fraction‐specific non‐coplanar dynamic trajectory, which consists of a 360° gantry arc with dynamic bi‐directional couch rotation and whose gantry‐couch path is automatically determined during the treatment plan optimization. To account for the fractionation effects when different non‐uniform dose distributions are delivered in distinct fractions, the cumulative biologically effective dose (BED) is optimized rather than the cumulative physical dose.

The proposed BED‐based optimization problem is solved using a DAO algorithm that has been implemented into our in‐house research optimization software. The algorithm combines a column generation based method to iteratively add apertures from promising beam orientations together with a gradient based method for DAO that refines MU weights and aperture shapes at each iteration. The optimization workflow is schematically illustrated in Figure 1 and further detailed in the following.¹

FIGURE 1.

(a) Workflow of the DAO algorithm. The index $\lambda$ refers to the iteration number in the column generation based method and $n$ is the total number of fractions in the treatment. (b) Schematic illustration of iteration $\lambda >n$ of the DAO algorithm for two exemplary fractions $t$ and $\stackrel{\sim }{t}$. The black stars indicate control points on a non‐coplanar dynamic trajectory for which an aperture has already been generated. First, candidate apertures are generated for each beam orientation that can still be efficiently reached by the non‐coplanar dynamic trajectories (colored squares). The apertures with the lowest price in each fraction are subsequently added to the treatment plan. Both the MU weights and shapes of all the already added apertures are finally refined. DAO, direct aperture optimization; MU, monitor unit.

2.1.1. Column generation based method for selecting apertures from promising beam orientations

Favorable non‐coplanar dynamic trajectories for the delivery of each treatment fraction are determined using a column generation based method. This is a heuristic algorithm which iteratively adds suitable apertures from promising beam orientations (defined by gantry angles $\theta$ and couch angles $\varphi$) to a treatment fraction, until all control points along the fraction‐specific non‐coplanar dynamic trajectories are populated. The algorithm starts with an empty set of apertures $K_t=\varnothing$ for all the fractions $t\in \{1,\text{…},n\}$ (where $n$ is the total number of fractions). Given a set $B$ of feasible beam orientations, the most promising multileaf collimator (MLC)‐based apertures per field $\xi \in B$ and per fraction $t\in \{1,\text{…},n\}$ are determined by considering the partial derivatives of the objective function $f$ with respect to the fluence of a bixel. Since a negative partial derivative indicates that increasing the fluence of the bixel improves the objective function, we determine the aperture that minimizes the sum of partial derivatives over all bixels contained in the aperture:

$$\begin{array}{c}\mathrm{minimize}\\ x_L^l,x_R^l\end{array}\sum _{l\in L_{\xi }}\sum _{p=x_L^l}^{x_R^l}\frac{\partial f}{\partial {\nu }^{lp}}$$ (1)

where ${\nu }^{lp}$ refers to the fluence of the bixel in leaf pair $l$ at position $p\in \{x_L^l,\text{…},x_R^l\}$, $x_L^l$ and $x_R^l$ are the left and right positions of the leaf pair $l$ (which corresponds to the optimization variables), respectively, and $L_{\xi }$ is the set of all MLC leaf pairs that apertures in field $\xi$ can take. This problem (referred to as pricing problem) can be solved independently for all leaf pairs $l\in L_{\xi }$ by looping through all bixels and fixing the left and right leaf positions such that the sum of the gradients contributions $\frac{\partial f}{\partial {\nu }^{lp}}$ of all the exposed bixels is minimal². Further descriptions on strategies to solve the pricing problem can be found in the work of Romeijn et al ²⁰ and Men et al. ²¹ The resulting candidate apertures have an associated price, given by the sum of the gradient contributions of each of the bixels not covered by the MLC, and the apertures with the lowest price in each fraction are added to the treatment plan at each iteration (see Figure 1b)³. During the first $\lambda \in \{1,\text{…},n\}$ iterations of the column generation method, only one single aperture is added to the treatment plan for fraction $t=\lambda$. This is necessary to allow for the dose distributions delivered in the different fractions to possibly diverge, as the gradient information for all fractions with an empty pool of apertures is the same.

Similar to the work of Mullins et al., ¹⁹ the set $B$ of candidate beam orientations (which is initially the same for all fractions) spans multiple potential control points for the non‐coplanar dynamic trajectories, each defined by different pairs of gantry‐couch angles. The search space for candidate beam orientations considered for the pricing problem varies dynamically with the addition of each new aperture. Both control points for which an aperture has already been defined and control points that cannot be reached efficiently by a dynamic trajectory due to delivery‐efficiency constraints (which are further detailed below) are excluded from the search space $B$. This makes solving the pricing problem more efficient after the addition of each new aperture. In this study, the control points' grid resolution is set to $\mathrm{\Delta }\theta =$5° for the gantry angles and to $\mathrm{\Delta }\varphi =$2.5° for the couch angles, with gantry angles ranging from −180° to +180° and couch angles ranging from −90° to +90°. Avoidance zones for gantry‐couch angle pairs leading to collisions are determined for typical patient geometries and treatment sites (i.e., they are not patient specific) and are excluded from the set of candidate beam orientations. The collimator angle is not optimized and is fixed to 0°.

2.1.2. Gradient based DAO method for apertures’ shape and MU weight refinement

After the addition of each aperture to the treatment plan, both the MU weights and the shapes of all the already defined apertures are refined using a gradient based DAO approach, aiming to minimize an objective function $f(b)$ evaluated for the cumulative BED $b$. As previously demonstrated in other studies, ²² , ²³ refining the shape of the apertures is particularly important to improve the plan quality. The apertures generated in the first few iterations of the column generation DAO approach, in fact, tend to be large. When more and more apertures are added to the treatment, smaller apertures with a finer modulation are instead preferred. In addition, the refinement of the aperture shapes allows to fine tune the leaf positions such that only a portion of the bixel might be exposed.

Formally, the treatment plan optimization problem reads:

$$\begin{array}{c}\mathrm{minimize}\\ x\end{array}f\left(b\right)$$ (2)

$$\mathrm{subject}\mathrm{to}{b}_i=\sum _{t=1}^n{d}_{it}\left[1+\frac{d_{it}}{{\left(\alpha /\beta \right)}_i}\right]\forall i$$ (3)

$$d_{it}=\sum _{k\in K_t}{\omega }_k\sum _{l\in L_k}{\mathrm{\Phi }}_i^{kl}\left(x_L^{kl},x_R^{kl}\right)\forall i,\forall t$$ (4)

$${\omega }^k\ge 0\forall k$$ (5)

$$\left|x_L^{\left(k-1\right)l}-x_L^{kl}\right|\le \mathrm{\Delta }{x}_{\max }\forall k\ge 1,\forall l$$ (6)

$$\left|x_R^{\left(k-1\right)l}-x_R^{kl}\right|\le \mathrm{\Delta }{x}_{\max }\forall k\ge 1,\forall l$$ (7)

$$x_L^{kl}\le x_R^{kl}\forall k,\forall l$$ (8)

where $b_i$ is the cumulative BED to voxel $i$, ${(\alpha /\beta )}_i$ is the $\alpha /\beta$‐ratio of the tissue that voxel $i$ belongs to, $d_{it}$ is the physical dose delivered to voxel $i$ in fraction $t$ by all MLC‐based apertures $k\in K_t$ belonging to that fraction, ${\mathrm{\Phi }}_i^{kl}$ is the dose contribution of the $l$‐th leaf pair of aperture $k$ to voxel $i$ per unit intensity and ${\omega }_k$ is the MU weight of aperture $k$. The parameters $x_L^{kl}$ and $x_R^{kl}$ describe the positions of the left and right MLC leaves in the $l$‐th leaf pair of aperture $k$, respectively, and $L_k$ is the set of all MLC leaf pairs in aperture $k$. The constraints in Equations ((6), (7)) are used to limit the MLC displacement between neighboring control points by a maximum distance $\mathrm{\Delta }{x}_{\max }$, as further detailed below⁴.

The proposed gradient‐based DAO approach, which is inspired by the work of Cassioli and Unkelbach, ²³ exploits gradient information to locally optimize the positions of the MLC leaves and the MU weights in an iterative way. At each iteration of the gradient based DAO method, a restricted optimization problem is solved, where the left and right MLC leaf positions for all leaf pairs $l\in L_k$ and for all apertures $k\in K_t$ are constrained to the bixels $b_L^{kl}$ and $b_R^{kl}$ they are positioned in. Under this assumption, the dose contribution ${\mathrm{\Phi }}_i^{kl}(x_L^{kl},x_R^{kl})$ to voxel $i$ in Equation (4) can be decomposed into three separate terms describing the dose contributions of the bixels that are in between the two leaves and are thereby completely exposed, and the dose contributions of the partially exposed bixels on the left and on the right, respectively,

$${\mathrm{\Phi }}_i^{kl}\left(x_L^{kl},x_R^{kl}\right)={\delta }_L^{kl}{D}_{i{b}_L^{kl}}+{\delta }_R^{kl}{D}_{i{b}_R^{kl}}+\sum _{j=b_L^{kl}+1}^{b_R^{kl}-1}{D}_{ij}$$ (9)

Here, ${\delta }_L^{kl}\in [0,1]$ and ${\delta }_R^{kl}\in [0,1]$ describe the fractional opening of the bixels that the left and right MLC leaves are restricted to, and $D_{ij}$ is the dose‐influence matrix term storing the dose contribution of bixel $j$ to voxel $i$ per unit intensity. As the dose contribution is linear to the fraction of the bixel area open, this leads to a well‐behaved optimization problem for the leaf positions and the aperture weights, which can be solved efficiently using standard gradient‐based algorithms. In this study, we use our in‐house implementation of the L‐BFGS quasi‐Newton method, ²⁴ where the constraints ${\delta }_L^{kl}\in [0,1]$ and ${\delta }_R^{kl}\in [0,1]$ are handled by projection methods.

If a MLC leaf is moved at the edge of a bixel after an iteration of the gradient based DAO method (e.g., ${\delta }_L^{kl}\in \{0,1\}$ or ${\delta }_R^{kl}\in \{0,1\}$), its position can be confined to the neighboring bixel in the next iteration (details on the leaf‐to‐bixel assignment procedure are discussed in the work of Cassioli and Unkelbach ²³ ). This allows for larger MLC leaf displacements over multiple iterations. In this work, the iterative refinement of the MLC leaf positions and the MU weights stops either after 20 iterations of the gradient based DAO algorithm or sooner if the decrease in the relative value of the objective function between two consecutive iterations is smaller than $\epsilon ={10}^{-6}$.

2.1.3. Delivery efficiency constraints

To keep the delivery time of non‐coplanar dynamic trajectories practical, delivery efficiency constraints are implemented. Similar to Peng et al., ²⁵ we first constrain the MLC leaf displacement $\mathrm{\Delta }{x}_{\max }$ between consecutive control points. The maximum MLC leaf position displacement from one control point to the neighboring one is determined by the maximum leaf speed $v_{\max }$ and the minimum time which the system composed by gantry and couch needs to move between the two control points. For the delivery of non‐coplanar dynamic trajectories, this is determined either by the inverse of the maximum gantry angular speed ${\omega }_g$ or by the inverse of the maximum couch angular speed ${\omega }_c$

$$\mathrm{\Delta }{x}_{\max }=\max \left(v_{\max }\frac{\mathrm{\Delta }\theta }{{\omega }_g},v_{\max }\frac{\mathrm{\Delta }\varphi }{{\omega }_c}\right)$$ (10)

where $\mathrm{\Delta }\theta$ and $\mathrm{\Delta }\varphi$ denote the difference in gantry and couch angles between neighboring control points, respectively⁵. In this study, the maximum MLC leaf speed is set to $v_{\max }$= 2.5 cm/s, the maximum rotation speed of the gantry is set to ${\omega }_g$= 6°/s and the maximum rotation speed of the couch is set to ${\omega }_c$= 3°/s. These values correspond to the dynamic parameters for a TrueBeam (Varian Medical Systems, Inc.) treatment unit equipped with a Millennium MLC 120 (Varian Medical Systems, Inc.). For the chosen gantry‐couch grid resolution (i.e., $\mathrm{\Delta }\theta =5^{\circ }$ and $\mathrm{\Delta }\varphi =2.5^{\circ }$), the travel time from one control point to the neighboring one corresponds to 0.83 s and the maximum leaf travel between consecutive control points is thereby limited to 2.08 cm. This constraint on the maximum MLC leaf displacement is incorporated both in the column generation based method (within the pricing problem) and in the gradient based DAO method (within the leaf‐to‐bixel assignment procedure).

Second, we constrain the delivery time for a non‐coplanar dynamic trajectory to be approximately the same as the delivery time for a full coplanar VMAT arc. The maximum couch rotation $\mathrm{\Delta }{\varphi }_{\max }$ between consecutive control points (defined by monotonically increasing gantry angles) is thereby restricted to

$$\mathrm{\Delta }{\varphi }_{\max }=\mathrm{\Delta }\theta \frac{{\omega }_c}{{\omega }_g}$$ (11)

To satisfy the constraint in Equation (11) and still obtain a continuous dynamic trajectory spanning a 360° gantry arc, control points which cannot be reached efficiently by a dynamic trajectory while avoiding the collision zones are excluded from the initial beam search space $B$.

In this work, we do not define any constraint on the maximum MU rate. It is therefore possible that the speed of the mechanical linear accelerator components must be adjusted if a too large number of MUs has to be delivered between consecutive control points.

2.2. In‐silico evaluation of STF treatments using fraction‐specific non‐coplanar dynamic trajectories

The proposed optimization approach has been retrospectively evaluated for two patients, who were previously investigated in the context of spatiotemporal fractionation research. ⁴ , ⁵ Patient 1 has four metastases of varying size located in the right lobe of the liver. The total GTV volume is 319 cc, and a 3 mm isotropic margin expansion is applied from the GTV to obtain the PTV. Patient 2 has a large AVM located in the right frontal lobe with a volume of 30 cc.

2.2.1. Patient 1

For patient 1, a 3‐fraction STF SBRT plan is generated using the proposed planning approach. For each fraction, a different non‐uniform dose distribution is optimized which is delivered using two fraction‐specific non‐coplanar dynamic trajectories. Both non‐coplanar dynamic trajectories are constrained to share the same gantry‐couch path, which is automatically optimized along with the corresponding dose distribution and consists of 73 control points selected from 1463 feasible beam orientations. This plan, which is hereafter referred to as STF_non‐copl plan, is optimized for the following choice of planning objectives:

1. A BED₁₀ of 100.8 Gy is prescribed to the GTV (implemented via a quadratic penalty function), corresponding to a physical dose of 42 Gy in 3 fractions. A BED₁₀ exceeding 112.5 Gy (equivalent to 45 Gy physical dose in 3 fractions) is penalized quadratically.

2. A BED₁₀ of 79.2 Gy (equivalent to 36 Gy physical dose in 3 fractions) is prescribed to the PTV (excluding the GTV voxels), while BED₁₀ values exceeding 100.8 Gy are penalized quadratically.

3. The dose has to be conformal to the PTV. This is implemented via a normal tissue objective (NTO) which limits the dose to the normal tissue by steepening the dose gradient (as further detailed in Appendix B).

4. The mean BED₃ to the non‐involved liver (i.e., liver without PTV) is minimized.

The bixel size is set to 5 × 5 mm² and the photon energy is 6 MV for all candidate beam orientations. A non‐uniform dose grid size is used throughout the body, with small voxels of 5.1 × 5.1 × 2.5 mm³ in size that are used in the PTV and close to the PTV, where a larger dose gradient is expected. At a distance between 2 and 4 cm from the PTV edge, medium‐size voxels are used with 8‐fold volume, whereas at distances larger than 4 cm from the PTV edge large‐size voxels are used with 64‐fold volume. As previously shown by Mueller et al., ²⁶ the use of a non‐uniform dose grid size allows to considerably enhance the computational efficiency with negligible trade‐offs on the plan accuracy. All dosimetric results, however, are evaluated based on the finest (small) dose grid size.

2.2.2. Patient 2

Analogously, for patient 2 we generated a 4‐fraction STF SRT plan using the proposed planning approach, where different dose distributions are optimized for each fraction utilizing two fraction‐specific non‐coplanar dynamic trajectories (both defined through the same 73 control points selected from 3035 candidate beam orientations). The following planning objectives have been used to generate the plan:

1. A BED₄ of 77 Gy is prescribed to the AVM (implemented via a quadratic penalty function), corresponding to a physical dose of 28 Gy in 4 fractions. A BED₄ exceeding 140 Gy (equivalent to 40 Gy physical dose in 4 fractions) is penalized quadratically.

2. The volume of a healthy brain (i.e., brain excluding the AVM) receiving a BED₂ larger than 60 Gy is minimized. This corresponds to a physical dose of 10 Gy in a single fraction and is implemented via a continuous relaxation of a dose‐volume objective.

3. The dose has to be conformal to the AVM. This is implemented via a NTO which limits the dose to the normal tissue by steepening the dose gradient.

4. The mean BED₂ to the healthy brain (i.e., brain without the AVM) is minimized.

Similar as for patient 1, a non‐uniform dose grid size is used, where the size of the smaller voxels is 2.7 × 2.7 × 1.2 mm³. The bixel size is set to 5 × 5 mm² and the photon energy is 6 MV for all candidate beam orientations.

Additional details on the planning objectives and the optimization settings used to generate such plans are reported in Appendix B.

2.2.3. Plan comparisons

To quantify the potential benefit of using different fraction‐specific non‐coplanar dynamic trajectories for the delivery of spatiotemporal fractionation schemes, the STF_non‐copl plan is benchmarked against the following plans for both patients:

1. A coplanar uniformly fractionated plan that delivers the same dose distribution in every fraction, using two coplanar VMAT arcs at a fixed couch angle of $\varphi =0^{\circ }$ in every fraction. This plan, which is hereafter referred to as UF_copl plan, is optimized using a pre‐defined and fixed set of beam orientations $B_0=\{({\theta }_k,{\varphi }_k)|{\theta }_k=-{180}^{\circ }+k\ast 5^{\circ },{\varphi }_k=0^{\circ }\forall k\in \{0,1,\text{…},72\}\}$. The UF_copl plan mimics current state‐of‐the‐art practice for both SBRT of liver metastases and SRT of AVMs.

2. A coplanar STF plan that delivers different non‐uniform dose distributions in distinct fractions, but using two coplanar VMAT arcs at a fixed couch angle of $\varphi =0^{\circ }$ in every fraction. This plan is hereafter referred to as STF_copl plan and is again obtained using a pre‐defined and fixed set of beam orientations $B_0=\{({\theta }_k,{\varphi }_k)|{\theta }_k=-{180}^{\circ }+k\ast 5^{\circ },{\varphi }_k=0^{\circ }\forall k\in \{0,1,\text{…},72\}\}$. The STF_copl plan is used to quantify the dosimetric benefit of spatiotemporal fractionation alone without the use of non‐coplanar beams.

3. A non‐coplanar uniformly fractionated plan that delivers the same dose distribution in every fraction, using the same two optimized non‐coplanar dynamic trajectories in every fraction, whose gantry‐couch path has been optimized using the proposed DAO approach. This plan, which is hereafter referred to as UF_non‐copl plan, serves to investigate the dosimetric benefit of non‐coplanar beam orientations alone without the spatiotemporal fractionation.

All these plans have been obtained using the same optimization parameters and planning objectives detailed above.

3. RESULTS

3.1. Patient 1

Figure 2 shows the fractional dose distributions obtained for patient 1 with the STF_non‐copl plan along with the corresponding gantry‐couch paths used in each fraction that define the distinct non‐coplanar dynamic trajectories. The dose distributions and gantry‐couch paths for the UF_copl, STF_copl and UF_non‐copl plans are reported in Figures C1, C2 and C3 in Appendix C, respectively. The STF_non‐copl plan delivers highly non‐uniform dose distributions in different fractions, each treating complementary parts of the PTV to a high dose of up to 20 Gy. In this way, the prescribed BED₁₀ to the PTV is achieved with a lower physical dose (the mean physical dose within the PTV is 42.0 Gy for the UF_copl plan and 39.4 Gy for the STF_non‐copl plan). At the same time, a similar dose bath is delivered to the healthy liver in every fraction, thereby exploiting the fractionation effect. Along with the use of different non‐coplanar dynamic trajectories in different fractions, which have been optimized to treat specific regions of the PTV, this results in a net reduction of the mean BED₃ to the healthy liver. Compared to the conventional UF_copl plan, the mean liver BED₃ is substantially reduced from 45.4 Gy to 32.9 Gy (−27.5%), for a similar target coverage.⁶ This dosimetric benefit is considerably larger than the benefit achieved by either using only spatiotemporal fractionation without non‐coplanar beam orientations, or only non‐coplanar beam orientations without spatiotemporal fractionation. For a similar target coverage, the STF_copl plan achieves a mean liver BED₃ of 38.1 Gy, and the UF_non‐copl plan a mean liver BED₃ of 38.5 Gy.

FIGURE 2.

Fractional dose distributions and the corresponding gantry‐couch paths for the STFnon‐copl plan for patient 1. The dark grey regions on the gantry‐couch map refer to beam orientations leading to collisions between the gantry and the couch. STF, Spatiotemporally fractionated.

The dosimetric results obtained for patient 1 using all the different plans considered in this study are summarized in Table 1. Figure 3 shows the comparison of the dose‐volume histograms for the PTV and healthy liver.

TABLE 1.

Dosimetric results for the different treatments plans generated for patient 1. Target coverage is expressed as the ratio of the PTV volume V_Dpres receiving a BED₁₀ larger than the prescribed BED₁₀ of 79.2 Gy and the total PTV volume V_PTV.

	UFcopl plan	STFcopl plan	UFnon‐copl plan	STFnon‐copl plan
Mean physical dose to the PTV (Gy)	42.0	38.2 (−9.0%)	41.9 (−0.2%)	39.4 (−6.2%)
Mean BED10 to the PTV (Gy)	101.2	99.9 (−1.3%)	100.6 (−0.6%)	100.1 (−1.1%)
PTV coverage – VDpres/VPTV (%)	94.8	92.7 (−2.2%)	94.6 (−0.2%)	92.7 (−2.2%)
Mean BED3 to the healthy liver (Gy)	45.4	38.1 (−16.1%)	38.5 (−15.2%)	32.9 (−27.5%)

Abbreviations: AVM, arteriovenous malformation; BED, biologically effective dose; STF, spatiotemporally fractionated.

FIGURE 3.

Dose‐volume histograms for the different plans generated for patient 1, evaluated for the equieffective dose. ²⁷ The equieffective dose $\mathrm{EQDX}=\frac{b}{x/\alpha /\beta }$ corresponds to the total physical dose that needs to be delivered in a uniformly fractionated treatment with a dose per fraction X to achieve a BED $b$, and is linearly proportional to the BED. For patient 1, we assumed X = 12 Gy for the PTV and X = 2 Gy for the healthy liver. BED, biologically effective dose.

3.2. Patient 2

The dose distributions and the corresponding gantry‐couch paths obtained for patient 2 using the STF_non‐copl plan are shown in Figure 4, while the dose distributions and gantry‐couch paths for the UF_copl, STF_copl and UF_non‐copl plans are reported in Figures C5, C6 and C7 in Appendix C, respectively. The STF_non‐copl plan delivers different dose distributions in different fractions, which hypofractionate the dose in the AVM and at the same time achieve more uniform fractionation in the normal brain. Similar as for patient 1, very different non‐coplanar dynamic trajectories are utilized in different fractions, depending on which regions of the AVM are targeted. The STF_non‐copl plan improves on the conventional UF_copl plan by reducing the mean brain BED₂ from 7.01  to 5.15 Gy (−26.5%), for a similar target coverage. The volume V₆₀ of healthy brain exposed to a BED₂ larger than 60 Gy is also reduced from 28.4  to 21.2 cc (−25.4%). Again, the dosimetric benefit of the STF_non‐copl plan over the STF_copl and UF_non‐copl plans is substantially larger. For a similar dose coverage of the AVM, the STF_copl plan achieves a mean brain BED₂ of 6.31 Gy and a brain volume V₆₀ exposed to high doses of 26.2 cc, while the UF_non‐copl plan achieves a mean brain BED₂ of 6.31 Gy and a brain volume V₆₀ exposed to high doses of 26.2 cc.

FIGURE 4.

Fractional dose distributions and the corresponding gantry‐couch paths for the SRT_FV plan for patient 2. The dark grey regions on the gantry‐couch map refer to beam orientations leading to collisions between the gantry and the couch. SRT, stereotactic radiotherapy.

The dosimetric results achieved by all plans generated for patient 2 are summarized in Table 2, while Figure 5 shows a comparison of the dose‐volume histograms for the AVM and the healthy.

TABLE 2.

Dosimetric results for the different treatments plans generated for patient 2. Target coverage is expressed as the ratio of the AVM volume V_Dpres receiving a BED₁₀ larger than the prescribed BED₁₀ of 77 Gy and the total AVM volume V_AVM.

	UFcopl plan	STFcopl plan	UFnon‐copl plan	STFnon‐copl plan
Mean physical dose to the AVM	32.8	28.8 (−12.2%)	32.9 (+0.3%)	23.7 (−27.7%)
Mean BED4 to the AVM	100.5	98.6 (−1.9%)	100.9 (+0.4%)	102.0 (+1.5%)
AVM coverage – VDpres/VAVM (%)	96.2	96.4 (+0.2%)	96.3 (+0.1%)	96.4 (+0.2%)
Mean BED2 to the healthy brain	7.01	6.31 (−10.0%)	6.79 (−3.1%)	5.15 (−26.5%)
Volume of the healthy brain receiving a BED2 larger than 60 Gy (cc)	28.4	26.2 (−7.8%)	26.2 (−7.8%)	21.2 (−25.4%)

Abbreviations: AVM, arteriovenous malformation; BED, biologically effective dose; STF, spatiotemporally fractionated.

FIGURE 5.

Dose‐volume histograms for the different plans in patient 2, evaluated for the equieffective dose. For patient 2, we assumed X = 7 Gy for the AVM and X = 2 Gy for the healthy brain. AVM, arteriovenous malformation.

4. DISCUSSION

Previous studies demonstrated that spatiotemporal fractionation schemes are a promising treatment approach, with improvements of 10%–15% in the ratio of tumor to normal tissue BED compared to conventional uniform fractionation schemes. In this work, we demonstrated that the dosimetric benefit of STF treatments can be considerably enhanced by using different non‐coplanar dynamic trajectories in distinct fractions, which are optimized together with the fractional dose distribution to treat specific regions of the target volume.

The use of different beam orientations in distinct fractions has previously been investigated by other groups. O'Connor et al. ²⁸ and Gu et al. ²⁹ showed for both non‐coplanar intensity modulated radiotherapy (IMRT) and intensity modulated proton therapy (IMPT) treatments that the use of different beam orientations in distinct fractions may either improve the dosimetric quality of a treatment while keeping the delivery time per fraction viable, or alternatively can maintain a good plan quality while lowering the delivery time. A similar approach was also investigated by Rossi et al. ³⁰ for SBRT. In all these studies, however, it was still assumed that a similar dose distribution is delivered within the target volume in every fraction. The benefit of varying the beam orientations in between different fractions thus mainly relied on the use of overall more beam orientations over the course of the treatment, and not in the selection of specific beam orientations that are beneficial to treat specific regions of the target volume.

The delivery of different non‐uniform dose distributions in distinct fractions using fraction‐specific non‐coplanar dynamic trajectories is technically feasible using conventional C‐arm linear accelerators. However, the following aspects must be carefully considered prior to an eventual clinical implementation of the proposed method:

• First, in this work, we assumed that all dose distributions are delivered as planned and add up to the prescribed BED in every part of the target volume. Inter‐ and intra‐fraction setup and motion uncertainties may, however, lead to misalignments of the different dose distributions when different parts of the target volume are treated in distinct fractions, thereby potentially causing target over‐ or under‐dosage. Also, uncertainties in the assumed BED model may impact the therapeutic efficacy. The use of image‐guided radiotherapy ³¹ and robust optimization approaches ⁶ , ⁹ may contribute to mitigating the uncertainties related to the delivery of non‐uniform dose distributions in distinct fractions.

• Second, although the simultaneous rotation of gantry and couch is supported by most commercial C‐arm linear accelerators, the clinical implementation of non‐coplanar dynamic trajectories still faces some potential problems, like the increased risk of collision between the gantry head and the system composed by couch and patient, and intra‐fraction patient motion due to the continuous couch rotation. Development of more accurate collision prediction models and the use of adequate immobilization devices may be warranted for the clinical use of non‐coplanar dynamic trajectories. Alternatively, BED‐based fraction‐variant beam orientation optimization may be used in combination with different delivery approaches that present less hurdles for clinical implementation, such as non‐coplanar IMRT ²⁸ or the use of multiple VMAT arcs at different static couch angles. ³²

• Finally, both the computation time needed for generating multiple fraction‐specific treatment plans⁷ and the time necessary for performing quality assurance of the different daily plans are increased. However, this is unlikely to be an insurmountable difficulty. The continuous development in computational hardware, in fact, makes optimization increasingly faster, while the need for plan‐specific quality assurance may be reduced through the use of artificial intelligence. ³³

Despite these challenges, STF treatments using fraction‐specific non‐coplanar dynamic trajectories show a potential to considerably improve the quality of radiotherapy compared to state‐of‐the‐art treatments that deliver the same dose distribution in every fraction using fixed and manually defined beam directions in every fraction. Future studies may be conducted to integrate previously developed robust optimization approaches ⁶ , ⁹ into the proposed DAO algorithm.

5. CONCLUSION

Spatiotemporal fractionation schemes are a promising approach for improving the therapeutic ratio in SBRT of liver metastases and SRT of large AVMs. The potential of spatiotemporal fractionation schemes to reduce normal tissue exposure can be considerably enhanced by selecting different beam orientations in different fractions, which are beneficial to treat specific regions of the target volume.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

APPENDIX A. POSSIBLE EXTENSION OF THE DAO ALGORITHM TO IMRT PLANNING

A.1.

The column generation‐based DAO algorithm proposed in Figure 1 has been specifically designed to determine fraction‐specific non‐coplanar dynamic trajectories. However, the algorithm could be modified to allow the optimization of IMRT plans. A possible modification to the algorithm workflow is shown in Figure A1, where promising beam orientations (defined through gantry angles $\theta$ and couch angles $\varphi$) are selected for each fraction from a discrete set of feasible beam orientations. Different from the DAO algorithm in Figure 1, multiple apertures can be delivered from each beam orientation. Also, delivery‐efficiency constraints must not be explicitly considered, what means that any feasible beam orientation can be selected, independently of which other beam orientations have previously been used. To anyway constrain the delivery time of IMRT plans, a maximum number $B_{\max }$ of different beam orientations can be specified for each fraction. Once for a given fraction multiple apertures have been generated for $B_{\max }$ different beam orientations, then new apertures can be added only for the already selected beam orientations in that fraction. The iterative generation of apertures will stop once a user‐defined number of apertures $N_{\max }$ is reached for each fraction.

FIGURE A1.

Example of a modified version of the DAO algorithm's workflow, which can be used to generate IMRT plans. Differences from the workflow proposed in Figure 1 are highlighted by the red contours. DAO, direct aperture optimization; IMRT, intensity modulated radiotherapy.

APPENDIX B. TREATMENT PLAN OPTIMIZATION

In this section, we further detail the treatment plan optimization methods. We provide the mathematical formulation of the planning objectives used in the treatment plan optimization problem, and discuss the dose calculation algorithm which has been used for computing the dose‐influence matrices.

Patient 1

For patient 1, the objective function in Equation (2) reads as follows:

$$\begin{array}{ccc}\hfill f\left(b\right)& =& \frac{1}{\left|GTV\right|}\sum _{i\in GTV}\left[{\omega }_1^{\mathrm{lower}}{\left(100.8-b_i\right)}_{+}^2\right.\hfill \\ & & \left.+{\omega }_1^{\mathrm{upper}}{\left(b_i-112.5\right)}_{+}^2\right]\hfill \end{array}$$ (B1)

$$+\frac{1}{\left|\stackrel{\sim }{PTV}\right|}\sum _{i\in \stackrel{\sim }{PTV}}\left[{\omega }_2^{\mathrm{lower}}{\left(79.2-b_i\right)}_{+}^2+{\omega }_2^{\mathrm{upper}}{\left(b_i-100.8\right)}_{+}^2\right]$$ (B2)

$$+\frac{{\omega }_3}{\left|NT\right|}\sum _{i\in NT}{\left(b_i-b_i^{\max }\right)}_{+}^2$$ (B3)

$$+\frac{{\omega }_4}{\left|L\right|}\sum _{i\in L}{b}_i$$ (B4)

where $GTV$ denotes the set of voxels belonging to the GTV, $\stackrel{\sim }{PTV}$ is the set of voxels belonging to the PTV excluding the GTV voxels, $NT$ denotes the set of voxels belonging to the normal tissue (i.e., the entire body except for the PTV) and $L$ is the set of voxels belonging to the healthy liver (i.e., liver without the PTV). The planning objective in Equation (B3) corresponds to the NTO implemented in the Eclipse Treatment Planning System (Varian Medical Systems, Inc.), where $b_i^{\max }$ is a voxel‐dependent value defined as:

$$b_i^{\max }=\left\{\begin{array}{c}{b}_0,if{x}_i<x_0\hfill \\ b_0{e}^{-\kappa \left(x_i-x_0\right)}+b_{\infty }\left(1-e^{-\kappa \left(x_i-x_0\right)}\right),if{x}_i\ge x_0\hfill \end{array}\right.$$

Here, $x_i$ indicates the distance of a normal tissue voxel $i$ from the PTV edge and we set $b_0=131.7\mathrm{Gy}$, $b_{\infty }=15.75\mathrm{Gy}$, $x_0=0.5\mathrm{cm}$ and $\kappa =0.6\mathrm{c}{\mathrm{m}}^{-1}$. The penalty value $b_i^{\max }$ as a function of $x_i$ is schematically illustrated in Figure B1.

FIGURE B1.

BED gradient as a function of the distance from the PTV edge required by the NTO in Equation (B3). BED, biologically effective dose; NTO, normal tissue objective.

The priorities ${\omega }_1^{\mathrm{lower}},{\omega }_1^{\mathrm{upper}},{\omega }_2^{\mathrm{lower}},{\omega }_2^{\mathrm{upper}},{\omega }_3$ and ${\omega }_4$ for the planning objectives in Equations (B1)–(B4) are set differently for the different treatment plans in order to achieve a similar target BED10 coverage and similar dose conformity in all plans:

$$\begin{array}{ccc}& & \mathrm{UFcopl}:{\omega }_1^{\mathrm{lower}}=20,{\omega }_1^{\mathrm{upper}}=2,{\omega }_2^{\mathrm{lower}}=10,{\omega }_2^{\mathrm{upper}}=1,{\omega }_3=7.5,{\omega }_4=6\hfill \end{array}$$

$$\begin{array}{ccc}& & \mathrm{STFcopl}:{\omega }_1^{\mathrm{lower}}=20,{\omega }_1^{\mathrm{upper}}=2,{\omega }_2^{\mathrm{lower}}=12,{\omega }_2^{\mathrm{upper}}=1.2,{\omega }_3=7.5,{\omega }_4=4.5\hfill \end{array}$$

$$\begin{array}{ccc}& & \mathrm{UFnon}-\mathrm{copl}:{\omega }_1^{\mathrm{lower}}=8,{\omega }_1^{\mathrm{upper}}=0.8,{\omega }_2^{\mathrm{lower}}=8,{\omega }_2^{\mathrm{upper}}=0.8,{\omega }_3=10,{\omega }_4=2.5\hfill \end{array}$$

$$\begin{array}{ccc}& & \mathrm{STFnon}-\mathrm{copl}:{\omega }_1^{\mathrm{lower}}=10,{\omega }_1^{\mathrm{upper}}=1,{\omega }_2^{\mathrm{lower}}=10,{\omega }_2^{\mathrm{upper}}=1,{\omega }_3=10,{\omega }_4=2\hfill \end{array}$$

Patient 2

For patient 2, the objective function in Equation (2) is expressed as follows:

$$f\left(b\right)=\frac{1}{\left|AVM\right|}\sum _{i\in AVM}\left[{\omega }_5^{\mathrm{lower}}{\left(77-b_i\right)}_{+}^2+{\omega }_5^{\mathrm{upper}}{\left(b_i-140\right)}_{+}^2\right]$$ (B5)

$$+\frac{{\omega }_6}{\left|B\right|}\sum _{i\in B}\frac{1}{1+e^{-\left(b_i-60\right)/0.5}}$$ (B6)

$$+\frac{{\omega }_7}{\left|NT\right|}\sum _{i\in NT}{\left(b_i-b_i^{\max }\right)}_{+}^2$$ (B7)

$$+\frac{{\omega }_8}{\left|B\right|}\sum _{i\in B}{b}_i$$ (B8)

where $AVM$ denotes the set of voxels belonging to the AVM, $B$ is the set of voxels belonging to the healthy brain (i.e., brain without the AVM) and $NT$ denotes the set of voxels belonging to the normal tissue (i.e., the entire body except for the AVM). The planning objective in Equation (B6) is used to control the brain volume exposed to BED₂ values larger than 60 Gy. Different than classical dose‐volume objectives, which are evaluated using the Heaviside step function $\mathrm{\Theta }(x)$, the objective function in Equation (B6) is evaluated using a continuously differentiable logistic sigmoid function ${\zeta }_{\epsilon }\left(x\right)$ (Figure B2a), where $\epsilon$ is called smoothness parameter. ³⁴ For $\epsilon \ne 0$, we can define a smooth approximation of a dose‐volume objective with a non‐vanishing gradient around $b$ = 60 Gy (Figure B2b). In this work, we set the smoothness parameter $\epsilon$ to 0.5 (note that ${\zeta }_{\epsilon }\left(x\right)\to \mathrm{\Theta }\left(x\right)$ for $\epsilon \to 0$).

FIGURE B2.

(a) Logistic sigmoid function and (b) its derivative. In (a), the red solid line is obtained for a smoothness parameter $\epsilon =0.5$ and corresponds to the continuously differentiable function that is used for the treatment plan optimization in this work, whereas the black dashed line (obtained for $\epsilon \to 0$) corresponds to the Heaviside step function.

The parameters for the NTO in Equation (B7) are set to $b_0=115\mathrm{Gy}$, $b_{\infty }=13.125\mathrm{Gy}$, $x_0=0.5\mathrm{cm}$ and $\kappa =0.6\mathrm{c}{\mathrm{m}}^{-1}$. Analogously to patient 1, the priorities ${\omega }_5^{\mathrm{lower}},{\omega }_5^{\mathrm{upper}},{\omega }_6,{\omega }_7$ and ${\omega }_8$ for the planning objectives in Equations (B5)‐(B8) are set differently for the different treatment plans in order to achieve a similar BED₄ coverage of the AVM in all plans:

$$\begin{array}{ccc}& & \mathrm{UFcopl}:{\omega }_5^{\mathrm{lower}}=10,{\omega }_5^{\mathrm{upper}}=1,{\omega }_6=100,{\omega }_7=1,{\omega }_8=10\hfill \end{array}$$

$$\begin{array}{ccc}& & \mathrm{STFcopl}:{\omega }_5^{\mathrm{lower}}=10,{\omega }_5^{\mathrm{upper}}=1,{\omega }_6=100,{\omega }_7=2,{\omega }_8=10\hfill \end{array}$$

$$\begin{array}{ccc}& & \mathrm{UFnon}-\mathrm{copl}:{\omega }_5^{\mathrm{lower}}=10,{\omega }_5^{\mathrm{upper}}=1,{\omega }_6=100,{\omega }_7=1,{\omega }_8=10\hfill \end{array}$$

$$\begin{array}{ccc}& & \mathrm{STFnon}-\mathrm{copl}:{\omega }_5^{\mathrm{lower}}=10,{\omega }_5^{\mathrm{upper}}=1,{\omega }_6=100,{\omega }_7=4,{\omega }_8=5\hfill \end{array}$$

Dose calculation algorithm

Calculation of the dose‐influence matrix elements $D_{ij}$ is performed with the open‐source radiotherapy planning research platform CERR ³⁵ using a quadrant infinite beam (QIB) algorithm. ³⁶

APPENDIX C. RESULTS

Patient 1

Results for the UF_copl, STF_copl, and UF_non‐copl plans

The dose distributions and gantry‐couch paths for the UF_copl, STF_copl and UF_non‐copl plans for patient 1 are shown in Figures C1, C2, and C3, respectively. Interestingly, the spatial dose distributions are quite different between the STF_copl and STF_non‐copl plans, indicating that the optimal dose compartmentalization of the target volume depends on the field setup used for the treatment. Also, the gantry‐couch path used to deliver the UF_copl plan presents differences compared to the gantry‐couch paths used to deliver each fraction of the STF_non‐copl plan.

FIGURE C1.

Fractional dose distribution and the corresponding gantry‐couch path for the UFcopl plan for patient 1.

FIGURE C2.

Fractional dose distributions and the corresponding gantry‐couch paths for the STFcopl plan for patient 1. STF, spatiotemporally fractionated.

FIGURE C3.

Fractional dose distribution and the corresponding gantry‐couch path for the UFnon‐copl plan for patient 1.

Cumulative EQD12 distributions

Figure C4 shows the cumulative EQD12 distributions achieved with all plans generated for patient 1. Although the STF plans deliver non‐uniform doses to the target volume in distinct fractions, the sum of all fractional doses results in the same prescribed BED10 within the PTV as in the uniformly fractionated plans.

FIGURE C4.

Cumulative EQD2 distributions for the UFcopl, STFcopl, UFnon‐copl, and STFnon‐copl plans for patient 1. STF, spatiotemporally fractionated.

Patient 2

Results for the UF_copl, STF_copl, and UF_non‐copl plans

The dose distributions and gantry‐couch paths for the UFcopl, STFcopl, and UFnon‐copl plans for patient 2 are shown in Figures C5, C6, and C7, respectively.

FIGURE C5.

Fractional dose distribution and the corresponding gantry‐couch path for the UFcopl plan for patient 2.

FIGURE C6.

Fractional dose distributions and the corresponding gantry‐couch paths for the STFcopl plan for patient 2. STF, spatiotemporally fractionated.

FIGURE C7.

Fractional dose distribution and the corresponding gantry‐couch path for the UFnon‐copl plan for patient 2.

Cumulative EQD7 distributions

Figure C8 shows the cumulative EQD7 distributions achieved with all plans generated for patient 2. Although the STF plans deliver non‐uniform doses to the target volume in distinct fractions, the sum of all fractional doses results in the same prescribed BED₄ within the AVM as in the uniformly fractionated plans. AVM, arteriovenous malformation; BED, biologically effective dose; STF, spatiotemporally fractionated.

FIGURE C8.

Cumulative EQD2 distributions for the UF_copl, STF_copl, UF_non‐copl and STF_non‐copl plans for patient 2.

Torelli N, Unkelbach J. Fraction‐variant beam orientation optimization for spatiotemporal fractionation schemes. Med Phys. 2025;52:4723–4741. 10.1002/mp.17798