Development and experimental validation of an in-house treatment planning system with greedy energy layer optimization for fast IMPT

Abstract

Background:

Intensity-modulated proton therapy (IMPT) using pencil beam technique scans tumor in a layer by layer, then spot by spot manner. It can provide highly conformal dose to tumor targets and spare nearby organs-at-risk (OAR). Fast delivery of IMPT can improve patient comfort and reduce motion-induced uncertainties. Since energy layer switching time dominants the plan delivery time, reducing the number of energy layers is important for improving delivery efficiency. Although various energy layer optimization (ELO) methods exist, they are rarely experimentally validated or clinically implemented, since it is technically challenging to integrate these methods into commercially available treatment planning system (TPS) that is not open-source.

Purpose:

This work develops and experimentally validates an in-house TPS (IH-TPS) that incorporates a novel ELO method for the purpose of fast IMPT.

Methods:

The dose calculation accuracy of IH-TPS is verified against the measured beam data and the RayStation TPS. For treatment planning, a novel ELO method via greed selection algorithm is proposed to reduce energy layer switching time and total plan delivery time. To validate the planning accuracy of IH-TPS, the 3D gamma index is calculated between IH-TPS plans and RayStation plans for various scenarios. Patient-specific quality-assurance (QA) verifications are conducted to experimentally verify the delivered dose from the IH-TPS plans for several clinical cases.

Results:

Dose distributions in IH-TPS matched with those from RayStation TPS, with 3D gamma index results exceeding 95% (2mm, 2%). The ELO method significantly reduced the delivery time while maintaining plan quality. For instance, in a brain case, the number of energy layers was reduced from 78 to 40 (reduction of 38 layers), leading to a 62% reduction in total delivery time. Patient-specific QA validation with the IBA Proteus^®ONE proton machine confirmed a >95% pass rate for all cases.

Conclusions:

An IH-TPS equipped with a novel ELO algorithm is developed and experimentally validated for the purpose of fast IMPT, with enhanced delivery efficiency and preserved plan quality.

1. Introduction

Intensity-modulated proton therapy (IMPT) based on the pencil beam scanning (PBS) technique can deliver highly conformal dose to tumor targets, effectively reducing the exposure of surrounding healthy tissues when compared to photon therapy or passive scattering proton therapy [1]. The fast treatment delivery of IMPT has several benefits, such as enhanced patient comfort, reduced motion impact [2,3], and increased patient throughput, allowing more efficient use of limited proton treatment resources and expanding access of proton therapy [4,5].

The IMPT treatment scans the tumor target layer by layer, and then spot by spot [6,7]. The delivery time of PBS based IMPT mainly consists of three parts: dose delivery time, spot scanning time, and energy layer switching time (ELST). The ELST constitutes a significant portion of the overall plan delivery. For instance, in cyclotron-based and synchrotron-based systems like the IBA’s proton machines, the average ELST accounts for more than half of the total plan delivery time [8,9]. This motivates the reduction of energy switching time in the pursuit of delivery efficiency.

Several energy layer optimization (ELO) methods have been proposed to reduce the energy switching time. Van De Water et.al [10] minimized the logarithm of total spot weight per energy layer and then excluded the energies with low-weighted value. Cao W et.al [11] formulated the problem as a mixed integer problem and iteratively deleted one energy at a time. Jensen et.al [12] introduced the root mean squared regularizer and excluded energy with lowest root mean squared value. Lin Y et.al [13] proposed MMSEL, which addressed energy reduction through group sparsity and minimum monitor unit constraints. Lin B [14] developed so-called the CARD method, applying the cardinality-sparsity constraint to the energy vector. However, these ELO methods are rarely experimentally validated or clinically implemented, as it is technically challenging to integrate these methods into commercially available treatment planning system (TPS) that is not open-source.

Studies referenced in [10–14] primarily evaluate their algorithms using self-developed IH-TPS, often without validation within clinical software. Other studies have demonstrated experimental validation by implementing spot-reduction methods directly in clinical systems [15], although these typically do not involve modifications to the underlying optimization algorithms. While each approach offers valuable insights, our work aims to contribute by integrating both algorithm design and experimental verification in a unified framework.

2. Methods

2.1. IH-TPS: dose engine

The dose engine of IH-TPS is based on a modified version of pencil beam dose calculation algorithm [16]. The pencil beam model is a simplified method for dose calculation in which each proton beam is treated as a narrow, focused spot that deposits dose according to a predefined distribution, typically modeled as a Gaussian or double Gaussian function. In our work, we use double Gaussian function. In proton therapy, the source-to-axis distance refers to the distance between the proton beam source and the central axis of the treatment beam, which is typically the axis around which the beam is directed toward the target. To model our IBA’s Proteus^®ONE proton machine, two virtual source-axis-distance (SAD) values are used. The physical reasoning behind the use of two SADs within our beam model is related to the two different beam scanning magnets that are situated at different locations along the proton beamline [17–18]. To improve the accuracy of dose calculation in patient, we convert the Hounsfield Unit values to mass densities using CT calibration measurements obtained from our CT scanner.

In order to ensure the clinical deliverability and accuracy of IH-TPS, the machine parameters and the configuration of IH-TPS are matched to our proton machine. The beam model consists of the parameters from all energy layers. To create the beam data for IH-TPS, RayStation’s Monte Carlo (MC) dose engine is used to simulate the 3D dose of different energy layers in a 40×40×40 cm³ water phantom. The base data consists of single-spot dose results at 2.5 MeV intervals, ranging from 70 to 225 MeV. In RayStation’s MC, each energy layer is delivered as a single spot with one monitor unit, with the beam isocenter set at the surface of the water phantom. Then we can get a 3D dose of single spot shooting in water. These base data are used to calculate the beam parameters for all energy layers, including integrated depth dose (IDD), range, peak, and spot sizes in water at different depths. The IDD is directly obtained by integrating the dose with a 1 mm interval. The range is calculated as the position where the IDD decreases to 80% of the peak dose. The peak position is determined through curve fitting (using the formula in [19]). The spot sizes are calculated using a double Gaussian fitting method. By adjusting these beam parameters, a new beam model of 63 energy layers is developed for IH-TPS.

2.2. IH-TPS: inverse planning method

2.2.1. Energy layer optimization

The inverse planning of IH-TPS with ELO is to solve the following optimization problem

$$\begin{array}{l}\underset{x\in {\mathbb{R}}^n}{\mathrm{m}\mathrm{i}\mathrm{n}}f(d,\mathrm{\Omega })\\ \mathit{s}.t.\left\{\begin{array}{l}d=Dx\\ x_j\in \left\{0\right\}\cup \left[G_{\mathrm{m}\mathrm{i}\mathrm{n}},+\mathrm{\infty }\right),j\le n\end{array}\right.\end{array}$$ (1)

where $d$ is dose vector, is dose influence matrix and is beam intensity. The variable $x$ satisfies the minimum-monitor unit constraint (MMU) with MMU threshold $G_{min}$. $f(d,\Omega )$ is dose-volume histogram (DVH) [18,19] related plan objective, enforcing the consistency between the prescribed dose and optimized dose $d$. Based on quadratic model, the $f(d,\Omega )$ has the following form

$$f(d,\mathrm{\Omega })=\sum _{i=1}^{N_{L2}}{\omega }_{1,i}{‖d_{{\mathrm{\Omega }}_{1,i}}-d_{1,i}‖}^2+\sum _{i=1}^{N_{DNH-\mathrm{m}\mathrm{a}\mathrm{x}}}{\omega }_{2,i}{‖d_{{\mathrm{\Omega }}_{2,i}}-d_{2,i}‖}^2+\sum _{i=1}^{N_{DVH-\mathrm{m}\mathrm{i}\mathrm{n}}}{\omega }_{3,i}{‖d_{{\mathrm{\Omega }}_{3,i}}-d_{3,i}‖}^2$$ (2)

The first term is the $L_2$ type objectives (for target and region-of-interest (ROI)) that sums the squared deviation between optimized dose and prescription dose over voxels under consideration. The second term is DVH-max constraint objective that limits the upper bound of ratio of overdosed voxels in ROIs. The third term is DVH-min constraint objective that sets the lower bound of ratio of underdose voxels inside the target. ${\omega }_{1,i}$, ${\omega }_{2,i}$ and ${\omega }_{3,i}$ are positive objective weights and $\Omega$ is the indexes of voxels that violate the DVH constraints [20,21].

The number of columns in $D$ is determined by the number of spots for each energy layer, the number of energy layers for each delivery angle, and the total number of delivery angles. As previously mentioned, the number of energy layers plays a crucial role in minimizing the overall delivery time, to improve delivery efficiency, the number of energy layers should be optimized.

Firstly, the influence matrix $D$ can be decomposed by the energy variable as the follows

$$D=\left[D_1,D_2,\cdots ,D_{NE}\right]$$ (3)

where $D_i$ is the influence matrix for $i^{\mathit{th}}$ energy, and the number of columns of $D_i$ is determined by the number of spots. The beam intensity $x$ can be similarly decomposed by

$$x={\left[x_1,x_2,\cdots ,x_{NE}\right]}^T$$ (4)

Each $x_i$ represents a given block of $x$, and their indexes are given by

$$[1,2,\cdots ,N]=\mathrm{\Lambda }=\left[{\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2,\cdots ,{\mathrm{\Lambda }}_{NE}\right]$$ (5)

Limiting the number of energies is equivalent to restricting the number of nonzero blocks of $x$ in (4). Denote the $\Vert \mathrm{x}{\Vert }_{\text{BS}}$ as the number of nonzero blocks, then the energy layer constraint can be modeled as

$$\Vert x{\Vert }_{BS}\le N$$ (6)

where $N<N_E$ is predefined number of energies to be used.

Combing (1) and (6), we get the following energy layer optimization model

$$\begin{array}{l}\underset{x\in {\mathbb{R}}^n}{\mathrm{m}\mathrm{i}\mathrm{n}}f(d,\mathrm{\Omega })\\ s.t.\left\{\begin{array}{l}d=Dx\\ x_j\in \left\{0\right\}\cup \left[G_{\mathrm{m}\mathrm{i}\mathrm{n}},+\mathrm{\infty }\right),j\le n\\ \Vert x{\Vert }_{BS}\le N\end{array}\right.\end{array}$$ (7)

2.2.2. Optimization algorithm

The optimization algorithm for solving (7) is based on the block orthogonal matching pursuit (BOMP) method [30,31], and its workflow is illustrated in Figure 1. The process successively selects energy layers as follows: (1) Utilize the BOMP method to identify the potential energy levels that contribute most significantly to enhancing plan quality. (2) Inverse planning based on the selected energies. (3) Evaluation of the plan quality for the newly obtained solution, where a candidate energy block is retained only if the plan quality improves. This iterative process continues until the predefined number of energy layers is reached or a planning quality parameter, such as Dmax, is satisfied. A strict condition-checking mechanism ensures that each iteration results in a monotonic decrease in the objective function value. The final output is an optimal IMPT plan that satisfies clinical constraints. The details of the algorithm can be found in supplementary material.

Figure 1. The optimization algorithm workflow.

2.3. Materials

Three clinical cases, including abdomen, brain, and lung, were utilized to validate and evaluate the proposed IH-TPS in terms of both accuracy and efficiency. The dose influence matrix was generated on a 1 mm³ dose grid. Clinically used beam angles were used for each case: (90°, 180°, 270°) for the abdomen, (45°, 135°, 225°, 315°) for the brain, and (0°, 60°, 90°, 330°) for the lung. All OAR planning constraints are shown in Table S1 of the supplementary material. For all cases, both the conventional plan and the ELO plan were calculated. The conventional plan refers to the plan without energy layer optimization.

When using the ELO method to compute treatment plans with different numbers of energy layers, the process begins with a single energy layer and gradually increases, adding one layer per step. In this study, for cases with fewer energy layers, such as the abdomen case, a step size of 5 energy layers was selected. Conversely, for the lung case, a step size of 10 energy layers was used. At each step, Dmax was evaluated to determine whether it satisfied the criterion |Dmax - 120%| ≤ 2%. If this condition is met, the calculation process is terminated.

Clinically-used DVH plan objectives were utilized, and all plans were normalized to D95% = 100% in CTV. The conformity index (CI) was evaluated, defined as ${\mathrm{V}}_{100,\mathrm{C}\mathrm{T}\mathrm{V}}^2/({\mathrm{V}}_{\mathrm{C}\mathrm{T}\mathrm{V}}\times {\mathrm{V}}_{100})$. (${\mathrm{V}}_{100,\mathrm{C}\mathrm{T}\mathrm{V}}$: CTV volume receiving at least 100% of prescription dose; ${\mathrm{V}}_{\mathrm{C}\mathrm{T}\mathrm{V}}$: CTV volume; ${\mathrm{V}}_{100}$: total body volume receiving at least 100% of prescription dose; ideally CI = 1). The dose quantities are in percentage with respect to the prescription dose; the values of objective value and CI are unitless. All the plan-quality dosimetric quantities of OARs are in percentage with respect to the prescription dose.

The experimental validations were conducted: (1) proton spot weights and locations were scripted as a csv file to be imported to RayStation TPS for delivery on our proton machine and recalculating 3D dose in RayStation for comparison; (2) experimental validations were performed with patient-specific quality-assurance plan delivery in phantoms; (3) the absolute proton dose was measured via MatriXX ONE 2D ionization chamber array; (4) the gamma indexes was calculated to quantify the delivery accuracy, using the clinical standard for the passing criterion (≥95%, with 2%/2mm and a 10% threshold). During the experimental validation, we recorded the delivery time for each plan on the machine, which can be directly obtained from the IBA machine.

3. Results

The results are divided into five sections: the first three focus on validating the effectiveness of IH-TPS and the ELO method, while the last two cover robust analysis and comparisons with other method. Figure 2 illustrates the roadmap of the section 3. To validate IH-TPS, CONV plans were generated using IH-TPS and recalculated in RayStation for dose comparison. The effectiveness of the ELO method was then evaluated by comparing ELO plans from IH-TPS with CONV plans. Finally, the ELO plans were recalculated in RayStation and delivered on an actual machine to experimentally validate the ELO method and demonstrate its efficiency. To mitigate the uncertainties in proton therapy, robust optimization was applied in section 3.4. To evaluate the effectiveness of our proposed method, we compared it with the iterative mixed-integer programming (MIP) optimization approach in section 3.5.

Figure 2. Roadmap of the section 3.

3.1. IH-TPS v.s. RayStation

To compare IH-TPS and RayStation (which is clinically used at our institution), we conducted analyses under two scenarios: (1) to examine the IDD, central depth dose profile, lateral beam profile at beginning and peak, and three-dimensional gamma-analysis of a single spot irradiation in water; (2) to compare optimized plans (without ELO) for three clinical cases. For each scenario, three-dimensional gamma analysis [37], was employed with a distance to agreement (DTA) of 2 mm or 3 mm and a dose difference (DD) criterion of 2% or 3%. Only dose points exceeding 10% of the maximum dose value were included in the gamma analysis.

3.1.1. Single spot

IDD of single spot for three different energy layers (with peak positions at 5 cm, 15 cm, and 25 cm) were given in Figure 3(a). The central depth dose profiles were given in Figure 3(b). The lateral dose profiles were displayed (c) and (d). All the results were scaled to spread them for visualization and utilized identical image dimensions and dose resolutions (x = 1 mm, y = 1 mm, z = 1 mm). The results demonstrated a good agreement between IH-TPS and RayStation. The global gamma analyses (dta = 2 mm, Dd = 2%) pass rates for the three energy layers are 100%, 99.98%, and 96.51%, respectively. The detailed error maps are presented in Figure S1.

Figure 3. Validation of beam model for IH-TPS.

Dose distributions of single spot for different energy layers with peak positions at 5 cm, 15 cm, and 25 cm. (a). Integral depth dose. (b). Central depth dose profiles. (c)-(d). Lateral dose profiles at the beginning (2 cm) and the peak. All the results were rescaled with maximum dose 2, 1, and 0.5.

3.1.2. Different cases

There were small differences between IH-TPS and RayStation as shown in Figure 4–6. For instance, the dose difference heatmaps in Figure 4–6(c) show that the dose discrepancies within the target between IH-TPS and RayStation were within 10%. Specifically, the lung case exhibited the smallest differences, while the abdomen and brain cases showed relatively larger discrepancies. However, almost all plan quality parameters were within 10%. As shown in Table 1 and Table S4, the CI and target maximum dose for IH-TPS vs. RayStation in the lung case were 0.74 and 113.1% vs. 0.72 and 113.9%, respectively. The maximum dose differences for key OARs, such as the chest wall, heart, and spinal cord, were 1.5%, 3.2%, and 2.9%, respectively. Regarding target coverage, the CI and target maximum dose for IH-TPS vs. RayStation were 0.90 and 110.8% vs. 0.84 and 112.9% for the abdomen, and 0.72 and 113.1% vs. 0.64 and 116.1% for the brain. Compared to the lung case, the abdomen and brain cases exhibited more noticeable differences, which can also be observed in Figure 4–6(c). In addition, as presented in Table 1 and Tables S2–S3, the dose differences for most oars were minimal. For example, in the abdomen case, the mean kidney dose, the maximum dose to the spinal cord, and the bowel V_100% were 49.1% vs. 48.1%, 103.1% vs. 103.7%, and 5.2% vs. 5.9%, respectively. Similarly, for the brain case, the mean brainstem dose and the maximum dose to the spinal cord were 25.5% vs. 26.0% and 108.6% vs. 110.4%, respectively.

Figure 4. Abdomen.

(a)-(b). Dose plots of IH-TPS and RayStation; (c). The heatmap of dose difference between IH-TPS and RayStation; (d)-(e). DVH plots of target and kidney. The dose plot window is [0%, 120%] of the prescription dose, with 80% (yellow) and 100% (red) isodose lines and CTV (green) highlighted in dose plots of (a) and (b).

Figure 6. Lung.

(a)-(b). Dose plots of IH-TPS and RayStation; (c). The heatmap of dose difference between IH-TPS and RayStation; (d)-(e). DVH plots of target and esophagus. The dose plot window is [0%, 120%] of the prescription dose, with 80% (yellow) and 100% (red) isodose lines and CTV (green) highlighted in dose plots of (a) and (b).

Table 1. IH-TPS v.s. RayStation.

The plan-quality dosimetric quantities from left to right: optimization objective value f, conformity index (CI), max target dose Dmax, mean OAR dose Dmean_oar and max OAR dose Dmax_oar. All the plans were generated with the CONV method. The OARs for the abdomen, brain, and lung cases are the kidneys, brainstem, and esophagus, respectively.

Case	Model	f	CI	Dmax	Dmax_oar	Dmean_oar
Abdomen	IH-TPS	2.23	0.90	110.8%	108.1%	49.1%
	RayStation	2.42	0.84	112.9%	109.6%	48.1%
Brain	IH-TPS	2.21	0.72	113.1%	97.7%	25.5%
	RayStation	2.88	0.64	116.1%	102.7%	26.0%
Lung	IH-TPS	1.15	0.74	113.1%	106.8%	16.9%
	RayStation	1.32	0.72	113.9%	104.0%	15.4%

Additionally, the 3D gamma index results for the three cases were as follows: 95.59% (2mm, 2%) and 99.43% (3mm, 3%) for the abdomen; 97.09% (2mm, 2%) and 99.38% (3mm, 3%) for the brain; and 95.77% (2mm, 2%) and 99.45% (3mm, 3%) for the lung. The dose and DVH plots in Figures 3–5 further validate these results. Lastly, Figure 7 presents the QA results from one beam in each case. The gamma index was 97.2% (with 100% for the other two beams) for the abdomen, 100% (98.15%, 98.18%, 95.92% for the remaining beams) for the brain, and 100% (100%, 97.96%, and 97.56% for the other beams) for the lung.

Figure 5. Brain.

(a)-(b). Dose plots of IH-TPS and RayStation; (c). The heatmap of dose difference between IH-TPS and RayStation; (d)-(e). DVH plots of target and brainstem. The dose plot window is [0%, 120%] of the prescription dose, with 80% (yellow) and 100% (red) isodose lines and CTV (green) highlighted in dose plots of (a) and (b).

Figure 7. QA results.

The measured ((a), (d), (g)), IH-TPS ((b), (e), (h)), and gamma index ((c), (f), (i)) for the abdomen, brain and lung, respectively (2%/2mm, 10% threshold).

3.2. Energy layer optimization via IH-TPS

We compared the results without (CONV) and with ELO (using the modified BOMP) in IH-TPS. The results with ELO required fewer energy layers to achieve similar plan quality, as shown in Table 2. For instance, with the target maximum dose kept at an acceptable level, ELO reduced the number of energy layers by 50% (from 50 to 25) for the abdomen, 62% (from 78 to 30) for the brain, and 30% (from 100 to 70) for the lung. In terms of target coverage, ELO showed comparable CI, such as 0.90 vs. 0.84 for the abdomen, 0.72 vs. 0.66 for the brain, and 0.74 vs. 0.70 for the lung. Additionally, ELO achieved similar OAR sparing, with mean kidney doses of 49.1% (CONV) vs. 50.3% (ELO) for the abdomen, mean brainstem dose of 25.5% vs. 29.7% for the brain, and mean esophagus dose of 16.9% vs. 18.2% for the lung. These findings are further validated by the dose plots in Figure 8 and DVH plots in Figure 9. Additional plan quality results can be found in Tables S2–S4.

Table 2. CONV v.s. ELO using IH-TPS.

The plan-quality dosimetric quantities from left to right: number of used energies, optimization objective value f, conformity index (CI), max target dose Dmax, mean OAR dose Dmean_oar and max OAR dose Dmax_oar. The OARs for the abdomen, brain, and lung cases are the kidneys, brainstem, and esophagus, respectively.

Cases	Methods	Energy	f	CI	Dmax	Dmax_oar	Dmean_oar
Abdomen	CONV	50	2.23	0.90	110.8%	108.1%	49.1%
	ELO	30	2.70	0.86	115.4%	112.5%	51.1%
		25	3.12	0.84	120.6%	114.3%	50.3%
		20	3.83	0.83	130.7%	114.7%	50.7%
Brain	CONV	78	2.21	0.72	113.1%	97.7%	25.5%
	ELO	40	2.79	0.69	117.8%	100.5%	26.3%
		30	3.82	0.66	120.1%	100.0%	29.7%
		20	6.34	0.61	125.9%	104.2%	29.7%
Lung	CONV	100	1.15	0.74	113.1%	106.8%	16.9%
	ELO	80	1.38	0.72	115.1%	106.8%	17.8%
		70	1.54	0.70	118.4%	108.1%	18.2%
		60	1.86	0.69	122.5%	109.1%	19.0%

Figure 8. Dose plots of CONV and ELO with different energy layers.

(a)-(d). Dose plots of CONV and ELO with different energy layers for the abdomen; (e)-(h). Dose plots of CONV and ELO with different energy layers for the brain; (i)-(l). Dose plots of CONV and ELO with different energy layers for the lung; The dose plot window is [0%, 120%] of the prescription dose, with 80% (yellow) and 100% (red) isodose lines and CTV (green) highlighted in dose plots.

Figure 9. DVH plots for CONV and ELO with different energy layers.

(a)-(b). The DVH plots of the target and kidneys for abdomen. (c)-(d). The DVH plots of the target and brainstem for brain. (e)-(f). The DVH plots of the target and esophagus for lung.

3.3. Experimental validation of IH-TPS

To demonstrate the effectiveness of the method, all newly generated plans using IH-TPS were recalculated in RayStation. The energy layers and monitor units (MU) used in each plan can be found in Figures S5–S7. As shown in Tables 2 and 3, the plan quality for the abdomen and brain cases slightly declined after recalculation in RayStation. For instance, in IH-TPS, the number of energy layers was reduced from 50 to 25 for the abdomen case (reduction of 25 layers), from 78 to 30 for the brain case (reduction of 48 layers) and from 100 to 70 (reduction of 30 layers) for the lung case. But in RayStation, the number of energy layers was reduced to 30 for the abdomen and 40 for the brain (all plans were recalculated in RayStation). Nevertheless, the time reduction remained significant based on the actual delivery time recorded during machine delivery. The abdomen, brain, and lung cases achieved delivery time reductions of 18%, 49%, and 28.7%, respectively.

Table 3. CONV v.s. ELO in RayStation.

The plan-quality dosimetric quantities from left to right: number of used energies, optimization objective value f, conformity index (CI), max target dose Dmax, mean OAR dose Dmean_oar and max OAR dose Dmax_oar. The OARs for the abdomen, brain, and lung cases are the kidneys, brainstem, and esophagus, respectively.

Cases	Methods	Energy	f	CI	Dmax	Dmax_oar	Dmean_oar
Abdomen	CONV	50	2.42	0.84	112.9%	109.6%	48.1%
	ELO	30	3.06	0.82	119.3%	113.4%	50.3%
		25	3.82	0.81	124.8%	104.0%	49.7%
		20	5.26	0.79	136.1%	118.9%	50.3%
Brain	CONV	78	2.88	0.64	116.1%	102.7%	26.0%
	ELO	40	3.44	0.62	119.8%	100.0%	24.8%
		30	4.89	0.60	124.6%	99.4%	28.2%
		20	6.54	0.58	124.6%	101.5%	28.3%
Lung	CONV	100	1.42	0.72	113.9%	104.0%	15.4%
	ELO	80	1.61	0.69	116.1%	105.5%	15.9%
		70	1.89	0.67	118.7%	107.6%	16.3%
		60	2.25	0.64	122.6%	110.0%	17.6%

Additionally, to verify the deliverability of the plans, we conducted QA and all cases exceeded the 95% pass rate. For ELO30 plan of the abdomen, beam 1 achieved 99.28%, beam 2 reached 100%, and beam 3 achieved 99.34%. For ELO40 plan of the brain, beam 1 and beam 2 both achieved 100%, while beam 3 and beam 4 attained 97.96% and 97.56%, respectively. For ELO70 plan of the lung, beam 1 achieved 98.99%, beam 2 and beam 3 both reached 100%, and beam 4 achieved 99.24%.

3.4. Robustness

To address the uncertainties inherent in proton therapy, robust optimization (Here the robustness is evaluated during the energy layer optimization) was implemented in this study. Setup uncertainty was modeled by applying a rigid shift (e.g., ±3 mm) in the axial, sagittal, and coronal directions, while range uncertainty was accounted for by uniformly scaling the CT values (e.g., ±5%). For each case, both the CONV plan and the plan optimized using the ELO method were calculated following the approach in section 2.3. The dose influence matrix was generated on a 3³ mm³ dose grid. The robustness of the plan is also quantitatively evaluated using the robustness variance (RV). The RV at a given dose level, denoted as RVp for p% dose, is calculated as the average variation between the maximum and minimum percentage volume observed at dose levels (p-1)%, p%, and (p+1)%.

The DVH curves from all uncertainty scenarios for the CTV and an OAR in the lung case are shown in Figure 10(e)–(h). These results indicate that the robustness of the plan is only slightly affected when the number of energy layers is reduced. For instance, when the number of energy layers is reduced to 55, the RV₉₈ of the CTV for the ELO55 and CONV plans is 9.8 vs. 8.5, while the RV₅₀ of the OAR remains unchanged at 14.4 for both plans. Similar trends can be observed in the abdomen and brain cases, as shown in supplementary Figures S8 and S9. The plan quality of nominal plans with different energy layers, after robust optimization, is summarized in Table 5 and supplementary Tables S5 and S6.

Figure 10. Robust optimization for lung.

(a)-(d). The dose plots of the CONV and ELO with different energy layers. (e)-(h). The DVH plots of the target and esophagus.

Table 5. Robust optimization for lung.

The plan-quality dosimetric quantities from left to right: number of used energies, max target dose Dmax, conformity index (CI), mean OAR dose Dmean_oar, max OAR dose Dmax_oar and volume percentage receiving more than 40% of the prescription dose V_40%. OAR1 to OAR6 correspond to the esophagus, chest wall, great vessels, heart, healthy lungs, and spinal cord, respectively.

Plan	Dmax	CI	Dmax_oar1	Dmean_oar1	Dmax_oar2	Dmax_oar3	Dmax_oar4	V40%_oar5	Dmax_oar6
CONV(103)	110.3%	0.80	104.8%	21.7%	104.0%	110.3%	56.6%	5.9%	8.3%
ELO-65	118.8%	0.77	104.7%	20.1%	104.8%	112.4%	59.1%	6.1%	7.7%
ELO-55	120.7%	0.76	105.9%	19.6%	105.4%	116.3%	59.4%	6.3%	7.1%
ELO-45	123.0%	0.74	107.1%	19.5%	106.8%	115.6%	66.0%	6.2%	7.6%

3.5. Comparisions

To demonstrate the effectiveness of the method we developed, we conducted a comparison with the iterative mixed-integer programming (MIP) optimization method [11] where we replace the algorithm for subproblem from branch and bound by CARD [15] for computational efficiency. We conduct the experiment on lung case to evaluate the methods. The MIP method decreases the number of energy layers iteratively and terminate the iteration when max target dose Dmax beyond 120%. In contrast, our method began with a single energy layer and incrementally increased until Dmax first fell under 120%. Both methods utilized the same planning constraints, as detailed in Table S1. The dose influence matrix was generated on a 3³ mm³ dose grid, and the robust settings described in section 3.4 were applied to both approaches.

As shown in Figure 11 and Table 6, the proposed energy layer optimization algorithm demonstrates significantly higher efficiency compared to the MIP method. The BOMP method successfully reduces the number of energy layers to 55, whereas the MIP method requires 70 energy layers to achieve a comparable plan quality. Specifically, the Dmax and CI for BOMP55 vs. MIP70 are 120.7% and 0.76 vs. 120.8% and 0.77, respectively. Furthermore, the ELO method achieved similar OAR sparing, with the mean esophagus dose at 19.6% vs. 21.0%, the maximum chest wall dose at 105.4% vs. 109.6%, and the V_40% of healthy lungs at 6.3% vs. 6.2%.

Figure 11. Comparisons.

(a)-(b). The dose plots of BOMP and MIP. (c). The DVH and robustness variance of target and esophagus.

Table 6. Comparisons.

The plan-quality dosimetric quantities from left to right: number of used energies, max target dose Dmax, conformity index (CI), mean OAR dose Dmean_oar, max OAR dose Dmax_oar and volume percentage receiving more than 40% of the prescription dose V_40%. OAR1 to OAR6 correspond to the esophagus, chest wall, great vessels, heart, healthy lungs, and spinal cord, respectively.

Plan	Dmax	CI	Dmax_oar1	Dmean_oar1	Dmax_oar2	Dmax_oar3	Dmax_oar4	V40%_oar5	Dmax_oar6
BOMP-55	120.7%	0.76	105.9%	19.6%	105.4%	116.3%	59.4%	6.3%	7.1%
MIP-70	120.8%	0.77	109.4%	21.0%	109.6%	112.5%	52.4%	6.2%	3.9%

4. Discussion

This paper introduces and experimentally validates an IH-TPS with greedy energy layer optimization for fast IMPT. Experiments on abdomen, brain and lung cases demonstrate the accuracy and efficiency of the proposed IH-TPS. Compared to the standard IMPT plan, the treatment delivery time for brain and lung cases using ELO-based IMPT via IH-TPS was reduced by 48.6% and 28.7%, respectively.

As shown in Table 3, when plans generated by IH-TPS are recalculated in RayStation, the plan quality for the abdomen and brain cases were slightly degraded. It may be attributed to the difference in dose engine, i.e., the pencil beam algorithm in IH-TPS and the MC method in RayStation. This also leads to a reduced capability in decreasing the number of energy layers. For example, in the abdomen case, the number of energy layers can be reduced from 50 to 25 in IH-TPS, but only to 30 after recalculating in RayStation.

ELO generally resulted in the reduction of beam delivery times across multiple treatment anatomies. Among the cases investigated, we note that the treatment time reductions are the most efficacious for lung treatments. Within our clinic, lung proton treatments are performed using spirometer-gated breath-holds as a motion management strategy with each breath-hold lasting approximately 25 seconds in duration. A reduction of delivery times from 178 s to 127 s will result in the reduction in the required number of breath-holds from 7 to 5 breath holds. This will lead to better overall patient comfort, a greater accessibility and tolerability of breath-hold treatments to a wider patient selection and a much faster overall delivery time than the 51 s of beam delivery times reported.

As shown in Table 4, the number of energy layers was reduced by 40% in the abdomen case, but the delivery time decreased by less than 20%. This limited reduction in the total delivery time by only reducing number of energy layers shows that the other time components (i.e., dose delivery time and spot scanning time) need to be accounted for as well. In the future work, we will integrate a comprehensive modeling of total delivery time [8,9] into IH-TPS to accurately model and optimize the total delivery time.

Table 4. Delivery time.

The unit is seconds.

Cases	Methods	Energy layers	Delivery time
Abdomen	CONV	50	110
	ELO30	30	90
Brain	CONV	78	105
	ELO40	40	54
Lung	CONV	100	178
	ELO70	70	127

As shown in Figure S2–S4, a noticeable change in dose distribution outside the target can be observed when the number of energy layers is reduced. These differences are mainly reflected in the ELO plan, where the dose from the beam with the shallowest beam entrance depth is significantly higher than that in the CONV plan, while the dose from beams with larger beam iso-centers is lower compared to the CONV plan. These results can be attributed to the fact that the reduced energy layers are predominantly from beams with larger beam iso-centers. Consequently, to maintain the prescribed dose distribution within the target, a substantial increase in weight is required for the shallowest beam. One advantage of this outcome is that the increased dose outside the target in the ELO plan is mainly concentrated in regions without OARs, as illustrated in Figures S2(e) and S3(d)–S3(f). Furthermore, for beams passing through OARs, the reduction in energy layers does not result in a significant increase in dose to the OARs.

The proposed greedy method for ELO may be applicable to other inverse optimization problems in RT, such as the MMU problem [38,39] and beam angle optimization [40,41]. For example, when the high-dose-rate delivery is desirable (e.g., for the purpose of fast IMPT or FLASH [42,43]), it was shown in [38] that a modified orthogonal matching pursuit method can mitigate the failure of conventional optimization algorithms such as ADMM in handling the large-MMU problem and improve the plan quality.

5. Conclusion

An IH-TPS integrated with a novel ELO algorithm has been experimentally validated to enable rapid IMPT while significantly reducing the number of energy layers required, thereby improving delivery efficiency without compromising clinical feasibility.