New energy layer reduction method for rapid intensity-modulated proton therapy (IMPT) using a ridge filter

Abstract

Background:

Efficient delivery of intensity-modulated proton therapy (IMPT) plays a pivotal role in improving the effectiveness of proton therapy. Accelerating delivery not only increases patient throughput but also improves the ability to manage dynamic target motion and reduces uncertainties, thereby enhancing treatment accuracy. Additionally, rapid delivery enables the effective use of breath-hold techniques, allowing for smaller target margins and providing better protection for healthy tissues. Since energy layer switching time constitute the majority of delivery time, reducing the number of energy layers can be highly beneficial for improving delivery efficiency.

Purpose:

This work proposes a novel energy layer reduction method for rapid IMPT using a ridge filter.

Methods:

The width of the Bragg peak (BP) is extended using a ridge filter, referred to as the extended Bragg peak (EBP). The machine file of the generated EBP is subsequently incorporated in our in-house treatment planning system for inverse planning. The extended peak width reduces the number of energy layers required for target coverage. However, relying solely on EBP can compromise the accuracy of energy modulation. To address this limitation, pristine BP is utilized to fill small gaps left by EBP and ensure rapid dose fall-off at the distal and proximal edges of the target. To optimize the selection of BP and EBP energies, the problem is formulated as a mathematical model incorporating a group sparsity constraint. The model is then solved by a variant of block orthogonal matching pursuit algorithm, which selects BP and EBP energies in a greedy manner.

Results:

The combined use of BP and EBP, referred to as the BP-EBP, considerably reduced the number of energy layers required compared to conventional IMPT. For example, in an abdomen case, the BP-EBP method decreased the energy layers from 79 to 28 (11 for BP and 17 for EBP) while maintaining comparable plan quality. Similarly, in a lung case, it reduced the energy layers from 70 to 30 (9 for BP and 21 for EBP), achieving slightly better plan quality than the conventional approach. Moreover, the BP-EBP reduced delivery time by more than 50% in both cases compared to conventional IMPT. Furthermore, the feasibility of BP-EBP was experimentally demonstrated on the Proteus^®ONE proton therapy system.

Conclusions:

A novel energy reduction method is proposed and experimentally demonstrated that leverages BP and EBP generated by a ridge filter that significantly decreases the number of energy levels required in IMPT, thereby enhancing delivery efficiency.

1. Introduction

Intensity-modulated proton therapy (IMPT) with pencil beam scanning (PBS) technique provides highly precise dose conformity to tumor targets while reducing radiation exposure to surrounding healthy tissues compared to photon therapy or passive scattering proton therapy [1]. Shortening the delivery time in IMPT offers multiple benefits, including improved patient comfort and reduced motion-related uncertainties [2,3]. Additionally, it enhances patient throughput, reduces treatment costs, and promotes more efficient utilization of limited proton therapy resources, thereby expanding access to this advanced treatment modality [4,5].

The treatment time for IMPT can be divided into three components: spot delivery time, spot switching time , and energy layer switching time (ELST). Among these, ELST is particularly time-consuming in the widely used cyclotron-based and synchrotron-based systems, often accounting for more than half of the total delivery time (all beams in a fraction, without considering gantry rotation time) [6,7]. However, using pristine Bragg peaks (BP) with narrow peak widths to ensure adequate target coverage requires multiple energy layers, which inevitably increases the delivery time. This underscores the importance of strategies to reduce the number of energy layers, thereby enhancing delivery efficiency.

A promising approach to reduce the number of energy layers while maintaining target coverage is to extend the BP (EBP) width using a ridge filter (RF). For instance, [8] proposed a ripple filter to expand the width of the pristine BP, successfully decreasing the required number of energy layers. However, this approach increases the distal fall-off width, which can compromise the protection of organs at risk (OAR). To address this limitation, [9] and [10] introduced patient-specific multi-layer energy filters designed to minimize the required energy layers. Building on this, [11] and [12] proposed more universal ridge filters that reduce ELST and overcome the constraints of patient-specific customization. Furthermore, [13] presented a universal and dynamic RF capable of modulating the characteristics of the BP in real-time, improving dose shaping while further reducing energy layers. However, these ridge filters are rarely implemented for real cases in treatment planning system (TPS), leaving uncertainty about the actual effectiveness in practical cases.

This study aims to develop an in-house treatment planning system (IH-TPS) incorporating a RF for fast IMPT. To achieve this, we developed an RF that extends the width of the original BP by 3.5 cm, The method combines the use of pristine BP and EBP, where EBP reduces the number of required energy layers, and BP addresses the potential distal fall-off to protect OAR. To efficiently select energy layers between BP and EBP, we proposed a novel optimization method using a greedy approach. Experimental results confirmed the deliverability, accuracy, and efficiency of this method.

2. Methods

2.1. The RF and EBP

The key principle for generating the EBP lies in creating a range of rays with varying penetration depths. As the rays pass through the RF, they lose energy determined by the thickness of the RF material they go through. Rays traversing the thicker regions of the RF experience greater energy loss, resulting in stopping at shallower depths within the target. Conversely, rays passing through thinner regions retain more energy, allowing them to penetrate deeper into the target. This controlled energy modulation creates an extension of peak width, enabling target dose coverage with fewer energy layers. Based on the RF design proposed in [13], we utilize an RF with the shape and configuration depicted in Figure 1. The size of the RF is 12.9×12.9 cm, consisting of 43 pins as shown in Figure 1a. The RF has a total thickness of 3.8cm, with its design incorporating two sets of pins. The thickness of the support of the pins is 1cm and the distance between consecutive pins (RF unit) is 0.3 cm (Figure 1b). The RF was 3D printed using stereolithography (SLA) technology on a Formlabs Form 3+ 3D-printer, with resin as the printing material (density: 1.09 g/cm³). To increase the printing success rate and improve the integrity of the edges, a 2 mm-thick wall was added to both sides of the RF, as shown in Figure 1c.

Figure 1. Configuration of RF.

(a) The top view of RF. (b) The geometric parameters of the RF. (c) The side view of the RF.

To evaluate the RF, we conducted experiments on an IBA Proteus^®ONE proton system at the University of Kansas Medical Center. The experiments were divided into two parts: delivering a single spot and delivering a 10.5×10.5 cm beam. For single spot experiment, integrated depth doses (IDD) were measured using a 12 cm electrode diameter ionization chamber (IBA Stingray) in a water tank (Figure 2a), and spot sizes were measured using a charge coupled device (CCD) camera and films (Figure 2b and 2c). For beam delivery, CCD camera was used to measure the lateral beam profiles (Figure 2b), and a 3 cm size PPC05 ionization chamber was used to measure the absolute dose (Figure 2d). For all experiments, the machine isocenter was positioned at the surface of the detection device, with the RF placed 6 cm upstream and its center aligned with the laser. This 6 cm spacing ensures that the 3D dose distribution across all energy layers passing through the RF maintains uniformity within the peak region.

Figure 2. Experimental setup.

(a) IDD experiment using water tank. (b) Spot size and lateral beam profile experiments using CCD camera. (c) Spot size experiment using films. (d) Absolute dose experiment using PPC05.

As shown in Figure 3a, the RF successfully produced a 3.5 cm EBP (defined as the distance between the proximal and distal 80% dose levels) across different proton energies (100, 150, and 200 MeV). The results indicate that after passing through the RF, the single spot initially exhibits a non-Gaussian dose distribution near the entrance of the material (Figure 3b). However, as the beam propagates further, the dose distribution gradually broadens and conforms to a two-dimensional Gaussian profile at greater depths (Figures 3c). As shown in Figures 3d–l, uniform lateral beam profiles were achieved at the peak depth for different energy layers.

Figure 3. Experimental results.

(a) Relative IDD. (b) Dose measured by film. (c) Spot size at a depth of 12 cm. (d) Lateral beam profile of 80 MeV at a depth of 3 cm. (e-f) X-horizontal and Y-horizontal profile of (d). (g) Lateral beam profile of 150 MeV at a depth of 12 cm. (h-i) X-horizontal and Y-horizontal profile of (g). (j) Lateral beam profile of 220 MeV at a depth of 28 cm. (k-l) X-horizontal and Y-horizontal profile of (j).

The process for calculating the absolute dose IDD for an energy layer passing through the RF involved the following steps: (1) Relative dose IDD was measured by Stingray in water tank; (2) Simulate the 3D dose distribution of a 10.5×10.5 cm beam after passing through the RF, using the relative dose IDD and sigma in water. (3) Compare the simulated 3D dose distribution with the absolute dose values measured by the PPC05 ion chamber in Figure 2(d). (4) Finally, convert the relative dose IDD into the absolute dose IDD by calculating the dose scaling factor.

2.2. IH-TPS with RF

2.2.1. Dose engine

The dose engine employes the pencil beam dose calculation algorithm [14]. This dose engine is based on beam parameters including IDD, range, peak position, and spot sizes in water at various depths. The dose calculation is performed under two scenarios: with and without RF. For the beam parameter calculations without RF, the Monte Carlo dose engine in RayStation is employed to simulate the 3D dose distributions for different energy layers in a 40×40×40 cm³ water phantom. The energy used ranges from 70 to 225 MeV with 2.5 MeV intervals. The beam parameters of RF were collected from experiments shown in section 2.1 and described as the follows. Spot sizes in water were calculated using film measurements, as illustrated in Figure 2c. The EBP has two peak positions, located at 80% of the peak value during the ascending and descending phases. As the depth of 70 MeV energy is too shallow to generate the whole 3.5cm EBP, we excluded it and used the energies ranging from 80 to 220 MeV with 10 MeV intervals.

2.2.2. Mathematical Model

With the dose engine in 2.2.1, we can calculate the dose influence matrices for the cases with and without RF. Then, we are in position to build up the mathematical model as the follows

$$\begin{array}{l}\underset{x\in {\mathbb{R}}^{n_1},x_{RF}\in {\mathbb{R}}^{n_2}}{\mathrm{m}\mathrm{i}\mathrm{n}}f(d,\mathrm{\Omega })\\ \mathit{s.t.}\left\{\begin{array}{l}d=Dx+D_{RF}{x}_{RF}\\ \Vert x{\Vert }_{GS}+{‖x_{RF}‖}_{GS}\le N_E\\ x\left(j\right)\in \left\{0\right\}\cup \left[G_{\text{min}},+\mathrm{\infty }\right),j\le n_1\\ x_{RF}\left(j\right)\in \left\{0\right\}\cup \left[G_{\text{min}},+\mathrm{\infty }\right),j\le n_2\end{array}\right.\end{array}$$ (1)

where $D_{RF}$ and $D$ are the dose influence matrices with and without RF, respectively. $x$ and $x_{RF}$ are the corresponding beam intensity. Each entry $x(j),x_{RE}(j)$ of variable $x$, $x_{RF}$ satisfies the minimum-monitor unit constraint (MMU) [15–17] with MMU threshold $G_{\text{min}}$. $f(d_{.}\mathrm{\Omega })$ is dose-volume histogram (DVH) [18,19] plan objective, penalizing the deviation between the prescribed dose and optimized dose $d$. Using the quadratic objective function, the $f(d_{.}\mathrm{\Omega })$ takes the following form

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

The first term is the $L_2$ type objective 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 region-of-interest (ROI). 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 [18,19].

To enhance delivery efficiency by reducing the number of energy layers, it is necessary to impose a restriction on the number of energy levels. Mathematically, this restriction can be characterized using the concept of group sparsity, which is detailed as follows:

According to the nature of pencil beam scanning, for given dose influence matrix $D$, we can decompose it group-wisely by the energy layers as the follows

$$D=\left[D_1,D_2,\cdots ,D_N\right]$$ (3)

with the group indexed by

$$\left[1,2,\cdots ,n_1\right]=\mathrm{\Lambda }=\left[{\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2,\cdots ,{\mathrm{\Lambda }}_N\right]$$ (4)

The $D_i$ is the influence matrix for $i^{\mathit{th}}$ energy, and the number of columns of $D_i$ (i.e., cardinality of ${\Lambda }_i$) is the number of spots. The beam intensity $x$ can be analogously decomposed by

$$x={\left[x_1,x_2,\cdots ,x_N\right]}^T$$ (5)

where each group $x_i$ of $x$ represents the beam intensity of the spots corresponding to a specific energy layer. The total number of utilized energy layers is equal to the number of nonzero groups in $x$ which can be mathematically interpreted by the group sparsity norm, denoted as $\Vert x{\Vert }_{BS}$. Using the group sparsity norm, the second inequality in (1) indicates that the number of energy layers used, both with and without RF, does not exceed the predefined threshold $N_E$.

2.2.3. Optimization algorithm

Suppose the number of energy layers with and without RF are $N$ and $N_{RF}$, it is observed that the problem (1) is a combinatorial problem, as it involves selecting $N_E$ energy layers from $N+N_{RF}$. The problem is similar to the energy layer optimization (ELO) problem [20–25], except for considering additional energies of the RF, i.e., the mathematical model of ELO has the following form

$$\begin{array}{l}\underset{x\in {\mathbb{R}}^{n_1}}{\mathrm{m}\mathrm{i}\mathrm{n}}f(d,\mathrm{\Omega })\\ \mathit{s.t.}\left\{\begin{array}{l}d=Dx\\ \Vert x{\Vert }_{GS}\le N_E\\ x\left(j\right)\in \left\{0\right\}\cup \left[G_{\mathrm{m}\mathrm{i}\mathrm{n}},+\mathrm{\infty }\right),j\le n_1\end{array}\right.\end{array}$$ (6)

For (6), [20] proposes a modification of block orthogonal matching pursuit (BOMP) algorithm that selects energy layers in a greedy fashion. Let us briefly introduce the algorithm. Firstly, rewriting the L2 type objective in (2) concisely into normed squared yields

$$\Vert Ax-b{\Vert }_2^2$$ (7)

where, analogous to $D$, the $A$ can be decomposed as

$$A=\left[A_1,A_2,\cdots ,A_{NE}\right]$$ (8)

The algorithm updates as the follows

Input: $D,A,b,N,N_E,G_{\text{min}},f$ ($A,b$ are constructed as that in (7) using $D$ and $f$)

Initialize: $x_1^0=x_2^0=\mathbf{0},S_1^0=S_2^0=\mathrm{Ø}$ The number of activated energy layers $C_E=0$.

For $k=0,1,\dots ,N-1$

$$C_E=C_E+1;$$

$$j^{k+1}=\underset{j}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}\left\{{‖A_j^T\left(b-A{x}_1^k\right)‖}_2\right\}$$ (9a)

$$S_1^{k+1}=S_1^k\cup \{{\mathrm{\Lambda }}_{j^{k+1}}\};$$ (9b)

$$x_1^{k+1}=\underset{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathrm{z})\in S_1^{k+1}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}{\Vert b-Az\Vert }_2^2$$ (9c)

$$S_2^{k+1}=S_2^k\cup \{{\mathrm{\Lambda }}_{j^{k+1}}\};$$ (9d)

$$(f^{k+1},x_2^{k+1})=\left\{\begin{array}{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}f(Dx,\mathrm{\Omega })\\ x\in {\mathbb{R}}^n\\ \mathit{s.t.}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)\subset S_2^{k+1},x(j)\in \{0\}\cup \left[G_{\mathrm{m}\mathrm{i}\mathrm{n}},+\mathrm{\infty }\right),j\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)\end{array}\right.$$ (9e)

$$f^{k+1}\ge f^k$$ (9f)

$$S_2^{k+1}=S_2^k$$ (9g)

$$x_2^{k+1}=x_2^k$$ (9h)

$$C_E=C_E-1$$

End

If $C_E==N_E$

$$x^{\text{out}}=x_2^{k+1}$$ (9i)

Break

End

End

Output: $x^{\mathit{out}},C_E$.

The algorithm first performs generic BOMP to select a candidate energy layer (9a–9c). Subsequently, perform inverse planning on the selected energy layers to evaluate the plan quality (function value $f^{k+1}$) (9d–9e). After that, it adaptively includes or excludes the spots index pool ${S_2}^{k+1}$ of the select energy based on the $f^{k+1}$ and that of previous iteration (9f–9h). The iteration terminates when the number of the selected energy $C_E$ achieves the $N_E$ (9i) or the maximum iteration number $N$ (the number of all energy layers) is reached.

To solve (1), we propose a variant of the modified BOMP that select energy layers for the cases with and without RF sequentially. Denote the modified BOMP in (9a)–(9i) as $BOMP\left(D,A,b,N,N_E,G_{\text{min}},f\right)$, then algorithm updates as the follows

Input: $D,D_{RF},n_1,A,A_{RF},b,N,N_{RF},N_E,G_{\text{min}},f$.

$$x,C_E=BOMP\left(D_{RF},A_{RF},b,N_{RF},N_{RF},G_{min},f\right);$$ (10a)

$$b:=b-A_{RF}x;$$ (10b)

$$x_{RF},~=BOMP\left(D,A,b,N,N_E-C_E,G_{min},f\right)$$ (10c)

Output: $x,x_{RF}$

The main idea of BOMP is to determine suitable vectors $x$ and $x_{RF}$ such that $Ax+A_{RF}{x}_{RF}$ closely approximates the target vector $b$. To achieve this, BOMP is first applied to the RF case by selecting all potential energy layers, as described in equation (10a), to obtain an initial estimate of $x_{RF}$​ and the corresponding number of selected energy layers $C_E$. This step identifies the RF energy layers that best approximate $b$; however, this approximation may still be insufficient. Therefore, the residual $b-A_{RF}{x}_{RF}$ is computed, and BOMP is applied again to the non-RF case, using the remaining $N-C_E$ energy layers to further approximate the residual. Finally, the complete beam intensity vectors $x$ and $x_{RF}$ are obtained.

Remark:

The primary rationale for performing energy selection on the RF first is that EBP provides superior target dose coverage capability. This enhanced coverage enables the achievement of the desired target dose with fewer energy layers. However, using EBP alone may not ensure uniform target dose distribution or adequate OAR sparing, as EBP tends to be less uniform than expected and exhibits a more gradual dose falloff compared to BP. Therefore, the subsequent BP step in (10c) is employed to address these limitations by compensating for potential dose inhomogeneities within the target and enhancing dose conformity with a sharper dose falloff near the target, thereby improving OAR sparing.

2.3. Materials

A phantom with cubic target (configuration of phantom shown in Figure 4a) and two clinical cases (abdomen and lung) were used to the proposed method. The dose influence matrices were calculated with dose grid resolutions of 1 mm³ for the phantom and 3³ mm³ for the clinical cases. A single beam angle was used for the phantom plan, 60°, 150°, 240°, 330° and 0°, 60°, 120° were employed for the abdomen and lung.

Figure 4. Configuration and PDD of phantom.

(a) Configuration of phantom. The yellow region represents the CTV, while the blue region represents the ROI. (b) The PDD for different plans.

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.

The spot positioning and the dose calculation were calculated by our IH-TPS. The experimental validation was conducted by importing the spot weights and locations into our proton machine. The absolute dose was measured via MatriXX ONE to calculate the gamma index using the clinical standard for the passing criterion (≥90%, with 2%/2mm and a 10% threshold). During the experimental validation, the delivery time for each plan was recorded directly from the IBA machine. However, this measurement did not account for the time required for the RF to move in and out.

3. Results

3.1. Phantom

We conducted phantom experiments to evaluate the performance of the RF. The IH-TPS dose engine was employed to select candidate energies, with 14 (the number of energy layers used in IMPT) and 3 energies chosen for BP and EBP, respectively.

Treatment plans generated using IH-TPS with RF (EBP) and without RF (BP) were compared. As reported in Table 1, the incorporation of RF maintained the plan quality while reducing the number of energy layers required. For instance, EBP with only 2 energy layers (EBP2) achieved superior results compared to BP with 4 energy layers (BP4) and delivered comparable results to BP with 8 energy layers (BP8). Although IMPT without ELO outperformed EBP2 in terms of plan quality, it required more energy layers. Notably, when BP and EBP were combined, the resulting plan quality surpassed that of IMPT. For instance, the BP2-EBP2 plan reduced the objective function value from 7.7 to 5.4, the maximum target dose from 125.4% to 122.0%, and the mean body dose from 15.7% to 15.3%, while improving the CI from 0.91 to 0.93. Importantly, BP2-EBP2 achieved these improvements with only 4 energy layers, representing a 71.4% reduction in energy layers and a 54.6% reduction in delivery time compared to IMPT. When additional energy layers were employed, as in BP4-EBP2, the plan quality remained similar to BP4-EBP2 case. These results are further supported by the DVH and dose distribution plots presented in Figure 5.

Table 1. Phantom.

The dosimetric quantities from left to right: optimization objective value (f), max dose of the target (D_max), conformal index (CI), mean dose of ROI, (D_{mean_ROI}). The unit of time is in seconds.

Plan	f	D max	CI	D mean_ROI	Time (s)
IMPT	7.68	125.4	0.91	15.7	33
BP4	568.8	205.9	0.60	19.3	17
BP8	27.3	130.6	0.90	16.2	23
EBP2	26.5	135.1	0.88	15.8	8
BP2-EBP2	5.4	122.0	0.93	15.3	15
BP4-EBP2	4.1	121.7	0.93	15.3	16

Figure 5. Dose and DVH plots for phantom:

(a) Dose plot for IMPT; (b) Dose plot for BP using 4 energy layers; (c) Dose plot for BP using 8 energy layers; (d) Dose plot for EBP using 2 energy layers; (e) Dose plot for BP-EBP using 4 energy layers (2 energy layers for BP and 2 for EBP); (f) Dose plot for BP-EBP using 6 energy layers (4 energy layers for BP and 2 for EBP); (g) DVH for CTV; (h) DVH for ROI. The dose plot window is [0%, 200%], with 80% (blue), 100% (green), 130% (yellow) isodose line, and targets (magenta) highlighted in dose plots.

Furthermore, the IMPT, BP4, EBP2, and BP2-EBP2 plans were delivered on the treatment machine, with dose measurements obtained using the IBA Zebra detector. As shown in Figure 4(b), the percentage depth dose (PDD) results are consistent with the plan quality metrics in Table 1, with the PDD of BP2-EBP2 closely matching that of IMPT. Furthermore, all plans achieved a pass rate exceeding 90%. Specifically, BP14 and BP4 achieved a pass rate of 99.07%, EBP2 achieved 90.74%, and BP2-EBP2 achieved 92.59%. The QA results are detailed in Figure 6.

Figure 6. QA results for phantom.

The measured ((a), (d), (g), (j)), IH-TPS ((b), (e), (h), (k)), and gamma index ((c), (f), (i), (l)) for different plans.

3.2. Abdomen

The previous experiments were conducted on relatively simple geometries. To further assess the effectiveness of the proposed method, we extended our evaluation to clinical cases. The first case focused on an abdominal treatment plan, where the number of candidate energy layers selected was 99 for BP and 19 for EBP. Employing EBP alone with 10 energy layers (EBP10) resulted in superior plan quality compared to BP with either 10 or 30 energy layers. Specifically, EBP10 reduced the objective function value from 3.7 (BP10) and 2.8 (BP30) to 2.4, decreased the maximum target dose from 125.4% (BP10) and 122.8% (BP30) to 117.8%, improved the conformity index (CI) from 0.81 (BP10) and 0.83 (BP30) to 0.85, and lowered the mean body dose from 6.35% (BP10) and 6.13% (BP30) to 4.68%.

When BP and EBP were combined, the BP-EBP plans with 19 (BP2-EBP17) and 28 (BP11-EBP17) energy layers achieved dosimetric quality comparable to IMPT. However, the BP-EBP approach required significantly fewer energy layers, leading to substantially shorter delivery times. For instance, BP11-EBP17 reduced the delivery time from 117 seconds to 41 seconds, representing a 65% reduction. The dose distribution and DVH plots in Figure 7 further corroborate these results, demonstrating the effectiveness of BP-EBP in enhancing treatment efficiency while maintaining plan quality.

Figure 7. Dose and DVH plots for abdomen.

(a) Dose plot for IMPT; (b) Dose plot for BP using 10 energy layers; (c) Dose plot for BP using 30 energy layers; (d) Dose plot for EBP using 10 energy layers; (e) Dose plot for BP-EBP using 19 energy layers (2 energy layers for BP and 17 for EBP); (f) Dose plot for BP-EBP using 28 energy layers (11 energy layers for BP and 17 for EBP); (g) DVH for CTV; (h) DVH for body. The dose plot window is [0%, 120%], with 80% (yellow), 100% (red) isodose line, and targets (green) highlighted in dose plots.

3.3. Lung case

Similar results were observed for the lung case. The number of energy layers selected was 73 for BP and 21 for EBP. As shown in Table 3, EBP10 and EBP20 achieved superior results compared to BP30 and BP40, demonstrating reduced plan objective values, improved target coverage, and enhanced OAR sparing. The combined BP-EBP method further demonstrated its efficiency, achieving comparable plan quality with fewer energy layers. For instance, BP9-EBP21 (30 energy layers) required less than half the energy layers to achieve similar plan quality as IMPT. This reduction also significantly decreased delivery time, from 129 seconds to 60 seconds. The dose distribution and DVH plots in Figure 8 corroborates these results.

Table 3. Lung.

The dosimetric quantities from left to right: optimization objective value (f), max dose of the target (D_max) , conformal index (CI), mean dose of esophagus (Eso). The unit of time is in seconds.

Plan	f	D max	CI	Eso	Time (s)
IMPT	2.55	117.6	0.91	28.1	129
BP20	11.3	134.2	0.72	30.4	64
BP40	3.94	120.9	0.87	28.3	96
EBP10	7.27	125.4	0.82	29.4	29
EBP20	3.16	119.4	0.87	27.8	41
BP9-EBP21	2.46	116.9	0.87	27.2	60

Figure 8. Dose and DVH plots for lung.

(a) Dose plot for IMPT; (b) Dose plot for BP using 20 energy layers; (c) Dose plot for BP using 40 energy layers; (d) Dose plot for EBP using 10 energy layers; (e) Dose plot for EBP using 20 energy layers; (f) Dose plot for BP-EBP using 30 energy layers (9 energy layers for BP and 21 for EBP); (g) DVH for CTV; (h) DVH for esophagus. The dose plot window is [0%, 130%], with 80% (yellow), 100% (orange). 100% (red) isodose line, and targets (green) highlighted in dose plots.

To verify deliverability, we conducted QA experiments on the lung case. All beams from all plans achieved a pass rate exceeding 90%. We provide the QA results of beam 1in Figure 9. The gamma index was 96.82% (95.33% and 99.41% for the other beams) for IMPT, 95.98% (100% and 96.11% for the other beams) for BP20, 97.28% (99.34% and 99.42% for the other beams) for EBP20, and 96.09% (97.95% and 98.81% for the other beams) for BP9-EBP21 which demonstrate the deliverability of the plans.

Figure 9. QA results for lung.

(a) The measured ((a), (d), (g), (j)), IH-TPS ((b), (e), (h), (k)), and gamma index ((c), (f), (i), (l)) for different plans.

3.4. Robustness

To address the uncertainties inherent in proton therapy, robust 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 IMPT plan and the BP-EBP plan 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) [25]. 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 abdominal case are shown in Figure 10(c)–(d). In the abdominal case, the RV₉₇ of the CTV for the IMPT and BP11-EBP17 plans is 19.0 vs. 18.8, while the RV₃₀ of the OAR is 9.0 vs. 6.3. However, BP-EBP is more affected by robustness. For instance, as shown in Table 4, the CI and $D_{mean\_oar1}$ for IMPT and BP11-EBP17 are 0.90 vs. 0.85 and 17.0 vs. 21.8. In the lung case, a slight reduction in robustness is observed after reducing the number of energy layers with the BP-EBP. Specifically, the RV₉₈ of the CTV is 8.1 vs. 10.3, and the RV₄₅ of the OAR is 13.7 vs. 11.8. The plan quality of the IMPT and BP-EBP plans following robust optimization is summarized in Table 4 and Supplementary Table S2.

Figure 10. Robust optimization for abdomen.

(a) Dose plot for IMPT; (b) Dose plot for BP-EBP using 28 energy layers (11 energy layers for BP and 17 for EBP); (c-d) The DVH plots of the CTV and bowel _large. The dose plot window is [0%, 120%], with 80% (yellow), 100% (red) isodose line, and targets (green) highlighted in dose plots.

Table 4. Robust optimization for abdomen.

The dosimetric quantities from left to right: optimization objective value (f), max dose of the target (D_max) , conformal index (CI), mean dose of body (BD), mean dose of bowel_large (D_{mean_oar1}).

Plan	f	D max	CI	BD	D mean_oar1
IMPT	1.85	112.9	0.90	4.49	17.0
BP11-EBP17	1.96	113.1	0.85	4.62	21.8

3.5. Comparisons

To demonstrate the effectiveness of the method we developed, we conducted a comparison with CARD [26] for computational efficiency. CARD iteratively selects several energy layers with the highest spot weights in each iteration, whereas BOMP adopts a greedy strategy to select energy layers one by one based on their contribution to improving plan quality.We conduct an experiment on abdomen case to evaluate the methods. The method proposed in this study, similar to CARD, iteratively reduces the number of energy layers and terminates the iteration when the difference in $f$ compared to IMPT is less than 5%. Both methods utilized the same planning constraints. 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 5, the BOMP plan demonstrates significantly higher efficiency compared to the CARD method. The BOMP method successfully reduces the number of energy layers to 29, whereas the CARD method requires 43 energy layers to achieve a comparable $f$. Specifically, the $f$ and CI for BOMP vs. CARD are 1.78 and 0.85 vs. 1.81 and 0.87, respectively. Furthermore, the BOMP method achieved similar oar sparing, with the mean body dose at 4.63% vs. 4.56%. Finally, the BOMP plan also exhibits improved robustness variance of the CTV, with a value of 18.8 compared to 20.0.

Figure 11. Comparisons.

(a)-(b). The dose plots of BOMP and CARD. (c). The DVH and robustness variance of CTV and bowel _large.

Table 5. Comparisons.

The dosimetric quantities from left to right: optimization objective value (f), max dose of the target (D_max) , conformal index (CI), mean dose of body (BD), mean dose of bowel_large (D_{mean_oar1}), used energy in plan (Energy).

Plan	f	D max	CI	BD	D mean_oar1	Energy
BOMP	1.78	113.0	0.85	4.63	22.5	BP12-EBP17
CARD	1.81	111.2	0.87	4.56	17.8	BP40-EBP3

4. Discussion

This work develops an energy layer reduction method that utilizes the characteristic of EBP generated by ridge filter. The proposed method considerably decreases the number of energy layers required compared to the conventional IMPT, thereby improve the delivery efficiency. Experiments on a water phantom, as well as abdomen and lung cases, demonstrated the accuracy and efficiency of this method.

As illustrated in Figure 3b, when a single spot passes through the ridge filter (RF), the dose distribution near the entrance deviates from an ideal Gaussian shape. As the depth increases, the dose gradually disperses and eventually approximates a Gaussian distribution, as shown in Figure 3c. Moreover, the dose distribution varies depending on the precise location of the spot within an individual RF unit. To simplify the dose calculation engine, we assume that each spot is positioned at the center of its respective unit (maintaining a fixed spot spacing of 0.3 cm), and the dose distribution follows a double-Gaussian model. While this assumption does not affect dose calculations within the target region, it may influence dose estimation in organs at risk (OARs) and limit the flexibility of spot placement. In future work, we plan to refine our model to account for these effects.

As shown in Figure 3a, there is a noticeable dip appears in the middle of the EBP. This dip may require supplementation with an additional BP, partially diminishing the effectiveness of the EBP. In our experiment, we find the shape of the EBP is sensitive to the curvature of RF unit . When testing the shape RF, we observed that even minor adjustments to the curvature of RF significantly impacted the flatness of EBP and, in some cases, even led to the absence of EBP. To address this problem, a simulation-based strategy could be explored to identify and optimize the shape of RF to achieve flatter EBP, and we left to the future work.

Furthermore, the broad depth coverage of the 2.5 cm extended EBP may impact performance when applied to smaller targets. To investigate this effect, we conducted an experiment on a prostate case, with results presented in the supplementary material (Figure S1 and Table S1). The results show that, for a given number of energy layers, the combination of BP and EBP yields better results compared to using BP alone. However, the improvement is not substantial, particularly when the number of energy layers is limited. A possible reason for this is that we employed only a single RF with a relatively wide EBP. The target with relatively small size may not be well benefit from the EBP. However, this limitation could be addressed by incorporating multiple RFs with varying peak widths. It will allow greater flexibility in dose shaping and better adaptation to different target sizes and, again, we left it for future work.

As shown in Tables 2 and 4, when the number of energy layers is kept constant, the BP-EBP plan is more sensitive to robustness compared to the IMPT plan. Table 5 further demonstrates that the CARD plan, which utilizes a larger number of BPs, achieves a significant reduction in the mean dose to the large_bowel. This may be attributed to the current limitations in the number and width of available EBP options. Specifically, the relatively large width and limited number of EBPs can result in some dose spill into the large bowel, thereby increasing its mean dose in the BP-EBP plan. In future work, we plan to incorporate EBPs with a wider range of widths and finer granularity to achieve improved dosimetric outcomes.

Table 2. Abdomen.

The dosimetric quantities from left to right: optimization objective value (f), max dose of the target (D_max) , conformal index (CI), mean dose of body (BD). The unit of time is in seconds.

Plan	f	D max	CI	BD	Time (s)
IMPT	1.8	113.9	0.93	4.34	117
BP10	3.7	125.4	0.81	6.35	19
BP30	2.8	122.8	0.83	6.13	52
EBP10	2.4	117.8	0.85	4.68	18
BP2-EBP17	2.0	115.3	0.86	4.58	18
BP11-EBP17	1.9	113.6	0.88	4.71	41

Finally, the delivery time for each plan was recorded directly from the IBA machine. However, this measurement did not account for the time required for the RF to move in and out. For the IBA machine used in our study, we envision that the RF, like the range shifter, can be mechanically controlled remotely by the system. In future designs, dose delivery for a single beam angle will be divided into two stages: after completing the without RF portion, the RF will be automatically inserted to deliver the remaining dose. Based on the insertion time of the range shifter in the current IBA machine, this process is expected to take approximately 3 seconds. Given that this switching time is negligible compared to the reduction in delivery time by half, its impact can be considered minimal.

5. Conclusions

This study introduces an energy layer reduction method in a IH-TPS that leverages the unique characteristics of energy deposition patterns (EBP) generated by ridge filters. The proposed approach significantly reduces the number of energy layers required compared to conventional IMPT, thereby enhancing delivery efficiency. Validation experiments conducted on a water phantom, as well as abdomen and lung cases, demonstrated the method's accuracy and efficiency.