Robust Optimization for Spot-Scanning Proton Therapy based on Dose-Linear-Energy-Transfer Volume Constraints

Abstract

Purpose:

Historically, spot-scanning proton therapy (SSPT) treatment planning uses dose-volume constraints and linear-energy-transfer (LET) volume constraints separately to balance tumor control and organs-at-risk (OARs) protection. We propose a novel dose–LET-volume constraint (DLVC)-based robust optimization (DLVCRO) method for SSPT in treating prostate cancer to obtain a desirable joint dose and LET distribution to minimize adverse events.

Methods and Materials:

DLVCRO treats DLVC as soft constraints that control the shapes of the dose–LET volume histogram (DLVH) curves. It minimizes the overlap of high LET and high dose in OARs and redistributes high LET from OARs to targets in a user-defined way. Ten patients with prostate cancer were included in this retrospective study. Rectum and bladder were considered as OARs. DLVCRO was compared with the conventional robust optimization (RO) method. Plan robustness was quantified using the worst-case analysis method. Besides the dose-volume histogram indices, the analogous LET-volume histogram, extrabiological dose (the product of per voxel dose and LET) volume histogram (xBDVH) indices characterizing the joint dose/LET distributions and DLVH indices were also used. The Wilcoxon signed-rank test was performed to measure statistical significance.

Results:

In the nominal scenario, DLVCRO significantly improved joint distribution of dose and LET to protect OARs compared with RO. The physical dose distributions in targets and OARs are comparable. In the worst-case scenario, DLVCRO markedly enhanced OAR protection (more robust) while maintaining almost the same plan robustness in target dose coverage and homogeneity.

Conclusions:

DLVCRO upgrades 2D DVH-based to 3D DLVH-based treatment planning to adjust dose/LET distributions simultaneously and robustly. DLVCRO is potentially a powerful tool to improve patient outcomes in SSPT.

Introduction

Spot-scanning proton therapy (SSPT) uses pencil-beam beamlets to precisely irradiate tumors in 3 dimensions^1-3 by its physical property of sharp dose fall-off distal to the target and its capability of effectively modulating beamlet intensities.^4,5 Despite its dosimetric advantages, SSPT faces several challenges, including plan robustness^6-11 and variable relative biological effectiveness (RBE).^12,13

Unlike photons, protons are characterized with high linear energy transfer (LET)^14,15 near the distal end of the Bragg Peak. In clinical practice, a constant RBE of 1.1 is widely adopted as a coarse approximation to describe the RBE effect of protons.¹⁶ To study the relation between RBE and high LET, numerous in vitro^16,17 and in vivo^14,15 studies have been conducted, and several phenomenologic^18,19 and mechanism-related^20,21 RBE models have been proposed. Most of these phenomenologic models were developed based on the linear-quadratic models, assuming the LET effect as a function of cell radiation sensitivity change (α and β values) because of protons. Consequently, in addition to the common dose-volume constraints (DVCs) guided treatment planning strategies,²² various LET-guided or RBE-guided treatment planning strategies have been developed to improve clinical outcomes.^6,7,13,17

Although the proton therapy community has well recognized the variation in RBE induced by high LET,¹⁶ the precise combined biological effects of dose and LET remain unclear.^18,23 Recent studies have begun to elucidate these combined effects on patient outcomes.^24-27 For example, the interplay between dose and LET have been crucial in initiating adverse events (AEs), as observed in studies concerning mandible osteoradionecrosis in head and neck cancer²⁶ and rib fractures in patients with breast cancer receiving SSPT.^27,28

Previously reported methods that optimize dose and LET distributions separately through DVCs and LET-volume constraints (LETVCs) cannot effectively address the combined biological effects between dose and LET in SSPT. To improve clinical outcomes and minimize AEs, the development of a new treatment planning method that considers the combined effects of dose and LET in SSPT is crucial.

The newly introduced dose–LET volume histogram (DLVH) serves as an effective tool for integrating physical dose and LET to assess their combined biological effects in SSPT.^25-27,29 This method improves on traditional dose-volume histograms (DVHs) by including LET as an independent variable alongside the physical dose. DVHs typically represent 3-dimensional (3D) dose distributions in a 2-dimensional (2D) format, simplifying spatial data at the cost of losing spatial information of the dose distribution.³⁰ The DLVH, along with dose–LET volume constraints (DLVCs), were proven to be predictive of clinical outcomes, such as rectal bleeding in patients with prostate cancer treated with SSPT.²⁵

SSPT is highly sensitive to range and setup uncertainties.^10,31,32 Neglecting these uncertainties can lead to significant degradation of dose distributions for SSPT treatment plans.^11,33 Additionally, LET distributions will also be altered in these uncertainty scenarios, which may result in unexpectedly high RBE doses to OARs.^7,17 Therefore, it is important to incorporate these uncertainties into the optimization of radiobiological doses, including both physical dose and LET distributions. Investigators have studied the impact of uncertainties on dose and LET distributions and developed dose-related^6,34 or LET-related robust optimization (RO) to improve the robustness of dose and LET distributions in the presence of uncertainties.^7,8

In this study, we introduced a novel RO method: the dose–LET volume constraint-based RO (DLVCRO) method. This method significantly advanced the traditional DVC and LETVC-based RO methods by employing constraints that account for the combined effects of both dose and LET, rather than addressing them separately. Building on prior studies that the derived DLVCs for rectal bleeding in patients with prostate cancer treated with SSPT,²⁵ we evaluated the feasibility of applying this method to patients with prostate cancer treated with SSPT. The proposed method successfully minimized the overlap of high LET and high dose in the rectum and bladder, redistributing high LET from these organs to the targets with minimal sacrifice to target physical dose and plan robustness.

Methods and Materials

Dose–LET volume histogram and extra biological dose

Dose–LET volume histogram (DLVH) has been recently introduced as a novel tool to facilitate analysis of the LET-enhancing effect for AE initialization in SSPT.^25-27,29 The DLVH is constructed as a 3D cumulative volume histogram that simultaneously displays the distributions of both dose and LET (dose-averaged) within a specified structure. Figure 1a illustrates a prototype of the 3D DLVH structure. In this histogram, the axes represent the physical dose (Gy) and the LET (keV/μm), whereas the third axis corresponds to the normalized volume of the structure. Of note, Gy (RBE = 1.1) is also used as the unit of relative biological effective dose, which numerically equals to 1.1 times the physical dose. The DLVH index quantifies the volume V_D,LET(d, l), defined as:

$$V_{D,LET}(d,l)=V(D\ge d,LET\ge l)$$

Fig. 1.

Dose-linear-energy-transfer (LET) volume histogram (DLVH). An example of DLVH in rectum. Three-dimensional DLVH is sketched in (a). DLv% lines (solid lines with different colors) are the isovolume contour of the 3D DLVH and represent the percentage volume of a structure that has a dose of at least d Gy and an LET of at least l keV/mm. The intersections of the 3D DLVH surface plot in the dose-volume plane is the dose-volume histogram (DVH), as shown in (d). The LET-volume histogram (LETVH) is shown in (b). The projected 2-dimensional DLVH with the corresponding DLv% lines are shown in (c). The intersections of DL lines with the dose and LET axes are Dv% in DVH and LETv% in LETVH, respectively. Each voxel of the structure was mapped to the 2D dose-LET plane indicated as gray dots in (c). Abbreviations: 3D = 3-dimensional; DL = dose-linear; DLv = dose-linear volume; Dv = dose volume; LETv = LET volume.

Here, V(D≥d, LET≥l) denotes the cumulative normalized volume of the structure receiving at least physical dose of d (Gy) and LET of l (keV/μm).

Analogous to the well-known DVH,³⁰ the DLVH extends the concept by incorporating LET. It distinctly advances the traditional DVH and LET-volume histogram (LETVH) by assessing the combined effects of dose and LET distributions and considering their potential interplay.

For effective visualization, a 2D isovolume contour can be derived from the 3D DLVH by projecting its isovolume curves onto the dose-LET plane as shown in Figure 1 (c). In this representation, the x-axis denotes the physical dose, and the y-axis represents the LET. For the specific structure, the contour lines on this plane illustrate varying percentage volumes. Figure 1 (c) displays contour lines for 5%, 20%, 50%, and 80% isovolumes, providing a clear and quantifiable depiction of how dose and LET interplay within the specific structure.

The product of dose and LET, known as extra biological dose (xBD), has been frequently used as a surrogate for describing the combined effects of high LET and dose in proton therapy plan optimization and evaluation.^13,35 In addition, xBD has been identified as a good dose and LET descriptive feature for AE initialization in SSPT.^26,27 Furthermore, xBD-volume constraints have been derived to predict mandible osteoradionecrosis for SSPT in the treatment of head and neck cancer.^26,27 Besides the DVH and the LETVH, the xBD-volume histogram (xBDVH) was also adopted in this work to evaluate the biological effect of the generated SSPT plans.

Dose–LET-volume constraints

Current DVH or LETVH-based SSPT treatment planning methods to mitigate the impact of high LET^{6,7,13,17,34,36} are essentially based on a 2D concept. These methods adjust the dose or LET distribution separately, not simultaneously, using a 2-way relationship either between dose d and volume V or between LET l and volume V, to achieve a desirable dose/LET distribution (Fig. 2a). We upgraded SSPT treatment planning from these 2D approaches to a more comprehensive 3D framework. This approach incorporates dose, LET, and volume through DLVH-based RO, enabling simultaneous adjustment of both dose and LET distributions (Fig. 2b).

Fig. 2.

DLVH-based optimization upgrades SSPT treatment planning from 2D to 3D. (a) DVCs in conventional DVH-based optimization, where only voxels with a dose between D₁ and D₂ are penalized as a soft constraint. (b) DLVCs in DLVH-based optimization. The yellow surface is the desired 3D DLVH surface, and the green surface is the current 3D DLVH surface. The black dash line indicates DLVC, ie, $V_{desired}\left(f_1(d,l)\right)\le V_1$ used in the optimization, where $f_1(d,l)$ could be derived from the DLVH-based patient outcomes study. The red dash $f_2(d,l)$ is the intersection of the current DLVH surface (green surface) and Volume = V₁ plane (purple plane). To implement the DLVC, penalties are applied to voxels with dose and LET values between the projected red and black solid lines in the dose-LET plane for each iteration. (c) Diagram of the DLVC-based optimization concept as shown in (b) in the 2D dose-LET plane. The voxels in the gray area will be penalized. Abbreviations: DLVC = dose-linear-energy-transfer volume constraint; 2D = 2-dimensional; 3D = 3-dimensional; DLVH = dose-linear-energy-transfer volume histogram; DVC = dose-volume constraint; DVH = dose-volume histogram; LET = linear-energy-transfer; SSPT = spot-scanning proton therapy.

In our method, dose–LET-volume constraints (DLVCs) were implemented as “soft constraints” within the objective function. This modifies the shape of the DLVH surface to satisfy DLVCs by penalizing only those voxels violating DLVCs. This approach reduces the number of potential seed spots and subsequently leading to a lower likelihood of the corresponding AEs. In Figure 2 (b), the yellow surface is the desired 3D DLVH surface, whereas the green surface is the current 3D DLVH surface. The red pentagram indicates the DLVC, defined as $V_{desired}\left(f_1(d,l)\right)<V_1$ in the optimization. Here, $f_1(d,l)$ is a curve related to the specified DLVC, derived from the DLVH-based patient outcomes study, represented by the black dash line. The DLVC limiting the overlap of high dose and high LET in OARs, as shown in Figure 2 (b), is specified as: $V_{desired}\left(f_1(d,l)\right)<V_1$. This means the relative volume receiving dose ≥d and LET ≥l, correlated by $f_1(d,l)$ (black dash line) should not exceed V₁.

Following the same concept of implementing DVCs based on DVHs to optimize physical dose distribution (Fig. 2b), the DLVCs were integrated into the objective function.²² During each optimization iteration, we identify a curve $f_2(d,l)$ (red dash line), which is the intersection line of the current 3D DLVH surface and the parallel plane defined by V = V₁(purple plane), so that Vcurrent $V_{current}\left(f_2(d,l)\right)=V_1$ in the current 3D DLVH surface. These 2 dash curves are then projected (vertical dash lines) onto the 2D dose-LET plane, forming 2 solid curves. Voxels with dose and LET values enclosed by these 2 curves and the axes of dose and LET comprise the voxel set VDLVC in the underlined term in Eq. (1). These voxels were penalized for this structure in the optimization. Additional DLVCs may be introduced as needed to further control the DLVH surface to achieve the desired dose and LET distributions simultaneously. Consequently, our model is able to redistribute dose and/or LET and minimize the possible incidence of AEs in OARs.

In practice, the curve $f_1(d,l)$ for the DLVC $V_{desired}\left(f_1(d,l)\right)<V_1$ was depicted through line segments defined by their endpoints. In our optimization program, the DLVCs take the form of V_desired[(d₁, l₁) to (d₂, l₂) to (d₃, l₃) to …] < V₁. Each pair (d_i, l_i) specifies a point at dose d_i and LET l_i. The critical volume V₁ of DLVCs is typically set as 0%, 1%, or 2% normalized by the volume of organs at risk (OARs). The DLVCs applied in this study are patient-specific and are detailed in the Table E1.

DLVCRO

The proposed DLVCRO is an enhancement of the voxel-wise worst-case RO method.^6,8,9,37,38 Twenty-one uncertainty scenarios were considered in the RO, including the nominal scenario, 2 scenarios because of range uncertainties, 6 scenarios because of setup uncertainties and their combination. Range uncertainties were modeled by shifting the relative-to-water stopping power ratio conversion curve by ±3%, simulating maximum and minimum range uncertainties.¹¹ Setup uncertainties involve rigid shifts of the isocenter by 5 mm in the anteroposterior, superoinferior, and left-right Cartesian directions.

An additional “biological surrogate” (BS) term^6,13 (underlined term in Eq. [1]) was introduced as indirect RBE optimization. The BS term, ${BS}_i^m=D_i^m+{\sum }_j{cL}_{i,j}^m{IM}_{i,j}^m{\omega }_j^2$, where the LET influence matrix^6,7 $L_{ij}^m$ describes the LET contribution of beamlet j of unit intensity to voxel i in uncertainty scenario m. $D_i^m$ is the dose at ith voxel for mth uncertainty scenario^39,40 and can be calculated as: $D_i^m={\sum }_j{IM}_{i,j}^m{\omega }_j^2$, where ${\omega }_j^2$ represents nonnegative intensity weight of beamlet j. The dose influence matrix ${IM}_{i,j}^m$ describes the dose contribution of beamlet j of unit intensity to voxel i in uncertainty scenario m.^37,41 c is a scaling constant and is set to be 0.04 μm/keV,^6,13 and the second term can be regarded as xBD compared to the physical dose $D_i^m$.

A standard quadratic objective function of our proposed DLVCRO is designed as follows:

$$\begin{gathered} F_{obj}^{LET}=\sum _{i\in CTV}\left(p_{CTV}^{max}{(D_i^{max}+D_{0,CTV}^{max})}_{+}^2+p_{CTV}^{min}{(D_{0,CTV}^{min}+D_i^{min})}_{+}^2\right) \\ +\sum _{i\in VDLVC}{p}_{OAR}^{BS}{BS}_i^{max}2 \end{gathered}$$ (1)

The first 2 terms in Eq. (1) ensure that minimum and maximum physical dose constraints for the clinical target volume (CTV) are met. The Heavyside function ($(x-x_0)$)₊ returns ($(x-x_0)$) if $x>x_0$, otherwise zero. Terms with subscript “0” ($D_{0,CTV}^{max}$ and $D_{0,CTV}^{min}$) are the prescribed dose. $p_{CTV}^{min}$, $p_{CTV}^{max}$, and $p_{0AR}^{BS}$ are penalty weights for each corresponding term. $D_i^{min}$, $D_i^{max}$, and ${BS}_i^{max}$ stand for the voxel-wise minimum or maximum value of the dose or BS among all the values corresponding to different uncertainty scenarios (voxel-wise worst-case dose/BS). An ad hoc parameter c = 0.04 μm/keV is used in the xBD term. The choice of c = 0.04 μm/keV is to have a RBE of 1.1 in the middle of spread-out Bragg peak.¹³ Unlike the previous xBD-guided optimization methods, the choice of c would not affect our results because the absolute values of RBE would not matter in our optimization.

The influence matrix ${IM}_{i,j}^m$, was calculated using an in-house semianalytical dose engine that employs a modified ray-casting pencil-beam algorithm with 3 lateral Gaussian components.⁴² The LET influence matrix $L_{i,j}^m$ was calculated using an in-house developed LET calculation code.⁴³ This study used a dose-averaged LET, calculated with restricted stopping powers. The methodology for calculating LET parallels the pencil-beam algorithm used in dose calculations. A 3D LET calculation kernel was created to replace the standard dose calculation kernel, using the precomputed LET data from Monte Carlo simulations.^43-45

Optimization was performed using the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm, which was enhanced through parallel computation via a message passing interface.⁴⁶ In the objective function, the physical DVCs for the first 2 terms of Eq. (1) were treated as “soft constraints.”^22,38 Because of the nonconvex nature of the objective function, a “trial-and-error” method was employed to adjust the initial optimization parameters until a clinically acceptable plan was achieved. Multifield optimization (MFO) was employed in both the RO and DLVCRO, enforcing a uniform target dose per plan basis.

Patient selection and machine configuration

This study was approved by Mayo Clinic Arizona institutional review board (IRB#: 24-011067). We generated 2 SSPT plans using both the RO and DLVCRO methods for each of 10 prostate cancer patients. The patients are treated at our institution from August 2016 to February 2024 with the treatment featuring antiparallel left and right fields. Patient selection criteria included the proximity of bladder and rectum as OARs to the CTV, where the impact of uncertainties, high LET and high dose on nearby OARs was most critical. And the patients with postprostatectomy or lymph node involvement were excluded in this study. For all cases, the bladder and rectum were selected for evaluation. Table E2 provides details on patient characteristics, prescribed doses, and fractionation schedules.

All plans began optimization with identical initial conditions regarding beamlet intensities, number, energy, and positions. Subsequently, only beamlet intensities were optimized using the 2 methods under study. The details of the treatment fields are provided in Table E3 and Figure E1.

Following optimization, a postprocessing procedure was applied to both the RO and DLVCRO plans to ensure deliverability by adhering to the minimum monitor unit (MU) limits. Specifically, any spot planned to deliver MUs below the minimum threshold (0.003 MU for our proton therapy machine) was adjusted upward or downward, depending on if it was above or below half the minimum MU limit. The dose grid voxel size was maintained at 2.5 × 2.5 × 2.5 mm throughout the study.

Plan evaluation and plan robustness

We evaluated CTV D_98%—the minimum dose normalized by the prescription dose to cover 98% of the target—and D_2% – D_98% for evaluating target dose coverage and homogeneity, respectively. To evaluate the OAR sparing, V40Gy (RBE = 1.1) and V65Gy (RBE = 1.1) of bladder and V70Gy (RBE = 1.1) of rectum were measured. To ensure a fair comparation, all the plans were converted to the same prescribed dose and fractions. CTV D_98% in the nominal scenario of each plan was normalized to the prescription dose, regardless of the optimization method used. LETVHs were used to compare LET distributions across different plans. Indices similar to those in DVH were employed to evaluate these distributions. We calculated CTV LET_98% and LET_mean to determine target LET coverage and average LET, respectively, where LET_98% is defined as the minimum LET required to cover 98% of the target. The maximum LET in OARs, specifically LET_max for the bladder and rectum, were also evaluated. xBD_mean was calculated for CTV to evaluate the biological effects. The xBD_max of bladder and rectum was calculated to indicating the biological effects. The DLVH index of bladder, V(50 Gy, 6 keV/μm), was calculated to evaluate the DLVH index improvement of bladder from RO to DLVCRO. For rectum, the DLVH index, V(67.8 Gy, 2.86 keV/μm), was calculated, which has been demonstrated as an important index for rectal bleeding from the reported literature.²⁵

In assessing plan robustness, we considered setup uncertainties of 5 mm and range uncertainties of 3%. A total of 21 uncertainty scenarios were analyzed, including nominal, minimum (−3%), and maximum (+3%) proton ranges, along with patient positions shifted in anteroposterior, superoinferior, and left-right directions (7 scenarios per proton range). We employed the worst-case scenario analysis method^8,47—recognized for its speed and efficiency—to evaluate plan robustness across these scenarios. The minimum CTV D_98% was used to assess the poorest target dose coverage, whereas the maximum CTV D_2% – D_98% evaluated the least homogeneity in target dose. Furthermore, the V40Gy (RBE = 1.1) and V65Gy (RBE = 1.1) of bladder and V75Gy (RBE = 1.1) of rectum were measured to determine OAR protection in the worst-case scenario. Corresponding LETVH, xBDVH, and DLVH indices were calculated to assess the robustness of the LET distribution, xBD distribution, and the dose-LET joint distribution in the face of these uncertainties.

Statistical analysis

The Wilcoxon signed-rank test was used to compare all evaluation metrics, with calculations performed using SciPy Python library. The 2-tailed test was conducted, and P values <.05 were deemed statistically significant. In the box-and-whisker plot, data points falling outside 1.5 times the interquartile range above the upper quartile or below the lower quartile were identified as outliers.

Results

Physical dose plan quality and plan robustness

We assessed plan quality in the nominal scenario between the 2 methods: RO and DLVCRO. As shown in Fig. 3a, both methods produced SSPT plans with comparable target dose homogeneity. Besides, DLVCRO provided comparable OAR protection in the bladder and rectum. DLVCRO seems to generate SSPT plans with the similar plan quality in target as RO and better dose distribution to protect OARs than RO. More details were provided in Tables E4 and E5.

Fig. 3.

Boxplots comparing dose-volume histogram indices of the treatment plans generated by RO and DLVCRO in the nominal scenarios and worst-case scenarios for all 10 patients. (a) Normalized CTV D_2% – D_98% in the nominal scenario. (b) Bladder V40Gy [RBE = 1.1], V65Gy [RBE = 1.1] and rectum V70Gy [RBE = 1.1] in the nominal scenario. (c) Normalized CTV D_98% in the worst-case scenario. (d) Normalized CTV D_2% – D_98% in the worst-case scenario. (e) Bladder V40Gy [RBE = 1.1], V65Gy [RBE = 1.1], and rectum V70Gy [RBE = 1.1] in the worst-case scenario. P values <.05 are highlighted, which are calculated from the Wilcoxon signed-rank test. Abbreviations: CTV = clinical target volume; DLVCRO = dose-linear-energy-transfer volume constraint robust optimization; RO = robust optimization.

With range and setup uncertainties accounted for, we calculated the DVH indices for the CTV and OARs in the worst-case scenario, as shown in Figure 3 (c-e). When compared with the RO method, plans developed using DLVCRO demonstrated similar robustness in target dose coverage and homogeneity. The OAR protection was comparable. The plan robustness of the plans generated by DLVCRO and RO was comparable for targets and OARs. More details were provided in Tables E4 and E6.

LET distribution

Figure 4 illustrates the LETVH indices of the plans generated using both methods under the nominal and worst-case scenarios. The corresponding data for these indices are presented in Tables E4, E7, and E8. In nominal scenarios, DLVCRO significantly reduced LET_max for OARs, as indicated by the indices (unit: keV/μm; rectum LET_max 8.76 vs 7.15, P = .0081), and similar target LET coverage, as indicated by the indices CTV LET_mean and CTV LET_98%.

Fig. 4.

Boxplots comparing LET-volume histogram indices of the treatment plans generated by RO and DLVCRO in the nominal scenarios and worst-case scenarios for all 10 patients. (a) CTV LET_mean and LET_98% in the nominal scenario. (b) Bladder LET_max and rectum LET_max in the nominal scenario. (c) CTV LET_mean and LET_98% in the worst-case scenario. (d) Bladder LET_max and rectum LET_max in the worst-case scenario. P values <.05 are highlighted, which are calculated from Wilcoxon signed-rank test. Abbreviations: CTV = clinical target volume; DLVCRO = dose-linear-energy-transfer volume constraint robust optimization; LET = linear energy transfer; RO = robust optimization.

In the worst-case scenarios, DLVCRO again showed improved LET protection for OARs (unit: keV/μm; rectum LET_max 10.94 vs 9.25, P = .0137). The target LET was also comparable, as shown by CTV LET_mean and CTV LET_98%.

Compared with RO, DLVCRO seems to generate SSPT plans with better and more robust LET distribution for OARs and similar LET distribution for targets. The plan generated by DLVCRO potentially had less toxicity according to high LET.

xBD distribution and DLVH indices

Figure 5 presents the xBDVH and DLVH indices for plans generated by the 2 competing methods under both nominal and worst-case scenarios. The corresponding data for these indices are presented in Tables E9, E10, and E11. Under nominal conditions, DLVCRO achieved superior xBD protection for OARs (unit: $\text{Gy}\cdot \frac{\text{keV}}{\mu \mathrm{m}}$; bladder xBD_max 499.55 vs 480.10, P = .0352, rectum xBD_max 414.20 vs 386.59, P = .0065), and equivalent target xBD coverage as indicated by CTV xBD_mean compared to RO. In the worst-case scenarios, DLVCRO continued to provide better OAR protection (unit: $\text{Gy}\cdot \frac{\text{keV}}{\mu \mathrm{m}}$; bladder xBD_max 721.58 vs 647.38, P = .0150, rectum xBd_max 663.88 vs 555.54, P = .0039), and similar target xBD coverages indicated by CTV xBD_mean. Compared to RO, DLVCRO seems to generate SSPT plans with better and more robust xBD distribution for OARs and similar xBD distribution for targets.

Fig. 5.

Boxplots comparing xBD-volume histogram indices and the DLVH indices of the treatment plans generated by RO and DLVCRO in the nominal scenarios and worst-case scenarios for all 10 patients. (a) CTV xBD_mean in the nominal scenario. (b) Bladder xBD_max and rectum xBD_max in the nominal scenario. (c) Bladder V(50Gy, 6keV/μm) and rectum V(67.8Gy, 2.86 keV/μm) in the nominal scenario. (d) CTV xBD_mean in the worst-case scenario. (e) Bladder xBD_max and rectum xBD_max in the worst-case scenario. (f) Bladder V(50Gy, 6keV/μm) and rectum V(67.8Gy, 2.86keV/μm) in the worst-case scenario. P values <.05 are highlighted, which are calculated from Wilcoxon signed-rank test. Abbreviations: CTV = clinical target volume; DLVCRO = dose-linear-energy-transfer volume constraint robust optimization; DLVH = dose-linear-energy-transfer volume histogram; RO = robust optimization; xBD = extra biological dose.

Figure 5 (c, f) presents the comparison of the DLVH indices generated by the RO and DLVCRO under both nominal and worst-case scenarios, respectively. The DLVH indices included here reflected the combined effect of dose and LET in AEs. Especially, V(67.8Gy, 2.86keV/μm) has been shown to be strongly related with rectal bleeding. In nominal scenarios, the DLVH index comparisons show that the DLVCRO provide better OAR protection by significantly reducing the corresponding DLVH indices in OARs (bladder $V\left[50Gy,\frac{6keV}{\mu m}\right]$ 1.41% vs 1.14%, P = .0374, rectum $V\left[67.8Gy,\frac{2.86keV}{\mu m}\right]$ 3.52% vs 3.00%, P = .0049). In the worst-case scenarios, the DLVCRO also provided better OAR protection (bladder $V\left[50.8Gy,\frac{6keV}{\mu m}\right]$ 12.54% vs 12.26%, P = .0333, rectum $V\left[67.8Gy,\frac{2.86keV}{\mu m}\right]$ 12.06% vs 11.42%, P = .0096). The better performance of DLVCRO on the rectum DLVH index, V(67.8Gy, 2.86keV/μm), shows the potential of DLVCRO in reducing rectal bleeding.

DLVH, DVH, and LETVH comparation

Figure 6 presents the comparison of the DLVHs for a representative patient, contrasting the RO (left) and the DLVCRO (right) methods. The gantry and couch angles used for this patient were: (1) field 1: couch 0° and gantry 95°; and (2) field 2: couch 180° and gantry 95°, as shown in Figure E2. In DLVCRO, DLVCs for $V_{desired}\left(f_i(d,l)\right)\le V_i$ at V_i = 0 are imposed on the corresponding OARs. These DLVCs are depicted as the red dashed lines in the DLVHs of both RO and DLVCRO results for a clearer comparison, as shown in Figure 6 (a-d). DLVCRO provides enhanced OAR protection by more effectively and precisely managing the overlap of high LET and high dose distributions within the OARs. As indicated by the yellow circles in Figure 6 (a-d), DLVCRO significantly reduced the number of voxels exposed to high dose and high LET. Additionally, the DLv% lines in the DLVH plot show a noticeable contraction in the high dose-high LET region, particularly for the DL1% lines indicated by black arrows. Figure 6 (e, f) provides the comparisons of the corresponding DVHs and LETVHs for this patient. DLVCRO resulted in lower LET exposure in OARs and increased LET within the CTV, while maintaining the similar physical dose distributions in the CTV comparable with those obtained with RO. The most significant differences in DVHs or LETVHs occur in the high-dose or high LET regions for both CTV and OARs, which underscores the effectiveness of the DLVCRO method in controlling the dose and LET joint distribution simultaneously.

Fig. 6.

Comparison of the LET and Dose in the DLVH, DVH, and LETVH of the treatment plan generated by RO and DLVCRO for a representative patient. DLVHs of the bladder are generated by (a) RO and (b) DLVCRO, respectively. DLVHs of the rectum are generated by (c) RO and (d) DLVCRO, respectively. In the DLVHs of bladder and rectum, red dash lines of $V\left(f(d,l)\right)=0\%$ indicate the DLVCs adopted in DLVCRO for bladder and rectum, respectively. Compared with RO, DLVCRO efficiently reduced the high LET and high-dose regions, which are indicated by the yellow circles. In the DVH and LETVH (e, f), yellow is clinical target volume, blue is bladder and green is rectum; the solid line is DLVCRO, and the dashed line is RO. Abbreviations: DLVCRO = dose–LET-volume constraint robust optimization; DLVH = dose–LET-volume histogram; DVH = dose-volume histogram; LET = linear energy transfer; LETVH = linear-energy-transfer volume histogram; RO = robust optimization.

Discussion

In this study, we introduced a novel optimization algorithm, DLVCRO, designed to robustly optimize both physical dose and LET distributions simultaneously and enhance OAR protection by implementing joint constraints of dose and LET. We employed DLVCRO to develop SSPT treatment plans for 10 patients with prostate cancer. Compared with the traditional RO methods, our approach adopted DLVCs, which constrain the joint distribution of dose and LET. All DLVCs can be customized according to user specifications. As a feasibility study, our work demonstrates equal or superior performance of the DLVCRO algorithm over the traditional RO methods, although our results only show a slight statistical benefit. By reducing the high dose and high LET overlap in nearby OARs, our approach could potentially reduce the possible incidence rates of the corresponding AEs. Specifically, DLVH indices corresponding to rectal bleeding are substantially reduced, such as V(67.8 Gy, 2.86 keV/μm), which has been demonstrated to be clinically relevant for rectal bleeding.²⁵

The central concept of the DLVCRO is to directly and precisely constrain the dose-LET joint distributions simultaneously using DLVCs. The reported xBD-volume histogram (xBDVH)³⁶ or LET-volume histogram (LETVH)-based SSPT treatment planning methods⁷ to mitigate the impact of high LET are essentially of a 2D concept. In this work, we developed SSPT treatment planning in 3D with dose, LET, and volume considered via DLVCRO. This method is fundamentally different from the previously reported LET-guided proton planning, in which the dose and LET were considered as 2 independent variables and the dose and LET-volume constraints were achieved separately with different sets of voxels penalized. Instead, the DLVCs was implemented as “soft constraints” in the objective function, which will modify the DLVH surface to modify the dose and LET distribution simultaneously to satisfy DLVCs by penalizing voxels violating joint dose and LET-volume constraints (ie, DLVCs, which can be derived from the retrospective patient outcomes studies). Thus, DLVCRO upgrades 2D DVH-based to 3D DLVH-based treatment planning to adjust dose/LET distributions simultaneously to reduce possible incidences of the corresponding AEs.

The xBD-guided proton planning to mitigate the impact of high LET used scalar values,²⁹ xBD, to indirectly represent the biological effects of the dose-LET joint distributions. This assumes that xBD is a good indicator of the synergistic biological impact of both dose and LET on AEs, although it might be true for some AEs, but not all. Therefore, the xBD-based constraints cannot precisely constrain the dose and LET distributions for some AEs. For example, the previous xBD-based method inadvertently penalized voxels with very low dose and high LET, which should not lead to the corresponding AEs because the possible RBE results from high LET could not be more than 1.7.¹⁶ In contrast, based on the DLVCs derived from the retrospective patient outcomes studies, DLVCRO eliminates ambiguity in the selection of voxels to be constrained, and thus would be specific to each adverse event if the adverse event-specific DLVC is applied in DLVCRO. The proposed DLVCRO would also be applicable to all AEs if adverse event-specific DLVC has been derived from the retrospective patient outcome study. This is very similar to our conventional use of DVCs to control the dose distribution, thus minimizing the possible incidence of the DVC-specific AEs in photon therapy treatment planning.

Additionally, within the framework of DLVC, the xBD term in the DLVCRO objective function is versatile and can be adapted into various forms relating to both dose and LET. Its primary purpose is to relate the dose and LET of 1 voxel in 1 single term to be penalized in the objective function. The xBD term was selected for its simplicity and can be certainly replaced by other forms. One potential and natural replacement of the xBD term in the objective function is the normal tissue complication probability model. Unfortunately, the normal tissue complication probability models for both rectum and bladder included in this study, with both dose and LET considered, are not available and thus further investigation is needed in this area.^{6,13,16,17,48} Moreover, our algorithm, which uses conventional nonlinear programming with a quadratic cost function, can be seamlessly integrated into the existing commercial treatment planning systems such as Eclipse (Varian Medical Systems), leveraging the same mathematical framework.

Compared with RO, DLVCRO achieved comparable plan quality in the nominal scenario with respect to target coverage and homogeneity (Fig. 3). And it has achieved comparable plan robustness for targets compared to RO (Fig. 3). Furthermore, our innovative DLVCRO method enhanced the protection of bladder and rectum in both the nominal and worst-case scenarios with respect to physical dose (Fig. 3), LET (Fig. 4), xBD (Fig. 5), and DLVH indices. In particular, the DLVCRO shows the ability to control the high dose-high LET voxels in OARs and has the potential to reduce the adverse effects of SSPT. Compared with the conventional DVC and LETVC-guided methods to constrain the physical dose and LET distributions separately, our approach more effectively minimizes the overlap of the high dose and high LET in OARs and redistributes high LET to targets. The DLVCs significantly reduce the high dose-high LET voxels in OARs to minimize the possible biological damage to OARs while minimally impacting target physical dose and robustness. Unlike the conventional DVC and LETVC-guided methods that often involve irrelevant voxels with lower-dose and high LET, as shown in Figure 6, our approach achieves similar outcomes more effectively.

The solution space for SSPT optimizations is highly degenerate. Many plans maintain similar physical dose distributions and robustness, yet they differ in LET distribution—some have the desirable high LET in targets and low LET in OARs—whereas others have the opposite. Our innovative concept of DLVC-based RO method, based on a novel tool of DLVH, steers the optimizer to generate SSPT plans with the optimal joint dose-LET distribution in a user-defined manner. Physical dose quantifies the energy deposited per mass unit, whereas LET measures the average energy loss per distance traveled by numerous particles.⁴⁹ This distinction allows for various joint dose-LET distributions without significantly altering the physical dose distribution.

The 10 patients with prostate cancer were treated with plans using antiparallel left and right fields. Because of the antiparallel treatment fields, which usually produce quite overall low LET distributions, and the anatomical geometric relationship between targets and OARs in prostate cancer, it is always challenging to further evaluate LET within the targets through optimization. Additionally, both DLVCRO and RO could generate highly modulated fields, which may heighten the susceptibility to the field misalignment. Therefore, it is important to consider the impact of the uncertainties through RO in both cases. The observed differences between RO and DLVCRO are primarily because of the distinct optimization methods employed, rather than variability or uncertainties in the optimization process itself. Both DLVCRO and RO use the same dose engine and the optimizer, but with different objective functions and constraints. Both methods also started with identical initial conditions, including beamlet intensities, number, energy, and positions.

In this study, MFO is employed for both RO and DLVCRO. Single-field optimization (SFO) is indeed commonly used in clinical practice for prostate cancer patients at our proton center. MFO is typically reserved for more complex cases including some complex prostate cancer patients. Because our manuscript is a feasibility study, the choice of MFO does not impact the evaluation of the proposed optimization method. Besides, demonstrating DLVCRO’s compatibility with MFO allows for potential future applications in some complex prostate cancer cases and more complex disease sites such as head and neck, which will benefit more from DLVCRO.

This work primarily serves as a feasibility study, demonstrating that the proposed optimization strategy is plausible and that the DLVCRO algorithm performs as well as or better than the traditional RO method. Other types of cancers, particularly those located in anatomically complex areas with radiosensitive organs such as head and neck cancer and neuro-oncology patients with tumors near radiosensitive structures in the brain, may benefit more significantly from the DLVCRO than the prostate cancer. The reasons why we selected prostate cancer as the example application in this study are: (1) the DLVCs for rectum bleeding have been derived from the existing study, which demonstrates the relationship between DLVH and AEs in proton therapy, thereby lending clinical significance to the DLVCRO methods. For other disease sites, for example, head and neck, the corresponding DLVCs are not available; and (2) because of its simplicity the prostate cancer is usually selected as the starting disease site to implement some new technologies clinically. By starting with prostate cancer, we aimed to establish the feasibility of DLVCRO. We are currently working on integrating a Monte Carlo dose engine^44,45 into DLVCRO and applying it to head and neck cancer.

This study has several limitations. As a feasibility study, it stems largely from a small number of prostate patients as the selected disease site and patient cohort and therefore has limited clinical benefits. Notably, the optimization framework did not consider beam angle selection, potentially limiting the optimality of the resulting SSPT plans with respect to both dose and LET distributions. Clinically, SFO is more commonly used in prostate cancer treatment. Although the choice between single-field or multifield optimization does not affect the evaluation of our method, future developing an SFO version of DLVCRO is necessary for broader clinical application. Furthermore, calculations were based on an analytical dose and LET calculation engine, which is acceptable in prostate cancer with overall homogeneous anatomies. In subsequent research, our aim is to include beam angle selection in the DLVCRO framework and to apply a Monte Carlo dose and LET calculation engine^44,45 to improve the dose and LET calculation accuracy in more inhomogeneous disease sites.

Conclusions

In this study, DLVCRO was distinguished from the traditional RO by optimizing the joint dose and LET distributions simultaneously to account for the combined effects of dose and LET in prostate cancer patients treated with SSPT. DLVCRO demonstrated superior performance, effectively minimized the overlap of high dose and high LET in OARs without sacrificing the physical dose distribution or its robustness in targets for patients with prostate cancer treated with SSPT. DLVCRO employed a user-specified tool of DLVC, based on DLVH, to strategically redistribute high LET from OARs to targets, thereby potentially reducing the possible incidence rate of AEs in prostate cancer patients treated with SSPT. DLVCRO upgraded 2D DVH-based to 3D DLVH-based treatment planning to adjust dose/LET distributions simultaneously and robustly^6,7 to reduce AEs in SSPT. DLVCRO is thus positioned as a front-runner for administering biologically optimized proton therapy.

Supplementary Material

Supplementary material associated with this article can be found in the online version at doi:10.1016/j.ijrobp.2024.11.068.