Biological optimization for hybrid proton-photon radiotherapy

Objective. Hybrid proton-photon radiotherapy (RT) is a cancer treatment option to broaden access to proton RT. Additionally, with a refined treatment planning method, hybrid RT has the potential to offer superior plan quality compared to proton-only or photon-only RT, particularly in terms of target coverage and sparing organs-at-risk (OARs), when considering robustness to setup and range uncertainties. However, there is a concern regarding the underestimation of the biological effect of protons on OARs, especially those in close proximity to targets. This study seeks to develop a hybrid treatment planning method with biological dose optimization, suitable for clinical implementation on existing proton and photon machines, with each photon or proton treatment fraction delivering a uniform target dose. Approach. The proposed hybrid biological dose optimization method optimized proton and photon plan variables, along with the number of fractions for each modality, minimizing biological dose to the OARs and surrounding normal tissues. To mitigate underestimation of hot biological dose spots, proton biological dose was minimized within a ring structure surrounding the target. Hybrid plans were designed to be deliverable separately and robustly on existing proton and photon machines, with enforced uniform target dose constraints for the proton and photon fraction doses. A probabilistic formulation was utilized for robust optimization of setup and range uncertainties for protons and photons. The nonconvex optimization problem, arising from minimum monitor unit constraint and dose-volume histogram constraints, was solved using an iterative convex relaxation method. Main results. Hybrid planning with biological dose optimization effectively eliminated hot spots of biological dose, particularly in normal tissues surrounding the target, outperforming proton-only planning. It also provided superior overall plan quality and OAR sparing compared to proton-only or photon-only planning strategies. Significance. This study presents a novel hybrid biological treatment planning method capable of generating plans with reduced biological hot spots, superior plan quality to proton-only or photon-only plans, and clinical deliverability on existing proton and photon machines, separately and robustly.


Introduction
Proton radiotherapy (RT) offers advantages in sparing organs-at-risk (OARs) and reducing integral dose compared to photon RT, attributed to the sharp dose falloff beyond the Bragg peak (Paganetti 2016).Protons exhibit a different biological effect than photons, which is quantified by the relative biological effectiveness (RBE).RBE is defined as the ratio of the equivalent photon dose to the physical dose of the particle required to achieve the same clinical endpoint.The RBE value for photons is 1 by definition, while the clinically used RBE value for protons is 1.1, i.e.
where d p physical is the physical dose (in Gy), and b 1.1 is the biological dose (in Gy(RBE)) with the constant RBE = 1.1.
However, this rule of thumb is oversimplified (Deng et al 2021).For instance, the RBE at the end of the Bragg peak is typically 1.3-1.5 or even higher (Unkelbach et al 2016, An et al 2017, Liu et al 2020).To address this complexity, various variable RBE models have been developed (Wilkens and Oelfke 2004a, Wedenberg et al 2013, Paganetti 2014, Wedenberg and Toma-Dasu 2014, McNamara et al 2015, Sanchez-Parcerisa et al 2016, Ödén et al 2017).Among these, a simplified RBE model that depends solely on linear energy transfer (LET) is widely employed (Kellerer and Rossi 1978, Wilkensa and Oelfke 2005, Unkelbach et al 2016), as RBE has been observed to monotonically depend on LET (Wilkens and Oelfke 2004a) where b p is the biological dose (in Gy(RBE)) defined by variable RBE, L is the dose-averaged LET (LET d ), and k is a scaling parameter typically set to a constant value of 0.04 µm keV −1 .This setting aims to achieve an RBE of 1.1 at the center of the spread-out Bragg peak with a modulation width of 5 cm and a range of 10 cm (Unkelbach et  (3) In equation (3), the stopping power S(E) represents the proton energy loss per unit length at energy E. The physical dose d p physical (E,z) = ϕ(E,z)S(E)/ρ(z) corresponds to the energy deposited per unit mass at location z (disregarding the neglectable change in proton flux that contributes to the dose, dϕ/dz = 0).Here, ϕ(E,z) denotes the proton energy spectrum.
Numerous studies (Wilkens and Oelfke 2004b, Wilkensa and Oelfke 2005, Giantsoudi et al 2013, Guana et al 2015, Unkelbach et al 2016, Cao et al 2018, Liu et al 2020, Li et al 2021, 2023a) have focused on optimizing LET to address the underestimation of the biological effect of protons on OARs near high-dose targets.However, uncertainties such as range uncertainty, setup uncertainty, and intra-and inter-fractional variation of patient anatomy can significantly compromise the dose coverage and uniformity of tumor targets in proton RT, even with robust optimization techniques (Unkelbach et al 2007, 2018a, Pflugfelder et al 2008, Fredriksson et al 2011, Liu et al 2012, 2013), due to the sharp dose fall-off of the Bragg peak.Furthermore, optimizing LET tends to further degrade target dose coverage and uniformity (Li et al 2023a).
Compared to proton RT, photon RT offers several advantages: (1) no concern for underestimated biological effect in OAR, (2) slowly-changing dose gradients along the beam direction, and (3) sharper dose gradient lateral to the beam direction, albeit with a tendency to increase the dose to normal tissues.The complementary nature of protons and photons has led to the development of combined proton-photon RT, aiming to synergize their strengths.This approach aims to achieve robust target coverage and optimized OAR sparing without risking biological overdose to the OARs.Specifically, protons are used to constrain the dose in the proximity of the target and reduce the integral dose, while photons enhance robustness to treatment uncertainties and compensate for potential loss of plan quality due to proton biological optimization.This combined approach helps eliminate hot spots near the target in terms of biological dose.
Current hybrid proton-photon RT is clinically utilized for treating spine sarcoma (Hug et al 1995, DeLaney et al 2009, 2014) and head and neck cancer (Furui et al 2023), where separate optimization of photon and proton plans is employed.Joint optimization of proton and photon variables (Unkelbach et al 2018b, ten Eikelder et al 2019, Fabiano et al 2020a, 2020b, Amstutz et al 2023) using non-uniform fractionation requires simultaneous delivery of proton and photon treatments.Methods such as fraction-wise target-dose uniformity regularization (Gao 2019) and stochastic optimization (Fabiano et al 2022) have been developed for robust delivery on proton and photon machines separately.However, these methods did not directly optimize the fraction ratio between photons and protons or automatically split the hybrid treatment into photon and proton fractions.Loizeau et al developed a fraction optimization method to minimize normal tissue complication probabilities, based on pre-optimized proton and photon plans (2021).Recently, joint optimization of the number of fractions and plan variables for protons and photons was developed (Li et al 2023b).This method ensures a uniform target dose per proton or photon fraction and can lead to optimized and automatic delivery of hybrid proton-photon RT using existing proton and photon machines.Nevertheless, the question remains open as to whether and how hybrid proton-photon RT can reduce the biological dose in OARs near the target, without affecting the overall dose distribution, compared to proton-only or photon-only RT.
This work proposes a novel biological optimization method for hybrid proton-photon RT that utilizes both protons and photons to jointly minimize the biological dose in OARs.It aims to maintain the advantages of hybrid proton-photon RT in terms of the physical dose distribution, specifically optimized target coverage and OAR sparing, compared to proton-only or photon-only RT.To ensure plan deliverability on existing proton and photon machines, the hybrid plans generated from this method will feature a uniform target dose per fraction.

Optimization objectives and formulation
The proposed biological optimization method for hybrid proton-photon RT involves optimizing the proton spot weight (x p , i.e. number of protons per spot), photon fluence (x γ ), and proton fraction ratio (r, i.e. number of proton fractions over total number of fractions, with the photon fraction ratio being 1-r).The hybrid proton-photon biological optimization can be formulated as the following inverse optimization problem: min In equation ( 4), the sum over j includes all the scenarios for the robust optimization (J = 9), and p j = 1/9 is the weight of each scenario.f represents the sum of planning objectives on the total dose, d p + d γ , where d p = b 1.1 , is the proton dose with constant RBE = 1.1, as defined in equation ( 1), with an objective dose d obj .The term g denotes the sum of planning objectives on proton dose d p or photon dose d γ with the objective dose rd obj or (1-r)d obj to ensure target dose uniformity per fraction.φ represents the sum of biological dose objectives on proton biological dose b p defined in equation (2), photon biological dose b γ , and total biological dose b p + b γ .Note that b γ is the same as the physical dose d γ of photons.C pj is the voxel-wise multiplication of LET and dose influence matrix.The objective weights for φ, w, are the same for proton, photon and hybrid plans, to analyze the effect of biological optimization without the bias from photon's contribution.To evaluate the effectiveness of biological optimization, three scenarios (s0, s1 and s5) are considered: w = 0 (i.e.without biological dose objectives in the optimization), w = 1 (i.e. with biological dose objectives), and w = 5 (i.e. with larger weighting for the biological dose objective term).The dose objectives are based on the dose-volume histogram (DVH) (table 1).

Robust optimization to setup and range uncertainties
In clinical practice, photon planning (without robust optimization) typically focuses on the planning target volume (PTV) (Khan and Gibbons 2014), which is an expansion of the clinical target volume (CTV) by the setup margin.In contrast, proton planning often targets the CTV, with robust optimization techniques (Unkelbach et al 2007(Unkelbach et al , 2018a) ) to account for setup and range uncertainties.Where various scenarios of uncertainties are modeled in the objective functions for proton planning.In this study, a robust optimization approach with probabilistic formulation is employed for photon, proton, and hybrid RT to ensure a fair comparison.
The setup uncertainty (for both protons and photons) is modeled by a rigid shift in the axial, sagittal, and coronal directions, while the range uncertainty (for protons only) is modeled by a uniform scaling of stopping power ratio values.This results in a total of 9 uncertainty scenarios, including the nominal scenario.The setup and range uncertainty parameters are listed in table 1.

An auxiliary OAR structure near target
For protons, the biological dose is approximately proportional to both the LET and the physical dose, as indicated by equation (2).Both high LET regions and high dose regions of normal tissues often occur near the tumor target.Similarly, for photons, the high dose region is frequently near the tumor target.To address this, an auxiliary OAR structure called PTV1cm is created as a 1 cm expansion surrounding the PTV, excluding the PTV itself, on which the biological dose is minimized.In this work, the PTV structure is defined as an expansion of the CTV by the setup margin, i.e.PTV = CTV + setup uncertainty.However, as mentioned in section 2.1.2,the treatment planning is still conducted with respect to the CTV via robust optimization.The choice of a 1 cm expansion empirically works well to cover the normal tissue region with high biological dose.

MMU constraints
The last constraint in equation ( 4) is known as the minimum monitor unit (MMU) constraint for the proton beams, represented by the MMU threshold h which is enforced to ensure that proton spots are deliverable on proton machines.It is applied in both proton-only planning and the proton component of hybrid planning.In this work, we utilize optimization methods that seamlessly integrate into the hybrid optimization framework described in equation (4).

Solution algorithm
Due to the nonconvex nature of the DVH dose objectives and the MMU constraint, we employ the iterative convex relaxation (ICR) method (Lin et al 2019, Gao et al 2020a, 2020b, 2022).The inner loops were solved using the alternating direction method of multipliers (ADMM) (Boyd et al 2011, Gao 2016).
To address the nonconvex nature of the DVH objectives, ICR is applied to equation ( 4).This approach lead to the derivation of the outer loop, which consists of: (5) Here, x is the concatenation of x γ and x p ,. as defined in equation ( 4).Ω denotes the active index set for each DVH objective, which includes the voxels violating the DVH constraint and thus need to be included and penalized during the next optimization iteration to meet the DVH constraint.Equation ( 5), with fixed active set Ω m from the last iteration can be efficiently solved by ADMM, which will be discussed next.Equation (6) updates the active set Ω m+1 based on x m+1 and r m+1 .
To solve the subproblem given by equation ( 5) using ADMM, we introduce dummy variables z and z r with dummy constraints z = x and z r = r, respectively.This decouples the non-negative constraint for x and the box constraint for r from the data fidelity term F(x,r).We then introduce additional auxiliary variables u and u r to reformulate the constraints z = x and z r = r as L2-norm penalty terms with corresponding weights λ and λ r , respectively.This allows us to derive the Augmented Lagrangian of equation ( 5) The inner loop for solving equation ( 5) via ADMM consists of the following six iterative steps.
x m+1 , r m+1 = arg min The existence of analytic solutions for the MMU constraint (the second equation in equation ( 8)) and the simple bound constraints (the third and fourth equation in equation ( 8)) is the motivation for the proposed solution algorithm using ADMM and ICR.The solution to the first equation of equation ( 8) can be obtained by taking the derivatives with respect to x and r respectively, using the conjugate gradient method.

Dose calculation and planning parameters
The dose influence matrix (A in equation ( 4)) for protons and photons, and the proton dose-LET-product matrix (C in equation ( 4)) are generated using MatRad (Wieser et al 2017).Here a 6 MV photon pencil beam kernel from LINAC data and a proton pencil beam model for energies from 31 MeV to 236 MeV are utilized.The dose-averaged LET is calculated based on the local mean of the proton stopping power, weighted by the local energy spectrum on the central axis of broad proton beams in water.The dose is calculated from a CT scan of the patients, and it is reported as dose to water.The MMU constraint for all cases assumes the threshold h = 5 × 10 6 protons.200 iterations are used for all the methods in all cases.

Hybrid optimization with fixed ratio of fractions
The hybrid method, which optimizes both x and r optimization ('Hybrid') (equation ( 4)) is being validated against the hybrid method which optimizes only x ('Hybrid-x') (equation ( 9)), with a fixed fraction ratio r.To determine the optimal r for Hybrid-x, it is solved multiple times for a range of r values from 0 to 1, encompassing all possible discrete values for proton fractions within a given total fraction number, selecting the r value with the smallest total objective.Despite being less efficient than the Hybrid method, this approach allows for obtaining the optimal r for Hybrid-x, The key distinction between Hybrid-x (equation ( 9)) and Hybrid (equation ( 4)) lies in their optimization approach for the fraction ratio r.Hybrid optimizes r, whereas Hybrid-x does not.In Hybrid-x, the plan with the smallest objective value (F opt ) is chosen as the solution of optimized fraction ratio (r opt ), and consequently, the optimized proton fraction number n (n opt ).For Hybrid, the optimized results are represented by F * , r * and n * respectively.Notably, Hybrid requires solving only one plan, while Hybrid-x needs to solve multiple plans to determine the best one.This efficiency makes Hybrid more practical than Hybrid-x.The efficacy of Hybrid in terms of plan quality is evaluated in comparison to Hybrid-x in section 3.4.
Additionally, the photon-only ('Photon') and proton-only optimization ('Proton') corresponds to Hybrid-x (equation ( 9)) with r = 0 and r = 1 respectively.To ensure a fair comparison, Proton and Photon plans are also optimized using ICR and ADMM, similar to equations ( 5)-( 8).All plans are normalized to maintain d98% in CTV equal to the prescription dose for the worst robustness scenario.

Clinical cases
Four clinical cases are used to validate the proposed method.

Head-and-neck (HN) case
This is a bilateral head and neck case.The px is 69.96Gy delivered in 33 fractions.The beam arrangement is the same as the brain case.

Abdomen case
The target is located in small bowel, between L1 and L3 spinal segment.The px is 55 Gy delivered in 25 fractions.The beam arrangement has three coplanar proton beams (0 • , 120 • , 240 • ), along with seven coplanar photon beams (0

Lung case
This is a left lung tumor case.The px is 60 Gy delivered in 30 fractions.The beam arrangement is the same as the abdomen case.
The planning objectives, setup and range uncertainty parameters for these cases are detailed in table 1.The prescription and OARs dose constraints correspond to a conventional fractionation schedule, i.e. 5 fractions per week.Setup and range uncertainties are set to be 3 mm and 3.5% for brain and HN, and 5 mm and 5%) for abdomen and lung.The larger range uncertainty for abdomen and lung is to account for larger differences in body volume and potential variations due to intrafractional motion.

Evaluation of biological hybrid optimization
The optimized plans with and without the biological objective φ (scenarios s0, s1) are compared in terms of DVH for physical dose d, biological dose b, and the dose plot for d, b and L for proton only, photon only (excluding L) and hybrid plans.
The tradeoff between the biological objective φ and other planning objectives is analyzed by comparing three different scenarios (s0, s1, s5) with weights, w, of 0,1 and 5, respectively, for the brain case.Additionally, the comparison between the Hybrid and Hybrid-x methods is presented for the brain case.
The hot spots for biological dose b in the PTV1cm are evaluated for clinical relevance, as the PTV1cm represents a high-dose region of normal tissues surrounding the target that is more likely to receive a high biological dose and be underestimated.For Hybrid and Proton plans, the hot spots are evaluated by plotting b in the 10% of PTV1cm volume receiving the highest b (V Max10 for PTV1cm), and the mean value of d and b in hot spot region (V Max10 ) in PTV1cm is also evaluated as (d Max10 or b Max10 ).Compared to commonly used DVH parameters like V110%, which can have different volumes for different methods, V Max10 , utilized in this study, has the constant volume, which is suitable for the comparison of high dose region across different methods.That is, we consider how the mean dose in this region (d Max10 or b Max10 ) of the same volume changes with respect to different optimization methods.

Hybrid plans v.s. Proton/Photon plans
The comparison of Hybrid plans with Photon and Proton plans is presented for all the clinical cases with biological optimization (s1), aiming to evaluate the potential improvement in plan quality offered by Hybrid over Photon or Proton.
In terms of target dose coverage and uniformity, the conformity index (CI) (Paddick 2000) and the maximum target dose (dmax CTV ) are evaluated for physical dose d and biological dose b respectively.CI is defined as V 100,CTV 2 /(V CTV × V 100 ) (where V 100,CTV is the CTV volume receiving at least 100% of prescription dose, V CTV is the CTV volume, and V 100 is the total body volume receiving at least 100% of prescription dose; ideally CI = 1).
In terms of OAR sparing, the evaluation primarily considers the high-dose OAR structure PTV1cm, including the maximum dose (dmax, bmax) mean dose (dmean, bmean) to the PTV1cm, and the PTV1cm volume within 90% of prescription isodose line (V90%) for d and b.Maximum LET (Lmax), mean LET (Lmeam) and mean LET in the hot spot region (L Max10 ) are also evaluated.To visualize the differences in the hot spots of b in PTV1cm between Hybrid and Proton, the difference between Hybrid and Proton is plotted within the 80% of prescription isodose line (V80%) for PTV1cm.

Results
The dose d and b plots for Photon, Proton and Hybrid plans are shown in figures 2, 4-6(i)-(n).

Biological dose in OARs
As w increased from 0, to 1 and 5 in the brain case for Hybrid and Proton plans, b in the PTV1cm decreased as expected (figure 1).In tables 2 and 3, the maximum Proton dose in PTV1cm (dmax and bmax) decreased for s1 in three cases (except Abdomen) and increased for s5 in all cases.Meanwhile, the hot spot region in PTV1cm evaluated by d Max10 , b Max10 and V90% (the volume receiving 90% prescription dose or higher for d or b) decreased for Hybrid and Proton (figure 1, tables 2 and 3).As w increased, dmax and bmax in CTV increased for all cases (tables 2 and 3).The maximum L in CTV and L Max10 in PTV1cm also increased as w increased from 0 to 1 (table 4).
However, as w increased from 1 to 5, certain trends reversed.The dose conformity (i.e.CI) to the target increased for s1 and remained similar or decreased for s5 for Proton, Photon and Hybrid plans (tables 2 and 3).The tradeoff of objective value between d and b was similar (table 5).Take the brain case as an example, for the Hybrid plan, the decreasing of optimized b objective value (F b /w) versus the increasing trend of d objective value (F d ) was (2.0-1.7 v.s.1.6-7.4)from s1 to s5.For the Proton plan, it was (2.0-1.7 v.s.1.5-7.0),and for the Photon plan: (2.4-2.3 v.s. 3.3-10.6).This indicated that compared to s1, s5 led to a greater loss in plan quality in the d for a smaller increase in b plan quality.

Target coverage
Optimization with φ objective (s1) improved the CI of d for all cases by 0.07-0.09for Proton plans, 0.02-0.08 for Photon plans, and 0.03-0.06for Hybrid plans (table 2).
The maximum target dose increased in all cases for all the s1 plans (table 2); however, dmax for CTV of all the Hybrid plans were within 112.2% of px, and all dmax of CTV of Proton plans except for lung case were within 109.7%.For Proton plan in lung case, the CTV dmax was 121.2% of px, while the increase due to the optimization of φ (compared with s0, 120.8%) was 0.4% of px (table 2).

Plan robustness
From the DVH figures (figures 3-6(i)-(n)), the target dose d in all Hybrid plans for all cases was similar or narrower than the Proton and Photon plans.For example, the volume (%) variance among all the uncertainty scenarios for 100% px dose coverage in CTV was 1.88%, 1.91% and 1.99% for Hybrid, Proton and Photon plans respectively for the brain case (figure 3(a)).
The robustness of b in PTV1cm in the Hybrid plans for all cases was significantly improved compared with Proton plans, while it was not as good as Photon plans.The volume variance for b ⩾ 90% px in PTV1cm was 9.7%, 11.8% and 3.7% for Hybrid, Proton and Photon plans respectively for the brain case (figure 3(d)).The variation of maximum b in PTV1cm among all uncertainty scenarios ranges from 102% to 112%, 102% to 115% and 115% to 117% for Hybrid, Proton and Photon plans respectively for the brain case (figure 3(d)).

Hybrid v.s. proton and photon
Although the biological optimization method for both Proton RT and Photon RT plans (scenario s1 v.s.s0) eliminated the b hot spot regions in PTV1cm (figures 1, 3-6, tables 2 and 3), The Hybrid-s1 plans were able to reduce the b hot spot in PTV1cm (Hybrid-s1 v.s.Hybrid-s0) in terms of b Max10 more than Proton and Photon for all cases except for the lung case.In the lung case, the Hybrid plan was better than the Proton plan, while compared to Photon which reduced the b Max10 for PTV1cm from 101.8% to 101.4%, Hybrid plan reduced it from 102.3% to 101.5%.It was evident that the Hybrid-s1 plans performed better than Proton and Photon in reducing the b hot spot in PTV1cm for all case.Particularly considering the b Max10 for PTV1cm in Proton plans reached 111.3% and 107.7% for s0 and s1, respectively, the Hybrid method was a significant improvement for Proton plan.
For each case, one of the OARs which was adjacent to the CTV and overlapping with PTV1cm was also evaluated in the DVH plots: the brainstem in the brain case, the larynx in the HN case, the healthy lung in the lung case and the bowel bag in the abdomen case (figures 3(e) and (f), 4-6(m) and (n)).The comparison of b among Photon, Photon and Hybrid plans was similar to that for PTV1cm, as expected.
Hybrid plans also outperformed Proton and Photon plans in the target dose plan quality when the biological optimization method was applied.As w increased from 0 to 1, dmax for CTV increased in all the Proton, Photon and Hybrid plans (table 2), which might raise concerns about the plan quality.Hybrid-s1 plans had a lower CTV dmax than Proton-s1 and Photon-s1 in all but the lung case, while maintaining the highest d conformity (CI) to the target.For lung case, Hybrid-s1 had a lower dmax for CTV than Proton, but higher than Photon dmax; however, compared to s0 plans, Photon-s1 increased CTV dmax by 2.7% px, Hybrid-s1 increased by 1.0% px.
Hybrid-s1 achieved a higher d-CI than Proton and Photon in the lung case (table 2).For the other three cases, Hybrid-s1 achieved the same CI as Proton-s1, while significantly improving CI compared to Photon-s1.

Hybrid v.s. Hybrid-x
For the brain and HN cases, the Hybrid optimized proton fraction number n * decreased (from 25 to 21, 26-23) as w increased from 0 to 5 (s0 to s5), while for the abdomen and lung cases, it increased (15-17, 11-17) (table 5).Hybrid and Hybrid-x plans were similar, as shown for the brain case in table 5.Both hybrid plans shared the same trend when comparing with Proton and Photon plans for s0, s1 and s5 optimization in terms of the objective values.
Hybrid was computationally more efficient than Hybrid-x, as Hybrid took similar amount of time as Hybrid-x for each given proton fraction number.Hybrid-x required multiple values of n to get the optimal n opt .

Discussion
The choice of expansion radius from PTV for the OAR structure PTV1cm was 1 cm, aimed to cover the high biological dose region adequately.An expansion that is too small might miss hot spots, while a larger volume could be unnecessary and increase computational burden.The ring OAR structure has been proven useful for characterizing the high-dose region in proton RT, and the selection of 1 cm as a rule of thumb has been validated (Gao 2019, Gao et al 2020b, 2022, Lin et al 2021b, Li et al 2023b).
When optimizing with φ objective (s1), the increase in CI for the Photon plans results solely from the inclusion of additional objectives, without requiring any biological dose optimization.It is analogous to the increase in maximum dose (dmax) within the CTV, the decrease in the maximum biological dose (b Max10 ) within the PTV1cm, and any other changes observed with s1.Therefore, to fairly assess the advantage or disadvantage of the Hybrid method in biological optimization, one should base the evaluation on the changes seen in the Photon plans.This approach helps to eliminate bias that may arise from simply adding an extra objective.
Another rationale for including the φ objective for the photon dose with the same weight w for the proton dose is to analyze the impact of biological optimization without bias from the photon contribution.In the Hybrid method, the proton partial plan and photon partial plan are optimized similarly, both driven by the same additional objective.Therefore, the comparison of parameters (such as the fraction ratio r between s1 and s0 for the Hybrid plan) fairly represents the effect of biological optimization, without the influence due to the addition of an extra objective to the proton dose.
Minimizing b during optimization, as shown in equation ( 2), penalized high-dose d and high-LET L, leading to hot spots in PTV1cm.As b in PTV1cm was minimized, the high-LET region was pushed towards the CTV, resulting in an increase in the maximum LET.This trend was confirmed by table 4: Lmax increased in CTV, Lmax and Lmean increased in PTV1cm for s1 compared to s0.This could explain why b in CTV was less robust than d as w increases, as seen in the DVH plots (figures 3(a) and (b), 4-6(i) and (j)).
For the brain and HN cases, the Photon plan exhibited poorer CTV robustness compared to the Proton or Hybrid plans (figures 3(a) and 4(i)).This observation did not contradict to a general experience that photon offers better robustness against proton plans.This was evident from the robustness DVH plots for the surrounding volume, PTV1cm, where Photon plans were significantly superior to Proton and Hybrid plans in terms of robustness (figures 3(c) and 4(k)).
Among all four cases, the lung case provided the strongest support for the notion that Hybrid could significantly improve plan quality, while Proton hardly reduced the hot spot region in PTV1cm in terms of V90% and max dose in CTV.The higher CTV dose in the Proton plan compared to Photon and Hybrid plans may be attributed to the majority of MUs being delivered through a single field (figures 5(b) and (c)).In comparison, Hybrid was able to leverage the target dose homogeneity from Photon as well as the lower integral dose from Proton.
Another justification for Hybrid to enhance plan quality was seen in the brain case.While the Photon plan was the most robust, with the smallest variation in bmax in PTV1cm among all uncertainty scenarios  (figure 3(d)), it significantly overdosed PTV1cm.In contrast, the Hybrid plan showed a lower bmax in PTV1cm (108.5%)compared to both Proton and Photon plans (109.9% and 116.9%, respectively).
As w increased, the optimized proton fractions n * decreased for brain and HN cases, while they increased for lung and abdomen cases.This trend could be explained as follows: an increase in the photon fraction introduced more integral dose to the healthy tissue compared to an increase in the proton fraction, especially for larger normal tissues (lung and abdomen cases).Conversely, for smaller normal tissues (brain and HN cases), the lower integral body dose allowed for more punishment to the biological dose in PTV1cm, resulting in fewer proton fractions.
The comparison between s0, s1 and s5 suggested an optimal choice for the weight w in terms of overall plan quality.However, it should be noted that the weighing parameter w was selected to be the same for all four cases, which may not be optimal.Future studies could investigate multi-criteria optimization for biological hybrid planning.
The uniformity of proton and photon fraction dose in the hybrid plan allowed for separate delivery of the hybrid plan, enabling the evaluation of photon and proton fractional effects independently which is similar to the combination of brachytherapy and photon external beam treatment.Therefore, the uniformity constraint for the fraction dose could potentially increase access to limited proton resources.The optimized plans generated using our method have previously demonstrated robustness to the fractionation effect (Unkelbach et al 2021) in terms of biological effective dose (BED), despite BED not being directly optimized during treatment planning (Li et al 2023b).
The chosen beam arrangements in this study followed the clinical recommendations at our institution as the initial template of beam arrangements.Although they were sufficient for the purpose of this study, the beam arrangements may be suboptimal.For example, during case-specific tuning of the initial template, proton beam arrangements may include a couch angle for brain and HN cases, in order to avoid certain critical OARs.On the other hand, beam angle optimization (BAO) methods could be explored in a future work to improve the optimality for treatment planning, e.g.our recently developed BAO method (Shen et al 2023).
Aside from the beam arrangement, the proton-only and photon-only plans were optimized according to clinical practice standards, ensuring target coverage, minimizing hot spots, and meeting OARs constraints, as illustrated in the DVH (figures 3-6(i)-(n)).
The photon planning (Photon or the photon component of Hybrid) in this study was based on intensity-modulated radiation therapy (IMRT) (with a few beam angles) for simplicity in this proof-of-concept study.The study with volumetric modulated arc therapy (VMAT) (with arc delivery) is ongoing and is not included in this study due to the technical complexity of VMAT and the belief that the qualitative efficacy of Hybrid should be independent of a specific photon delivery modality.However, it should be noted that the ICR and ADMM can handle the nonlinear and nonconvex constraints and objectives from direct aperture optimization of VMAT, which has been shown to work for complex treatment planning scenarios (Gao et al 2020b, 2022, Zhang et al 2023b, 2023c).Additionally, with IMRT, beam angles optimized via BAO (Shen et al 2023) could be performed to further improve Photon or the photon component of Hybrid.The replacement of IMPT by proton ARC (Zhang et al 2022(Zhang et al , 2023a) could potentially further improve Proton or the proton component of Hybrid.

Conclusion
A novel biological treatment planning method for hybrid proton-photon RT has been developed, capable of reducing biological hot spots in OARs and healthy tissue while achieving superior plan quality compared to proton-only or photon-only plans.Furthermore, the hybrid plans are clinically deliverable on existing proton and photon machines separately and demonstrate robustness.
Various methods have been developed to tackle the MMU problem, including preprocessing methods (Lin et al 2021a), postprocessing methods (Zhu et al 2010, Lin et al 2016, Gao et al 2019), and optimization methods (Cao et al 2013, Shan et al 2018, Lin et al 2019, Gao et al 2020a, Cai et al 2022, Zhu et al 2023, Li et al 2023a).

Figure 4 .
Figure 4. Dose and DVH plots for the HN case.Photon dose d (a), Proton d and b (b) and (c), Hybrid d and b (d) and (e) are plotted in the dose window [0%, 110%], with 50%, 80% and 100% isodose lines highlighted.b in VMax10 for PTV1cm, is plotted for Proton and Hybrid (f) and (g) in the dose window [75%, 110%], with 80% and 100% isodose lines highlighted.(h) is the b difference in PTV1cm between Hybrid and Proton in V80% plotted in the dose window [−15%, 15%].The DVH for d and b in CTV (i) and (j), PTV1cm (k) and (l) and larynx (m) and (n) are compared among Proton, Photon and Hybrid for w = 1 (s1).

Figure 5 .
Figure 5. Dose and DVH plots for the lung case.Photon dose d (a), Proton d and b (b) and (c), Hybrid d and b (d) and (e) are plotted in the dose window [0%, 110%], with 50%, 80% and 100% isodose lines highlighted.b in VMax10 for PTV1cm, is plotted for Proton and Hybrid (f) and (g) in the dose window [75%, 110%], with 80% and 100% isodose lines highlighted.(h) is the b difference in PTV1cm between Hybrid and Proton in V80% plotted in the dose window [−15%, 15%].The DVH for d and b in CTV (i) and (j), PTV1cm (k) and (l) and health lung (m) and (n) are compared among Proton, Photon and Hybrid for w = 1 (s1).

Figure 6 .
Figure 6.Dose and DVH plots for the abdomen case.Photon dose d (a), Proton d and b (b) and (c), Hybrid d and b (d) and (e) are plotted in the dose window [0%, 110%], with 50%, 80% and 100% isodose lines highlighted.b in VMax10 for PTV1cm, is plotted for Proton and Hybrid (f) and (g) in the dose window [75%, 110%], with 80% and 100% isodose lines highlighted.(h) is the b difference in PTV1cm between Hybrid and Proton in V80% plotted in the dose window [−15%, 15%].The DVH for d and b in CTV (i) and (j), PTV1cm (k) and (l) and bowel bag (m) and (n) are compared among Proton, Photon and Hybrid for w = 1 (s1).

Table 1 .
Planning objectives and robustness parameters.The objectives for CTV, body and PTV1cm are the same for all the cases: d(L2) or b(L2) = x% represents the least-square objective ||d-x% of px|| 2 with x as a percentile of the prescription dose.

Table 2 .
Dosimetric parameters for physical dose.Conformity index (CI), max dose (dmax, unit: % of px) in CTV, PTV1cm, mean dose (dmean) in the PTV1cm, mean d in the hot spot (dMax10) in PTV1cm, and V90% in PTV1cm are compared between Photon, Proton and Hybrid plans.dMax10 is the mean dose in the 10% volume of PTV1cm receiving higher d than the rest 90% volume of PTV1cm.

Table 3 .
Dosimetric parameters for biological dose.Conformity index (CI), max dose (bmax, unit: % of px) in CTV, PTV1cm, mean dose (bmean) in the PTV1cm, mean b in the hot spot (bMax10) in PTV1cm, and V90% in PTV1cm are compared between Photon, Proton and Hybrid plans.bMax10 is the mean dose in the 10% volume of PTV1cm receiving higher b than the rest 90% volume of PTV1cm.

Table 4 .
Dosimetric parameters for LET.Max LET (Lmax) in CTV, PTV1cm, mean LET (Lmean) in the PTV1cm, and mean LET of hot spot in PTV1cm (LMax10) are compared between Photon, Proton and Hybrid plans.LMax10 is the mean LET in the 10% volume of PTV1cm receiving higher L than the rest 90% volume of PTV1cm.

Table 5 .
Optimization result.n * or nopt is the optimized proton fraction number obtained by Hybrid method or Hybrid-x method.