Fraction optimization for hybrid proton-photon treatment planning

Abstract

Purpose:

Hybrid proton-photon radiotherapy (RT) can provide better plan quality than proton or photon only RT, in terms of robustness of target coverage and sparing of organs-at-risk (OAR). This work develops a hybrid treatment planning method that can optimize the number of proton and photon fractions as well as proton and photon plan variables, so that the hybrid plans can be clinically delivered day-to-day using either proton or photon machine.

Methods:

In the new hybrid treatment planning method, the total dose distribution (sum of proton dose and photon dose) is optimized for robust target coverage and optimal OAR sparing, by jointly optimizing proton spots and photon fluences, while the target dose uniformity is also enforced individually on both proton dose and photon dose, so that the hybrid plans can be separately and robustly delivered on proton or photon machine. To ensure the target dose uniformity for proton and photon plans, the number of proton and photon fractions is explicitly modeled and optimized, so that the target dose for proton and photon dose components is constrained to be a constant fraction of the total prescription dose while the plan quality based on total dose is optimized. The feasibility of the generation of clinically deliverable hybrid plans using the proposed method is validated with representative clinical cases including abdomen, lung, head-and-neck (HN), and brain.

Results:

For all cases, hybrid plans provided better overall plan quality and OAR sparing than proton-only or photon-only plans, better target dose uniformity and robustness than proton-only plans, quantified by treatment planning objectives and dosimetric parameters. Moreover, for HN and brain cases, hybrid plans also had better target coverage than photon-only plans.

Conclusion:

We have developed a new hybrid treatment planning method that optimizes number of proton and photon fractions as well as proton spots and photon fluences, for generating clinically deliverable hybrid plans that can be separately and robustly delivered on proton or photon machines. Preliminary results have demonstrated that hybrid plans generated via the new method have better plan quality than proton-only or photon-only plans.

1. Introduction

Compared to photon radiotherapy (RT), proton RT can often improve sparing of organs-at-risk (OAR) and reduce integral dose, owing to the sharp dose falloff beyond Bragg peaks [1]. Intensity modulated proton therapy (IMPT) via pencil beam scanning (PBS) is increasingly popular for delivering proton RT [2]. However, the treatment uncertainties (e.g., range uncertainty, setup uncertainty, intra- and inter-fractional variation of patient anatomy) can severely compromise dose coverage and uniformity of tumor targets for proton RT [1], even with robust optimization [3–8], which is also due to the sharp dose falloff beyond Bragg peaks. In contrast, with slowly varying dose gradient, photon RT by nature is less sensitive to treatment uncertainties and more robust in target dose coverage, which is however often at the expense of increased dose to OAR. The complementary nature of proton and photon beams makes it possible to consider the hybrid proton-photon RT that synergizes proton and photon RT, for robust target coverage and optimized OAR sparing when considering treatment uncertainties: (1) the use of protons to constrain the dose in proximity to tumor targets for sparing OAR and reducing integral dose; (2) the use of photons to shape the dose at tumor targets in the presence of treatment uncertainties.

Currently the hybrid proton-photon RT is clinically used for treating spine sarcoma [9–11], for which photon and proton plans are optimized separately: protons are used to improve the OAR sparing that allows for dose escalation to targets, while photons are used to reduce the skin dose. Recently various hybrid proton-photon treatment planning methods with joint optimization of proton and photon variables have been proposed with different motivations and outcomes [12–17]. To solve the problem that proton RT is a limited resource, Unkelbach et al proposed a hybrid planning method with the plan quality that is no worse than proton plans [12]. Using non-uniform fractionation, ten Eikelder et al demonstrated that better biological dose can be attained for hybrid planning than proton planning under certain conditions [13]. Motivated by the complementary nature of proton and photon beams, a hybrid planning method with uniformity regularization of target dose for proton/photon fractions provided better plan quality than either proton-only or photon-only planning, when accounting for treatment uncertainties [14]. Fabiano et al improved [12] by accounting for range uncertainties [15]. Fabiano et al proposed hybrid RT with simultaneous proton and photon treatment per fraction using a gantry-less proton beamline [16]. Loizeau et al developed a fraction optimization method to minimize normal tissue complication probabilities, which was based on pre-optimized proton and photon plans [17].

This work is primarily motivated by the complementary nature of protons and photons, considering both the robustness of target coverage and the optimality of OAR sparing, in the presence of delivery uncertainties (e.g., setup and range uncertainties). The major innovation of this work is the joint optimization of proton spots, photon fluences, and the number of proton/photon fractions. Specifically, this work will develop a new hybrid treatment planning method that (1) ensures the target dose uniformity of proton and photon dose components with hard constraints on proton and photon target doses to be a portion of the total prescription dose, i.e., the number of proton or photon fractions, and then (2) optimizes the number of fractions in addition to proton and photon plan variables. To the best of our knowledge, this is the first hybrid treatment planning method that can optimize the hybrid plan with needed target dose uniformity for proton and photon dose components, so that the hybrid plan can be separately delivered on proton or photon machine in the conventional fashion with uniform target dose per proton or photon fraction that is robust to day-to-day delivery uncertainties.

2. Methods and Materials

2.1. Hybrid proton-photon and fraction optimization

The proposed hybrid treatment planning method simultaneously optimizes proton spot x_p, photon fluence x_γ, and the proton fraction ratio r (i.e., number of proton fractions over total number of fractions, and therefore the photon fraction ratio is 1-r). The hybrid proton-photon and fraction optimization solves the following optimization problem

$$\begin{array}{c}\underset{(x,r)}{\min }F(x,r)=\underset{(x,r)}{\min }\sum _{j=1}^J{p}_{j}f\left(\left(d_{pj}+d_{\gamma j}\right),d\right)\\ +\sum _{j=1}^J{p}_{j}g\left(d_{pj},rd\right)+\sum _{j=1}^J{p}_{j}g\left(d_{\gamma j},(1-r)d\right).\\ \text{s.t.}\{\begin{array}{c}{d}_{pj}=A_{pj}{x}_p,d_{\gamma j}=A_{\gamma j}{x}_{\gamma }\\ x={\left[x_p{x}_{\gamma }\right]}^T\ge 0\\ 0\le r\le 1\end{array}\end{array}$$ (1)

In Eq. (1), f represents the sum of planning objectives on total dose d_p+d_γ with dose constraints d and is based on dose-volume constraints for clinical target volume (CTV) and OAR; g denotes the sum of planning objectives on proton dose d_p or photon dose d_γ with dose constraints rd or (1-r)d, which is to ensure the target dose uniformity per fraction so that the hybrid plans can be separately and robustly delivered on proton or photon machine. More details and explicit formulas for f and g can be found in Section 1 of the supplementary material.

Eq. (1) considers the robust optimization with respect to CTV, using probabilistic formulation [3,6]. The range and setup uncertainties are modeled by dose influence matrices A_p and A_γ: for example, A_pj is with respect to either range or setup uncertainty with its occurrence probability p_j. Note that the robustness of proton and photon dose components is also enforced, so that each fraction can be separately delivered with robustness to treatment uncertainties.

Since the photon dose falloffs are relatively slow, the photon planning with respect to planning target volume (PTV), i.e., an expansion of CTV with the same setup margin (e.g., PTV is a 5mm expansion of CTV for 5mm setup uncertainty), is clinically sufficient and used [30], instead of robust photon optimization with respect to CTV. This was also shown to hold for hybrid RT [14]. Therefore, without loss of generality and for computational efficiency, we solve the following reformulation of Eq. (1)

$$\begin{array}{c}\underset{(x,r)}{\min }F(x,r)=\underset{(x,r)}{\min }\sum _{j=1}^J{p}_{j}f\left(\left(d_{pj}+d_{\gamma }\right),d\right)\\ +\sum _{j=1}^J{p}_{j}g\left(d_{pj},rd\right)+g\left(d_{\gamma },(1-r)d\right)\\ \text{s.t.}\{\begin{array}{c}{d}_{pj}=A_{pj}{x}_p,d_{\gamma }=A_{\gamma }{x}_{\gamma }\\ x={\left[x_p{x}_{\gamma }\right]}^T\ge 0\\ 0\le r\le 1\end{array}\end{array},$$ (2)

where the setup uncertainties for photon dose are handled by PTV instead of robust optimization with respect to CTV. The details can be found in Section 1 of the supplementary material.

2.2. Optimization algorithm

The hybrid proton-photon and fraction optimization problem Eq. (2) is nonconvex. Here we solve it using iterative convex relaxation (ICR) [31–34], with inner loops solved by alternating direction method of multipliers (ADMM) [18,19].

To handle the nonconvexity of DVH constraints, we first apply ICR to Eq. (2), i.e.,

$$\begin{array}{c}\left(x^{m+1},r^{m+1}\right)=\underset{(x,r)}{\arg \min }F\left(x^m,r^m,{\text{Ω}}^m\right)\\ \text{s.t.}x\ge 0,0\le r\le 1\end{array},$$ (3)

$${\text{Ω}}^{m+1}=H\left(A{x}^{m+1},r^{m+1}\right).$$ (4)

where Eq. (3) is now a convex subproblem for a fixed active set Ω^m for DVH constraints and Eq. (4) updates the active set Ω^m+1 based on x^m+1 and r^m+1.

Next we provide ADMM solution algorithm for solving the convex subproblem Eq. (3). First, dummy variables z and z_r with dummy constraints z=x and z_r=x_r are introduced to decouple the non-negative constraint for x and the box constraint for r respectively from the data fidelity term F(x,r), and then additional auxiliary variables u and u_r are introduced for reformulating the constraints z=x and z_r=x_r to least-square penalty terms with corresponding regularization weights λ and λ_r. That is, the ADMM solution is to optimize the following augmented Lagrangian of. Eq (3)

$$\begin{array}{c}L\left(x,r,z,z_r,u,u_r\right)=F(x,r)+\lambda \parallel x-z+u{\parallel }^2+{\lambda }_r{\Vert r-z_r+u_r\Vert }^2\\ \text{s.t}\mathrm{.}\text{,}z\ge 0,0\le z_r\le 1\end{array}.$$ (5)

Based on Eq. (5), the subproblem Eq. (3) can be solved by the following inner loop

$$\{\begin{array}{c}\left(x^{n+1},r^{n+1}\right)=\underset{(x,r)}{\arg \min }L\left(x,r,z^n,z_r^n,u^n,u_r^n\right)\\ z^{n+1}=\max \left(0,x^{n+1}+u^n\right)\\ z_r^{n+1}=\min \left(1,\max \left(0,r^{n+1}+u_r^n\right)\right)\\ u^{n+1}=u^n+x^{n+1}-z^{n+1}\\ u_r^{n+1}=u_r^n+r^{n+1}-z_r^{n+1}\end{array}.$$ (6)

The solution to the first problem of Eq. (6) is provided in Section 2 of the supplementary material, and implementation details of the overall ICR algorithm with ADMM inner loop are provided in Section 3 of the supplementary material.

2.3. Validation and comparison of methods

Hybrid proton-photon and fraction optimization method (“Hybrid-F”) was validated against hybrid proton-photon optimization method without fraction optimization (“Hybrid”), where r is given before the optimization and a constant during the optimization by solving the following x-only problem

$$\begin{array}{c}\underset{x}{\min }F(x)=\underset{x}{\min }\sum _{j=1}^J{p}_{j}f\left(\left(d_{pj}+d_{\gamma }\right),d\right)\\ +\sum _{j=1}^J{p}_{j}g\left(d_{pj},rd\right)+g\left(d_{\gamma },(1-r)d\right)\\ \text{s.t.}\{\begin{array}{c}{d}_{pj}=A_{pj}{x}_p,d_{\gamma }=A_{\gamma }{x}_{\gamma }\\ x={\left[x_p{x}_{\gamma }\right]}^T\ge 0\end{array}\end{array}.$$ (7)

The only difference between Hybrid-F and Hybrid is that Hybrid-F also optimizes r. Therefore, a sequence of Hybrid optimization problems is solved with fixed r values from 0 to 1 respectively, and then the r value corresponding to the smallest planning objective F is selected as r_opt via solving Hybrid problems Eq. (7). On the other hand, the optimized r value via solving the Hybrid-F problem Eq. (2) is r*. The fraction optimization during Hybrid-F will be validated by the comparison between r_opt and r*.

On the other hand, the photon-only optimization method (“Photon”) corresponds to Hybrid with r=0, while the proton-only optimization method (“Proton”) corresponds to Hybrid with r=1. The plan-quality comparison will be made between Photon (r=0), Hybrid (r=r_opt) and Proton (r=1). For fair comparison, all treatment planning methods shared the same planning objective and robust optimization algorithm via ICR and ADMM.

Moreover, “Hybrid” will be compared with “P-Hybrid”, i.e., the pseudo-hybrid method, in which photon and proton plans are optimized separately and then the hybrid plan is formed as a fraction-weighted summation of proton and photon plans.

In the tables, doses and volumes are normalized to a percentage of the prescription dose and the total volume (e.g., CTV or OAR). The conformal index (CI) is evaluated, which is defined as V_100,CTV²/(V_CTV×V₁₀₀) (V_100,CTV: CTV volume receiving at least 100% of prescription dose; V_CTV: CTV volume; V₁₀₀: total body volume receiving at least 100% of prescription dose; ideally CI=1). Note that the dose parameters and normalization are with respect to probabilistically weighted total dose that accounts for all uncertainty scenarios when needed (i.e., for Proton, Hybrid and Hybrid-F).

2.4. Clinical cases

The following representative clinical cases are chosen to validate the proposed hybrid proton-photon and fraction optimization method and demonstrate the plan quality from hybrid treatment planning in comparison with photon-only and proton-only treatment planning.

Abdomen case

The target prescription dose is 55Gy in 25 fractions. the setup uncertainty is 5mm and the range uncertainty is 5%. The abdomen tumor is surrounded by the bowel, which is an OAR in this case and has a L2-norm planning constraint to minimize the bowel dose as low as reasonably achievable (ALARA), in addition to other DVH dose constraints (Table S1 of the supplementary material). There are 3 proton beams (0º, 120º, 240º) and 7 photon beams (0º, 50º, 100º, 150º, 210º, 260º, 310º).

Lung case

The target prescription dose is 60Gy in 30 fractions. the setup uncertainty is 5mm and the range uncertainty is 5%. The lung tumor is close to the heart, which is an OAR in this case and has a L2-norm planning constraint to minimize the heart dose ALARA. A list of planning constraints is summarized in Table S1 of the supplementary material. The beam angles for the lung case are the same as the abdomen case.

Head-and-neck (HN) case

The target prescription dose is 69.96Gy in 33 fractions. the setup uncertainty is 3mm and the range uncertainty is 3.5%. A list of planning constraints is summarized in Table S1 of the supplementary material, among which a L2-norm planning constraint is enforced for the larynx as an OAR for the HN case. There are 4 proton beams (45º, 135º, 225º, 315º) and 7 photon beams (0º, 50º, 100º, 150º, 210º, 260º, 310º).

Brain case

The target prescription dose is 60Gy in 30 fractions. the setup uncertainty is 3mm and the range uncertainty is 3.5%. The brain tumor is close to the brainstem, which is an OAR in this case and has a L2-norm planning constraint to minimize the heart dose ALARA. A list of planning constraints is summarized in Table S1 of the supplementary material. The beam angles for the brain case are the same as the HN case.

3. Results

3.1. Hybrid-F v.s. Hybrid

The difference between Hybrid-F Eq. (2) and Hybrid Eq. (7) is that the fraction ratio r is optimized in Hybrid-F, while r is a given value and remains a constant in Hybrid. This section validates Hybrid-F using Hybrid as the benchmark. Specifically, a sequence of Hybrid problems with various r values was solved, and the r value, namely r_opt, corresponding to the lowest optimized planning objective was identified; next Hybrid-F was solved with the optimized fraction ratio r*, which was compared with r_opt.

Note that (1) r* may not be exactly the same as r_opt and (2) the optimized planning objective from Hybrid-F, namely F*, may not be the same as that from Hybrid using r_opt, namely F_opt. This is due to differences in optimization algorithms for solving Hybrid-F and Hybrid, although the only difference in the formulation of Hybrid-F and Hybrid is whether the fraction ratio r is optimized or not.

3.1.1. Comparison of plan quality

Fig. 1 presents optimized planning objectives versus number of the proton fraction using Hybrid, and optimized planning objective and optimized number of proton fraction using Hybrid-F. The number of the proton fraction is denoted by n. As shown in Fig. 1, the n* value was the same as or close to the n_opt value, with n*=15, 11, 21, 21 and n_opt=15, 10, 18, 20 for abdomen, lung, HN and brain cases respectively; the F* value was also close to the F_opt value, with F*=6.5, 8.8, 16.2, 8.2 and F_opt=6.6, 8.5, 17.2, 8.6 for abdomen, lung, HN and brain cases respectively. For each sub-figure of Fig. 1, the leftmost plot (n=0) corresponded to the photon-only planning, while the rightmost plot corresponded to the proton-only planning. More details for the comparison of plan quality (e.g., DVH parameters for CTV and OAR) between Hybrid using r_opt and Hybrid-F are provided in Table S2 of the supplementary material, which also showed that two methods had comparable plan quality.

Figure 1. Hybrid-F v.s. Hybrid.

Hybrid-F optimizes r and x, while Hybrid optimizes only x for a given r. In each figure, the optimized plan objective F is plotted for a sequence of proton fraction n via Hybrid, i.e., the blue curve, and (F_opt, n_opt) corresponds to the smallest F and the corresponding fraction number, i.e., the blue solid dot; on the other hand, (F*, n*) denotes the optimized planning objective and fraction ratio via Hybrid-F, i.e., the orange solid dot. The proposed hybrid proton-photon and fraction optimization is validated through the comparison of Hybrid-F and Hybrid.

3.1.2. Comparison of computational efficiency

In terms of computational time, it took Hybrid-F about the similar computational time as Hybrid (for each n), i.e., 23 v.s. 21±2 minutes for abdomen, 37 v.s. 53±13 minutes for lung, 43 v.s. 36±9 minutes for HN, and 14 v.s. 12±1 minutes for brain. However, because one needs to solve Hybrid problems with all or many n values in order to determine (r_opt, F_opt), Hybrid-F is much more efficient since one needs to solve the Hybrid-F problem only once to obtain (r*, F*).

3.2. Hybrid v.s. Photon/Proton

3.2.1. Plan quality

Hybrid (r=r_opt) had smaller total planning objective value than Photon (r=1) or Proton (r=1), as shown in Fig.1 and Fig.2–5(f).

Figure 2. Abdomen.

Dose plots from (a) Photon, (b) Hybrid, and (c) Proton. DVH for (d) CTV and (e) bowel. (f) optimized planning objectives for CTV, OAR (including all OARs), and TOTAL (sum of CTV and OAR). The dose plot window is [0%, 120%]. 110%, 100%, 80%, 50% isodose lines and CTV are highlighted in dose plots. The robustness variance (RV) that measures the DVH variation across all uncertainty scenarios is indicated in DVH plots.

Figure 5. Brain.

Dose plots from (a) Photon, (b) Hybrid, and (c) Proton. DVH for (d) CTV and (e) brainstem. (f) optimized planning objectives for CTV, OAR (including all OARs), and TOTAL (sum of CTV and OAR). The dose plot window is [0%, 120%]. 110%, 100%, 80%, 50% isodose lines and CTV are highlighted in dose plots. The robustness variance (RV) that measures the DVH variation across all uncertainty scenarios is indicated in DVH plots.

As illustrated in Fig.2–5(f), (1) Hybrid greatly reduced CTV planning objective value from Proton, since the use of photon dose in Hybrid improved CTV coverage in the presence of setup and range uncertainty, while Hybrid was also better than Photon as well for HN and brain; (2) Hybrid had smaller OAR planning objective value than Photon or Proton, owing to increased degrees of freedom using both protons and photons to optimize the OAR sparing.

As demonstrated in Fig.2–5(d), (1) Hybrid had lower DVH CTV curves than Proton, and the improvement was substantial for lung; (2) Hybrid also had substantially lower DVH CTV curves than Photon for HN and brain. The CTV dosimetric parameters in Table 1 also provide quantitative evidences to two aforementioned observations: for example, Hybrid had larger CI values than Photon or Proton, which shows that Hybrid had the best target dose conformality.

Table 1. Dosimetric parameters.

From left to right: conformity index (CI), D₉₅ and D_max of CTV, D_mean of body, optimized objective value, D_max, and D_mean of OAR. Here OAR refers to bowel, heart, larynx, and brainstem for abdomen, lung, HN, and brain cases respectively. The dose is in percentile with respect to the prescription dose.

Case	Method	CI	D95, CTV	Dmax, CTV	Dmean, body	FOAR	Dmax, OAR	Dmean, OAR
Abdomen	Proton	0.63	100	104	0.6	6.0	100	4.4
	Photon	0.55	100	105	1.7	6.3	100	5.6
	Hybrid	0.68	100	104	1.0	5.4	96	5.3
Lung	Proton	0.66	97	114	3.6	3.3	98	1.8
	Photon	0.67	100	106	6.3	1.9	97	2.3
	Hybrid	0.74	99	107	5.2	1.7	92	1.8
HN	Proton	0.73	100	119	2.2	11.3	76	8.4
	Photon	0.69	100	133	3.7	14.7	77	14.0
	Hybrid	0.79	100	119	2.8	8.7	69	9.3
Brain	Proton	0.80	100	110	2.3	4.0	80	3.7
	Photon	0.75	100	113	4.7	4.7	72	7.8
	Hybrid	0.86	100	109	3.0	3.4	72	5.0

All OAR planning constraints in Table S1 of the supplementary material were satisfied, and therefore the only contributing OAR planning constraint to the final total planning objective is the L2-norm planning objective that minimizes OAR dose ALARA, i.e., on bowel, heart, brainstem and larynx for abdomen, lung, HN and brain case respectively. Therefore, the OAR in Fig.2–5(e) and Table S2 refers to bowel, heart, brainstem and larynx respectively. As shown in Fig.2–5(e), (1) Hybrid generally preserved the advantage of Proton for sparing OAR; (2) moreover, Hybrid provided the best overall OAR sparing, especially at the high-dose region. These observations were also quantitatively verified by the OAR dosimetric parameters in Table 1.

All these improvements in plan quality demonstrate the effectiveness of Hybrid to synergize protons and photons.

3.2.2. Robustness

The DVH curves from all uncertainty scenarios are plotted in (d) for CTV and (e) for an OAR in Fig. 2–5, which indicate that Hybrid had narrower DVH volume envelopes per fixed dose than Proton, e.g., the zoom-in plots in Fig. 2–5 (d) and (e).

The plan robustness is also quantitively evaluated using the so-called robustness variance (RV): RVp defined at p% dose is the averaged variation between maximum and minimum percentage volume at (p-1)%, p% and (p+1)%. Corresponding to the zoom-in plots of Fig. 2.−5(d), RV₁₀₀ of CTV for Hybrid and Proton is 9.9 v.s. 14.3 for abdomen, 14.3 v.s. 18.3 for lung, 5.7 v.s. 7.9 for HN, and 5.6 v.s. 7.4 for brain case. Corresponding to the zoom-in plots of Fig. 2.−5(e), RV₁₀ of OAR for Hybrid and Proton is 1.5 v.s. 2.5 in abdomen, 2.9 v.s. 5.2 in lung, 4.8 v.s. 7.4 in brain, and RV₂₀ of OAR in HN is 11.8 v.s. 16.8.

Both visual inspection and quantitative evaluation indicate that Hybrid was more robust than Proton.

3.2.3. Target dose uniformity per fraction for Hybrid

The dose parameters for photon dose component (H-photon) and proton dose component (H-proton) from Hybrid are summarized in Table S2 of the supplementary material, while CTV DVH from H-photon and H-proton are plotted in Fig. S1 of the supplementary material, in comparison with CTV DVH from Photon, Proton and Hybrid. Both quantitative evaluation with dosimetric parameters and visual inspection of CTV DVH conclude that Hybrid had uniform target dose per photon or proton fraction.

3.3. Comparison under robust photon optimization

Here the comparison was made between various methods with robust photon optimization, i.e., the photon component with the optimization target set to CTV and setup uncertainty accounted for by robust optimization (see Eq. (1)), while in previous sections the optimization target was set to PTV for the photon component without using robust photon optimization (see Eq. (2)).

3.3.1. Hybrid-F v.s. Hybrid

Similar to Section 3.1, as shown in Fig. 6, the comparison of Hybrid-F and Hybrid suggests that (1) the optimized proton fraction number n* via Hybrid-F was the same as exhaustively-searched n_opt from Hybrid; (2) the optimized plan objective F* via Hybrid-F was the same as the best plan objective F_opt from Hybrid. Therefore, the effectiveness of Hybrid-F for jointly optimizing fraction ratio and proton/photon variables is also validated using Hybrid as the benchmark for the scenario with robust photon optimization.

Figure 6. Comparison of optimization objective values for Hybrid-F, Hybrid and P-Hybrid.

Hybrid-F optimizes r and x; Hybrid optimizes only x for a given r; P-Hybrid is a r-weighted sum of proton and photon plans. The optimized plan objective F is plotted for a sequence of proton fraction n via Hybrid, i.e., the blue curve, and (F_opt, n_opt) corresponds to the smallest F and the corresponding fraction number, i.e., the blue solid dot; for P-Hybrid, the plan objective F from the r-weighted sum of proton and photon plans is plotted for a sequence of proton fraction n, i.e., the green curve, and (F_opt’, n_opt’) corresponds to the smallest F and the corresponding fraction number, i.e., the green solid dot; on the other hand, (F*, n*) denotes the optimized planning objective and fraction ratio via Hybrid-F, i.e., the orange solid dot. The proposed hybrid proton-photon and fraction optimization is validated through the comparison of Hybrid-F and Hybrid. Moreover, the comparison of P-Hybrid with Hybrid or Hybird-F suggests that the hybrid plan (Hybrid or Hybird-F) as a result of simultaneously optimized proton-photon plans has better plan quality than the pseudo-hybrid plan (P-Hybrid) with separately optimized proton-photon plans.

On the other hand, the comparison of Fig. 6 and Fig. 1(b) suggests that robust photon optimization to CTV had the similar results with non-robust photon optimization to PTV, which reassures the computational simplification using PTV (i.e., Eq. (2)) instead of CTV (i.e., Eq. (1)) for optimizing photon dose component [14].

3.3.2. Hybrid v.s. Photon/Proton

Similar to Section 3.2, as shown in Fig. 7 and Table S3 for the scenario of robust photon optimization, Hybrid substantially improved the target coverage from Proton, and attained better OAR sparing and overall planning objective values than either Photon or Proton. For example, as shown in Fig. 7(g), Hybrid has much smaller CTV objective value than Proton, and much smaller OAR objective value than Photon, while Hybrid was the smallest in both OAR and total objective values.

Figure 7. Comparison of Photon, Proton, Hybrid, and P-Hybrid for Lung with robust photon optimization.

Dose plots from (a) Photon, (b) Proton, (c) Hybrid, and (d) P-Hybrid. DVH for (e) CTV and (f) heart. (g) optimized planning objectives for CTV, OAR (including all OARs), and TOTAL (sum of CTV and OAR). (h) CTV DVH plots for proton (H-photon) and proton (H-proton) dose components jointly optimized via Hybrid. The dose plot window is [0%, 120%]. 110%, 100%, 80%, 50% isodose lines and CTV are highlighted in dose plots. The robustness variance (RV) that measures the DVH variation across all uncertainty scenarios is indicated in DVH plots.

3.4. Hybrid v.s. P-Hybrid

Hybrid and P-Hybrid are compared here to assess the improvement in plan quality from the pseudo-hybrid plan (P-Hybrid) with separately optimized proton-photon plans to the hybrid plan (Hybrid) from simultaneously optimized proton-photon plans.

As shown in Fig. 6 and Fig. 7(g), Hybrid had smaller total planning objective value than P-Hybrid; as indicated Fig. 7(g), Hybrid had smaller CTV and OAR planning objective value as well than P-Hybrid; as summarized in Table S3, Hybrid had better target dose conformality indicated by higher CI value than P-Hybrid, i.e., 0.71 v.s. 0.69. Therefore, Hybrid improved plan quality from P-Hybrid through the joint optimization of proton and photon variables.

4. Discussion and Conclusion

We have developed a hybrid proton-photon and fraction optimization method that can generate hybrid plans with uniform target dose per photon or photon fraction, for the purpose of safe and robust dose delivery of hybrid plans separately on existing photon and photon machines. And we have demonstrated that this hybrid planning method is better than either proton-only or photon-only treatment planning method, in overall plan quality, i.e., the combination of robust CTV coverage and optimized OAR sparing with joint optimization degrees of freedom via complementary proton and photon beams.

The need for optimizing the number of proton and photon fractions is shown in Fig. 1, where the curves of (n, F) suggest that the optimized planning objective F via hybrid planning depends on the number of proton fractions n. Without optimizing n, the determination of the optimal number of fractions n_opt is a combinatorial problem that is even time-consuming for the optimization algorithm (i.e., Hybrid), for which the proposed hybrid proton-photon and fraction optimization method (i.e., Hybrid-F) is much more efficient, e.g., with the computational time that is similar to the one-time computation of Hybrid. Moreover, for human planners, the exhaustive trial-and-error search for n_opt is extremely time-consuming and practically impossible, not even to mention that no currently-available treatment planning system (TPS) is capable of hybrid planning that jointly optimizes proton and photon variables for better plan quality than proton-only and photon-only planning.

The hybrid planning here assumes that proton and photon fractions are delivered separately, and thus enforces target dose uniformity and robustness for proton and photon dose components in Eq. (1). However, if proton and photon dose can be delivered in the same fraction (e.g., as hypothesized in [16]) so that the change in patient anatomy is negligible, it may not be necessary to regularize the uniformity of dose to targets in Eq. (1), and the removal of such component-wise uniformity regularization should further improve overall plan quality for CTV and OAR.

It is a conventional wisdom that PTV is sufficient to account for setup uncertainty for photon RT. Moreover, the benefits that the photon dose component targets at PTV instead of CTV are twofold: (1) if targeting at CTV, the setup uncertainty for photon dose and its contribution to total dose will need to be modeled and therefore increase the computational cost; (2) the daily setup uncertainties for proton and photon fractions may not have the same shifts, and the use of PTV for photon dose component mitigates the need of modeling different setup uncertainty scenarios in robust optimization, although this may be only technically relevant given that current robust modeling is ad-hoc, e.g., 6 rigid shifts along x, y, and z directions.

A limitation of this work is that we have not explicitly modeled the fractionation effect, which has been done for hybrid proton-photon RT [12,13]. However, because the target dose is constrained to be the same for both proton and photon dose per fraction in our method (where the proton dose is a multiple of physical dose by a relative biological effectiveness (RBE) constant 1.1), the modeling of fractionation effect may not be needed for the target. On the other hand, because the target dose is spatially constant, the entire proton and photon dose maps from our method are less inhomogeneous than other methods [12,13], and thus the impact of not modeling fractionation effect for normal tissues should be less. However, the actual impact and need of modeling fractionation effect for our method will need to be assessed for both target and OAR, which will be a future work because extensive efforts seem to be needed to address optimization challenges for our method (that optimizes the number of proton and photon fractions as well as proton and photon plan variables) with respect to biologically summed dose (which becomes quadratic with respect to dose per fraction).

Another limitation is the plan deliverability is not accounted for in this work. Here the hybrid planning is with respect to proton spot weights and photon fluences, which need to be further transformed respectively to proton spot weights subject to minimum monitor-unit constraint and photon MLC apertures for plans to be deliverable. The photon dose component can be modelled using VMAT instead, which is nowadays more common than IMRT in clinic.

The full picture of tradeoffs from hybrid planning in the context of multi-criteria optimization, in comparison with proton-only and photon-only planning, is needed to fully assess and utilize its potential for optimized combination of CTV coverage and OAR sparing.

The hybrid proton-photon and fraction optimization method proposed here is general, in the sense it should be applicable to other multi-modality RT problems, such as multi-ion RT [20–22] using proton, carbon or helium, photon-ion hybrid RT [23, 24], photon-electron RT [25], proton-photon-electron RT [26].

Figure 3. Lung.

Dose plots from (a) Photon, (b) Hybrid, and (c) Proton. DVH for (d) CTV and (e) heart. (f) optimized planning objectives for CTV, OAR (including all OARs), and TOTAL (sum of CTV and OAR). The dose plot window is [0%, 120%]. 110%, 100%, 80%, 50% isodose lines and CTV are highlighted in dose plots. The robustness variance (RV) that measures the DVH variation across all uncertainty scenarios is indicated in DVH plots.

Figure 4. HN.

Dose plots from (a) Photon, (b) Hybrid, and (c) Proton. DVH for (d) CTV and (e) larynx. (f) optimized planning objectives for CTV, OAR (including all OARs), and TOTAL (sum of CTV and OAR). The dose plot window is [0%, 120%]. 110%, 100%, 80%, 50% isodose lines and CTV are highlighted in dose plots. The robustness variance (RV) that measures the DVH variation across all uncertainty scenarios is indicated in DVH plots.