Single-field-uniform-dose-per-fraction simultaneous dose and dose rate optimization (SFUDPF-SDDRO) method for proton FLASH therapy

Abstract

Background:

The FLASH effect can significantly reduce radiation-induced normal tissue damage while maintaining tumor control, but requires ultra-high dose rates and high doses.

Purpose:

This work proposes a single-field-uniform-dose-per-fraction simultaneous dose and dose rate optimization (SFUDPF-SDDRO) method for proton FLASH radiotherapy to ensure both dose rate and dose meet FLASH effect thresholds.

Methods:

The SFUDPF method focuses on delivering the prescription dose for each fraction from only a single field instead of multiple fields, which inherently supports the ultra-high dose rate and high dose necessary for the FLASH effect. We performed retrospective FLASH treatment planning utilizing SFUDPF-SDDRO on 4 clinical head-and-neck (HN) cases for this study. SFUDPF planning involves delivering each prescription fraction (8 Gy x 5 fx) in 1 beam angle as opposed to multiple beam angles per fraction for IMPT. For each beam delivery, we maximized the FLASH effect in a 1 cm expansion of the HN CTV (CTV+1cm) by enforcing FLASH dose-rate and dose thresholds of 40 Gy/s and 5 Gy respectively in this region. The pencil-beam-scanning dose rate (PBSDR) was calculated voxel-wise by modeling the raster-scanning spot trajectory, while neglecting energy switching times under the assumption of a range modulator capable of expanding a single-energy beam into a spread-out Bragg peak (SOBP). Robust optimization at 3mm/3.5% was performed to address setup and range uncertainties. We employed iterative convex relaxation and alternating direction method of multipliers algorithms to solve the non-convex optimization problem posed by the SFUDPF-SDDRO model. The FLASH effect was modeled within this work by multiplying the proton dose with a constant 0.7 dose modification factor for voxels fulfilling the dose-rate and dose thresholds to obtain the FLASH effective dose (FED). Effects of FLASH sparing maximization via SFUDPF-SDDRO are verified by comparing with IMPT and VMAT on plan qualities such as (i) high-dose area sparing, (ii) conformity index (CI), and (iii) OAR doses.

Results:

FLASH RT via SFUDPF-SDDRO compared with IMPT and VMAT was evaluated for four clinical HN cases with different tumor geometries. When compared with their VMAT counterparts, SFUD-SDDRO achieved a considerable reduction of FED for OAR directly adjacent to the CTV. Specifically in case 1, the brainstem D_1% decreased from 87.57% to 62.26%, and the spinal cord D_10% decreased from 87.36% to 60.74%; in case 2, the D_10% of the carotid decreased from 102.46% to 63.30%; in case 3, the D_10%of the oral cavity decreased from 94.72% to 62.66%, and the D_10% of the oropharynx decreased from 102.5% to 69.09%; in case 4, the D_10% of the oral cavity decreased from 88.56% to 59.81%. The SFUDPF-SDDRO achieved a satisfactory CI in terms of FED, indicating that conformity was not sacrificed to achieve the FLASH effect.

Conclusion:

The proposed SFUDPF-SDDRO method is feasible and shows potential clinical benefits for FLASH treatment planning. Maximizing the FLASH effect within a 1 cm ring around the target substantially limits high-dose spillage and enhances OAR sparing compared with conventional approaches.

1. Introduction

FLASH radiotherapy (FLASH-RT) refers to the delivery of radiation doses at ultra-high dose-rates (UHDR) to induce differential biological responses between normal and cancer cells [1–4]. This differential response is known as the FLASH effect (FE) and its induction has been demonstrated in pre-clinical radiobiological experiments [5,6]. FE can enhance the therapeutic ratio of RT and allow for new clinical possibilities such as dose escalation to eradicate radio-resistant tumors and/or reduced radiation-induced normal tissue toxicities [7,8]. While current technological advances have enabled dosimetrically accurate [9,10] FLASH deliveries using electrons, x-rays and protons [11], the exact radiobiological mechanisms of the FE are not well-understood [12] with several competing hypothetical physical-chemical and biological mechanisms proposed in literature [12,13].

Among different FLASH-RT radiation modalities, there have been an increased interest in proton FLASH deliveries due to their dosimetric advantages in inducing FE more effectively at deeper depths compared to clinical electron beams and the growing availability of FLASH-capable medical particle accelerators [10,11,14–18]. It’s clear that the key to achieving the FLASH effect relies on ultra-high dose rates. Existing treatment planning methods for CONV RT have no requirements for dose rate, so they only optimize the dose distribution without optimizing the dose rate distribution [19, 20]. Therefore, the research on optimizing the FLASH treatment plan must take into account the optimization of dose and dose rate distribution. Most of the relevant studies adopt the “dose-average dose rate” (DADR) proposed by Van de Water et al. [19] to quantify the dose rate. Although DADR can give a first order estimate of the dose rates of the treatment plans, it does not account for spot scanning trajectory and beam-off time between spots. To overcome this limitation, we applied the “PBS dose rate” (PBSDR), presented by Folkerts et al [21], in this study.

Proton FLASH deliveries are technologically challenging [5] which limits the realization of its full theoretical clinical benefits. For example, the use of transmission proton beams to achieve a high nozzle current leads to increased integral dose delivered to adjacent organs-at-risk (OAR) [22,23]. Recently proton Bragg peak conformal FLASH [24–26] was proposed to overcome this problem, which can provide better conformity index (CI) and OAR sparing compared with transmission proton beams by using experimental techniques such as patient-specific ridge filters. FLASH-specific simultaneous dose and dose rate optimization (SDDRO) methods [21,26–29] were proposed to generate FLASH plans with high plan quality and high FLASH effect coverage.

FLASH RT based on ultra-high dose rate (>40Gy/s) has recently been studied actively because of a range of in vivo studies indicating that it can improve the sparing of organs at risk (OAR) while keeping the same tumor control effect compared to conventional (CONV) dose rate RT [2–4,6,12,18,30–33]. Experiments by Brita Singers Sørensen et al. further showed that delivering excessively low single doses may compromise the FLASH effect [34]. However, multi-field FLASH plans still have trouble meeting thresholds of FLASH effect. However, conventional multi-field plans delivering the fractional dose through several fields often fail to meet the dose thresholds required to trigger the FLASH effect, unless very high prescription doses are applied [22]. This suggests that splitting the dose across multiple fields within each fraction may hinder the FLASH effect due to the still uncertain “dose threshold.” Therefore, we consider delivering the prescription dose for each fraction from only a single field instead of multiple fields, which inherently supports the ultra-high dose rates and high dose necessary for the FLASH effect.

In this work, we developed a new Bragg-peak FLASH treatment planning method which is termed single-field-uniform-dose-per-fraction SDDRO (SFUDPF-SDDRO). This new method aims to meet the high dose rates and high dose thresholds for FLASH effect while improving conformity of Bragg peak FLASH proton treatments with multiple gantry angles. Compared to the previous Bragg peak FLASH, our proposed method will deliver 100% of the prescription dose for each fraction from only one field instead of multiple fields. By setting up the treatment this way, we can increase the dose and dose rate received by OAR in a single fraction, making it easier to meet the FLASH effect threshold. Moreover, single field uniform dose per fraction can improve the treatment robustness. We also employ SDDRO technique to simultaneously optimize the dose and dose rate to maximize FE inductions in abutting OAR relative to CTV. This work evaluates relevant dose metrics such as FLASH effective dose [35], high-dose sparing, OAR sparing and CI values, in comparison with conventional IMPT and VMAT.

2. Methods and materials

2.1. SFUDPF-SDDRO

We develop SFUDPF-SDDRO which allows for the creation of robust and conformal proton FLASH plans using SFUD techniques. In this approach, pencil beam fluences are individually optimized for each field with the primary objective of achieving a homogeneous dose distribution across the target volume. We optimize the treatment plans by targeting the limited dose for each structure and the dose for each single field, incorporating constraints on the FLASH dose and dose rate and minimum monitor unit (MMU). The proposed new method SFUDPF-SDDRO model can be established:

$$\begin{gathered} \underset{x}{\mathrm{m}\mathrm{i}\mathrm{n}}{\sum }_{j=1}^{N_S}{\omega }_{1,j}{\Vert {\sum }_{i=1}^{N_f}{d}_i^j-d_0\Vert }_2^2+{\sum }_{i=1}^{N_f}{\omega }_{2,i}{\Vert d_i^{ctv}-d_{\mathrm{f}}\Vert }_2^2 \\ \left\{\begin{array}{c}\gamma \ge {\gamma }_c\\ d\ge d_c\\ x\in \left\{0\right\}\cup [g,+\mathrm{\infty })\end{array}\right. \end{gathered}$$ (1)

Herein, $x$ denotes the beam fluence map to be determined. $N_s$ represents the number of voxels for each structure and $N_f$ represents the number of fields. $d_i^j$ and $d_i^{ctv}$ denote the dose vector of each structure and CTV in $i^{th}$ field, respectively. ${\omega }_{1,\mathrm{j}}$ and ${\omega }_{2,\mathrm{i}}$ are objective weights. $d_0$ and $d_f$ denote the specified dose for each structure and field. The constraints in Eq. (1) are (1) the FLASH dose rate constraint on $\gamma$, (2) the FLASH dose constraint on $d$ and (3) the MMU constraint on the spot weight $x$. In this study, the optimization was based on physical dose and biological dose was not considered. The physical dose $d$ here can be calculated by the following equation,

$$d=\sum _{j=1}^{N_x}{D}_{ij}{x}_j,i\le N_d$$ (2)

In Eq. (2), $D$ represents the dose influence matrix, $N_x$ denotes the count of proton spots, and $N_d$ signifies the quantity of spatial voxels.

2.2. PBSDR

Folkerts et. al. have developed a method of calculating the dose rate distribution of a PBS proton field, pencil-beam-scanning dose rate (PBSDR). For illustrative purposes, a 2D plane near the surface of a PBS field was shown in Fig 1(a). Fig 1(b) shows the PBS dose rate at point x is the quotient of [$D_i-2d_{th}$] and [$t_1-t_0$].

$${\gamma }_i=\frac{D_i-2d_{th}}{t^{\mathrm{*}}}$$ (3)

$$t^{\mathrm{*}}=t_1-t_0$$ (4)

Figure 1.

Panel (b) illustration of dose accumulation at the position of the red cross ($\overrightarrow{x}$) in panel (a). The orange dots in panel (a) represent PBS spots that contribute a significant dose to $\overrightarrow{x}$.

Here, $D_i$ is the total dose at voxel $i,d_{th}$ is the dose thresholds (5% of prescription dose), and $t^{\mathrm{*}}$ is the delivery time between thresholds (starts at $t_0$, ends at $t_1$).

As shown in Figure 1, the dose rate at the red cross ($\overrightarrow{x}$) is calculated by considering the surrounding proton spots. Panel (b) illustrates the temporal dose accumulation at this point, normalized to the total dose that would be delivered. As seen in Equation 3, the PBSDR effective irradiation time for each voxel is defined to be the time elapsed between the cumulative dose delivery between a lower and upper cumulative dose threshold. A voxel-wise PBSDR distribution was computed for each treatment field using these definitions.

2.3. Optimization algorithm

The iterative convex relaxation (ICR) and alternating direction method of multipliers (ADMM) was used to solve the optimization problem posed in this study. ADMM is an optimization algorithm used to solve convex optimization problems that involve separable objective functions and equality as well as inequality constraints. The algorithm aims to decompose the original optimization problem into smaller, more manageable subproblems and iteratively solve them until convergence [35].

In addressing our problem, we initially endeavor to transform Eq. (1) through the adjustment of the active voxel index $\Omega$ and $\Theta$. followed by the introduction of auxiliary variables $z$,

$$\begin{array}{c}\underset{x}{\mathrm{m}\mathrm{i}\mathrm{n}}\Vert \mathrm{A}(\mathrm{\Omega })\mathrm{x}-\mathrm{d}(\Omega ){\Vert }_2^2+\omega \Vert \mathrm{B}(\Theta )\mathrm{x}-\mathrm{b}(\Theta ){\Vert }_2^2\\ \mathit{s.t.}\left\{\begin{array}{c}x=z\\ z\in \left\{0\right\}\cup [g,+\mathrm{\infty })\end{array}\right.\end{array}$$ (5)

Here, the FLASH dose rate and dose constraints delineated in Eq. (1) are integrated into the model framework. Consequently, the initial segment of Eq. (5) encapsulates the optimization of both the total dose and individual-field dose, while the subsequent segment addresses the optimization of the FLASH dose rate and associated dosimetry.

The update of the active index set $\Omega$ is performed according to the dose-volume-histogram (DVH) constraints. The update of the active index set $\Theta$ is performed according to the FLASH dose and dose rate constraint, which are applied individually. CTV1cm corresponds to the 1 cm expansion surrounding the CTV for the maximization of FLASH effect inductions using SFUD-SDDRO for all treatment fields delivered. By arranging the dose and dose rate within the voxel in the CTV1cm area in descending order, voxels failing to meet the constraints are excluded. The calculation of $\Omega$ and $\Theta$ is simplified as follows,

$$\left\{\begin{array}{c}\Omega =H^1(x)\\ \Theta =H^2(x)\end{array}\right.$$ (6)

Then we can define the augmented Lagrangian as the follows,

$$L\left(x,z,u,\Omega ,\Theta \right)={‖\mathrm{A}(\Omega )\mathrm{x}-\mathrm{d}(\Omega )‖}_2^2+{\omega ‖\mathrm{B}(\Theta )\mathrm{x}-\mathrm{b}(\Theta )‖}_2^2+\frac{\rho }{2}{‖\mathrm{x}-\mathrm{z}+\mathrm{u}‖}_2^2$$ (7)

ADMM solves Eq. (4) by (1) decoupling DVH constraint, FLASH dose and dose rate constraint and MMU constraint using iterative convex relaxation, then updating the $L(x,z,u,\Omega ,\Theta )$ alternatively. Assume that the number of iterations of the algorithm is $k$, the update rule of ADMM is given in Table 1. The main methodological contribution of this work lies in modeling the single-field-per-fraction delivery approach and incorporating both dose and dose-rate constraints into the optimization formulation. The ICR+ADMM algorithm used for solving the model has indeed been presented in our previous work [31–34]. Therefore, we simplified the solution process and the more detailed solution process is given in the supplementary material.

Table 1.

The update rule of ADMM for SFUD-SDDRO

Algorithm 1: ADMM for SFUD-SDDRO
Input: $A$, $b,B,d,x_0=\mathbf{0},$ choose proper ρ > 0, set iteration number $K$.
1.	For $k=1,2,\dots ,K-1$
2.	${\Omega }_k=H^1(x_k)$
3.	${\Theta }_k=H^2(x_k)$
4.	$x_{k+1}={\left({A\left({\Omega }_k\right)}^{T}A\left({\Omega }_k\right)+\omega {B\left({\Theta }_k\right)}^{T}B\left({\Theta }_k\right)+\rho I\right)}^{-1}({A({\Omega }_k)}^{T}d+\omega {B({\Theta }_k)}^{T}b+\rho (z_k-u_k)$;
5.	$z_{k+1}^i=\left\{\begin{array}{c}\max \left(g,x_{k+1}^i+u_k^i\right),x_{k+1}^i+u_k^i\ge \frac{g}{2}\\ 0,otherwise\end{array}\right.$, $i=1,2,\dots ,N_x$
6.	$u_{k+1}=u_k+x_{k+1}-z_{k+1}$
7.	End
Output: ${\tilde{x}}_K^i=\left\{\begin{array}{c}\max \left(g,x_K^i\right),x_K^i\ge \frac{g}{2}\\ 0,otherwise\end{array}\right.$

2.4. FLASH effective dose via DMF

The FLASH radiobiological effects were modeled within this work by performing FLASH effective dose (FED) scoring. The FED $d_e$ is computed voxel by voxel, by multiplying the original dose d with a constant DMF if the FE is induced within the voxel, which can be expressed as,

$$d_e=d·DMF$$ (8)

The DMF can be defined via the product of two indicator functions, that is,

$$DMF=1+(\mathrm{f}-1)·1_{\gamma \ge {\gamma }_0}·1_{d\ge d_0}$$ (9)

Obviously, DMF equals $f$ only for voxels where both dose and dose rate FLASH constraints are satisfied; otherwise, DMF equals 1, indicating no FLASH effect. Independent studies that have successfully demonstrated the FLASH effect have reported a dose-modifying factor (DMF) of approximately 20–40% in favor of FLASH-RT relative to CONV-RT [4]. Accordingly, a representative intermediate value of 0.7 was adopted in this study, i.e., $f=0.7$. Inside the CTV, DMF is set to 1. To sum up, Ep. (8) can be re-expressed as,

$$d_e=\left\{\begin{array}{c}\begin{array}{cc}0.7d& \gamma \ge {\gamma }_0,d\ge d_0\end{array}\\ dtarget,orbelowthreshold\end{array}\right.$$ (10)

2.5. Materials

Retrospective FLASH treatment planning utilizing SFUD-SDDRO were performed on four clinical HN cases and subsequently compared with their IMPT and VMAT conventional counterparts. As mentioned earlier, SFUD-SDDRO FLASH planning involves delivering each prescription fraction (8Gy × 5fx) in 1 beam angle as opposed to multiple beam angles per fraction for IMPT. The radiation fields and the fractions of each field for 4 cases are listed in Table 2. Within CTV1cm, the lower dose-rate and dose thresholds to trigger the FLASH effect were set at 40 Gy/s and 5 Gy respectively. Robust optimization for dose coverage was performed at 3.5%/3mm to address setup and range uncertainties for FLASH and IMPT plans. For VMAT plans, we optimized directly to the PTV which is a 3 mm expansion of the CTV. All plans were generated via the open-source radiation treatment planning software, MatRad [36]. The spot width was 5 mm on the influence matrices generation and the lateral space was 3 mm on 3 mm3 dose grid. The CI is defined in Equation 11 and scored for all VMAT, IMPT and FLASH plans. [37],

$$CI=\frac{V_{100}^2}{V\times V_{100}^{\prime }}$$ (11)

Table. 2.

The radiation fields and the fractions of each field for 4 cases via SFUDPF_SDDRO.

	case1		case2		case3		case4
Field	45°	315°	45°	315°	225°	315°	45°	135°
Fraction	3	2	3	2	3	2	2	3

Here, $V_{\mathit{100}}$ is the target volume receiving at least 100% of prescription dose; $V$ is the target volume; $V_{100}^{\prime }$ is the is the prescription isodose volume. The calculation was under the plan normalization D95% = 100%. The most ideal situation is that CI is equal to 1.

3. Results

3.1. High-dose sparing

Figure 2 illustrates the axial distributions of the proton dose and FLASH effective dose for the four clinical HN cases. As observed, there is a reduction in the prescription dose and high-dose spillage to CTV + 1 cm in the FED compared to the original dose, indicating the potential of FLASH dose sparing. Tables 3–6 summarize the corresponding FLASH dose metrics, showing consistent improvements in all evaluated metrics, particularly in reducing high-dose spillage. All values were calculated under the plan normalization of D95% = 100%.

Figure 2.

FLASH effective dose of four HN cases

Table. 3.

Physical dose parameters for case 1.

	Structure	Quantity	IMPT	VMAT	FLASH d(de)
HN01	PTV	CI	0.69	0.89	0.68(0.95)
		Dmax	110.73%	103.75%	107.53%
		Dmean	102.02%	101.40%	103.41%
	CTV+1cm	V70	45.75%	60.87%	62.76% (11.18%)
		D10%	98.03%	101.08%	99.50% (73.82%)
		Dmean	63.86%	75.38%	74.34% (56.90%)
	Brainstem	D1%	85.82%	87.57%	79.84% (62.26%)
		Dmean	18.30%	24.73%	22.09% (20.94%)
	Spinal Cord	D10%	81.61%	87.36%	79.84% (60.74%)
		Dmean	22.92%	29.54%	24.55% (21.13%)

Table. 6.

Physical dose parameters for case 4.

	Structure	Quantity	IMPT	VMAT	FLASH d(de)
HN04	PTV	CI	0.70	0.92	0.84(0.95)
		Dmax	110.09%	103.31%	110.50%
		Dmean	101.40%	100.97%	103.79%
	CTV+1cm	V70	55.82%	68.46%	61.04% (4.89%)
		D10%	98.92%	100.72%	96.39% (68.09%)
		Dmean	70.32%	78.02%	74.94% (57.29%)
	Oral Cavity	D10%	78.53%	88.56%	80.57% (59.81%)
		Dmean	21.70%	46.59%	27.92% (23.64%)
	L Parotid	D10%	19.27%	53.16%	43.46% (43.46%)
		Dmean	14.97%	43.09%	33.40% (33.40%)

3.2. Normal tissue sparing

Tables 3–6 contain the relevant abutting OAR for each of the four clinical HN cases and demonstrates OAR dose sparing for FLASH treatment plans for all cases. The greatest improvement in OAR sparing is seen in Case 2 for the carotid. Compared to VMAT, the D10% of the carotid artery decreased by 39.16% with FLASH. This reduction is primarily because a portion of the carotid artery is included in the CTV1cm, where the effective dose is reduced due to the FLASH effect. Similar decreases are observed in Case 1 for the brainstem and spinal cord (25.3% and 26.62%, respectively), in Case 3 for the oral cavity and oropharynx (32.06% and 33.41%), and in Case 4 for the oral cavity (28.75%). Figures 3–6 illustrate the dose volume histograms (DVHs) of CTV1cm and the relevant OAR of each HN case comparing between the different techniques. As seen, FLASH plans achieved a marked reduction in integral and high doses to the surrounding OAR.

Figure 3.

Case1: plot of FLASH effective dose, IMPT dose and VMAT dose in (a)-(c); DVH for PTV and CTV+1cm in (d); DVH for OAR in (e); DVH for Body in (f)

Figure 6.

Case4: plot of FLASH effective dose, IMPT dose and VMAT dose in (a)-(c); DVH for PTV and CTV+1cm in (d); DVH for OAR in (e); DVH for Body in (f)

3.3. Conformity index

As illustrated in Figure 3–6, FLASH plans achieved better dose conformity when compared with IMPT and VMAT. Their corresponding CI values are also presented in Tables 2–5 showing better CI scores as compared to IMPT and VMAT when the FLASH effect was considered. It should be noted that the calculation of the conformity index (CI) based on the FLASH effective dose represents an idealized best-case scenario. The greatest improvement in the CI values can be seen in Case 2 as compared to IMPT, due to the FLASH effect concentrating the high dose within the CTV area. The greatest improvement is 0.37, and in the other three cases, FLASH also improved the CI values by an average of 0.27. VMAT achieves a higher dose conformity due to the greater number of selected angles and higher degree of freedom during optimization. However, FLASH still shows an improvement in dose conformity compared to VMAT, with an average improvement of 0.075.

Table. 5.

Physical dose parameters for case 3.

	Structure	Quantity	IMPT	VMAT	FLASH d(de)
HN03	PTV	CI	0.58	0.81	0.65(0.95)
		Dmax	116.01%	107.04%	110.50%
		Dmean	102.67%	101.99%	102.94%
	CTV+1cm	V70	45.74%	66.04%	62.82% (6.81%)
		D10%	99.41%	102.27%	98.41% (68.98%)
		Dmean	62.38%	76.04%	75.82% (56.64%)
	Oral Cavity	D1%	88.78%	94.72%	89.26% (62.66%)
		Dmean	6.87%	26.98%	9.99% (8.88%)
	Oropharynx	D1%	94.52%	102.50%	98.70% (69.09%)
		Dmean	21.86%	34.96%	42.45% (37.73%)

4. Discussion

In this study, we employed the strategy of delivering uniform dose to the target from single field per fraction, called SFUDPF. The most important advantage of SFUDPF is that SFUDPF plans can meet the FLASH effect threshold more easily than MFUDPF plans. We set prescription dose as 8 Gy per fraction for the HN cases in this work to meet the 5 Gy dose threshold of FLASH effect.

In this study, we used the SDDRO method to optimize the spots weights, called SFUDPF-SDDRO. Specifically, this approach involves optimizing not only the conformity of target dose and FLASH effect percentage within specific regions for each single field plan, but also OAR and target doses for the total plan. The results demonstrate that the FLASH RT plans via SFUD-SDDRO effectively optimize the dose and dose rate in CTV1cm, reducing the FLASH effective dose in high-dose regions surrounding the target area. This may lead to better protection of the normal tissues around the target area. Compared to VMAT plans, the FLASH RT plans via SFUDPF-SDDRO offer superior protection for OAR, particularly in high-dose regions adjacent to the target area. Additionally, when compared to IMPT plans, FLASH RT plans via SFUD-SDDRO show the same advantage in terms of dose conformity to the target area. This advantage is also evident when compared to VMAT plans. In summary, compared to CONV-RT, FLASH RT via SFUD-SDDRO not only may enhances protection of high-dose regions near the target, but also improves dose conformity while reducing the high-dose exposure to OAR.

With the strategy of SFUDPF, there is great potential to improve the quality of the FLASH RT plans. We may consider more beam angles to improve the plan quality of the total plan without sacrificing the high dose rate. For the prescription of 8 Gy × 5 fractions in this study, we may employ the beam angle optimization to select the angles for 5 fractions to provide better total plan quality while maintaining the FLASH effect coverage.

It is noteworthy that this work is proof-of-concept, without considering the actual characteristics of the FLASH beam (e.g., the spot size)^[38,39] and the process using ridge filter to generate SOBP from a monoenergetic proton beam^[40], which may require the MMU optimization to improve plan quality^[41,42]. In addition, in the current SFUD-SDDRO model, FLASH dose and dose rate constraints are managed independently, implying that voxels satisfying one constraint may not necessarily fulfill the other. Considering that the FLASH effect must be generated with consideration of both dose and dose rate constraints, one may combine them to improve the FLASH effect coverage [35]. However, the joint imposition of these two FLASH constraints introduces mathematical non-convexity, which may bring additional challenges to optimization. The selection of beam angles and their assignment to each fraction in our method was manually specified, which may lead to suboptimal OAR sparing compared to IMPT, as observed in Case 4. In the future work, we will incorporate the optimization of both beam angles^[43] and fraction allocation into the planning process. Furthermore, this study utilized the DMF model and treated FLASH parameters as constants, but the broader framework of the DMF model equation accommodates heterogeneous FLASH parameters. Moreover, it could be more effective to integrate the DMF model into optimization ^[44].

5. Conclusion

We proposed a novel optimization method for proton FLASH plans. We employed both SFUDPF strategy and SDDRO technique to generate robust and conformal proton FLASH plans utilizing multiple gantry angles for four clinical HN cases. The FLASH effect was successfully modeled by a constant DMF within our work and showed great effectiveness in conferring radiobiological protection to adjacent OAR surrounding the CTV. When compared to IMPT, there is a significant improvement in CI and when compared to VMAT, there is a significant reduction in high-dose spillage to the surrounding OAR.

Figure 4.

Case2: plot of FLASH effective dose, IMPT dose and VMAT dose in (a)-(c); DVH for PTV and CTV+1cm in (d); DVH for OAR in (e); DVH for Body in (f)

Figure 5.

Case3: plot of FLASH effective dose, IMPT dose and VMAT dose in (a)-(c); DVH for PTV and CTV+1cm in (d); DVH for OAR in (e); DVH for Body in (f)

Table. 4.

Physical dose parameters for case 2.

	Structure	Quantity	IMPT	VMAT	FLASH d(de)
HN02	PTV	CI	0.65	0.88	0.71(0.95)
		Dmax	115.55%	106.80%	111.16%
		Dmean	101.91%	100.94%	105.12%
	CTV+1cm	V70	40.16%	58.67%	57.58% (7.81%)
		D10%	96.39%	100.49%	97.92% (69.07%)
		Dmean	63.21%	73.65%	74.03% (56.44%)
	R Parotid	D10%	7.50%	46.84%	22.47% (22.47%)
		Dmean	2.42%	24.30%	7.12% (7.01%)
	Carotid	D10%	74.52%	102.46%	90.42% (63.30%)
		Dmean	36.10%	60.06%	45.94% (36.55%)