Revisiting fractionation in prostate carbon ion therapy: first comparative analysis of biologically effective dose using mMKM with diverse parameters

Abstract

Background

Carbon ion radiotherapy (CIRT) for prostate cancer has evolved from conventional regimens [66 Gy(RBE)/20 fractions (66G20), 63 Gy(RBE)/20 fractions (63G20)] toward hypofractionated schedules [57.6 Gy(RBE)/16 fractions (57.6G16), 51.6 Gy(RBE)/12 fractions (51.6G12), and 40 Gy(RBE)/4 fractions (40G4)]. Whether these are biologically equivalent remains unclear. To our knowledge, this is the first study to systematically evaluate the biological effects of these regimens using the modified microdosimetric kinetic model (mMKM) across diverse parameter settings.

Methods

Biologically effective dose (BED) was calculated using four approaches: (1) photon linear-quadratic (LQ) model (2), modified microdosimetric kinetic model (mMKM) with human salivary gland parameters and fixed RBE of 2.41 (3), mMKM3.1, and (4) mMKM with varied domain radius (0.2–0.5 μm) and α/β (1.5–10 Gy). Tumor control probability (TCP) was further estimated based on a reanalysis of the Ion Prostate Irradiation trial.

Results

Photon LQ analysis suggested near-equivalent BEDs for 63G20, 57.6G16, and 51.6G12 (~ 200 Gy and ~ 130 Gy for α/β = 1.5 and 3 Gy), with 40G4 markedly higher. In contrast, all mMKM-based methods showed progressive BED reductions with hypofractionation versus 63G20 (–16.8% to − 11.3% for 51.6G12; − 30.4% to − 5.2% for 40G4). TCP decreased by 15.5% for 51.6G12 and 28.7% for 40G4. mMKM also yielded higher effective α/β values for CIRT, reflecting reduced fractionation sensitivity.

Conclusions

mMKM-based modeling challenges the assumption of dose equivalence in hypofractionated prostate CIRT, suggesting potential underdosing risks. This parameter-sensitive analysis underscores the need for biologically robust models to refine dose prescription and optimize outcomes in particle therapy.

Clinical trial number

Not applicable.

Supplementary Information

The online version contains supplementary material available at 10.1186/s13014-026-02790-4.

Introduction

Prostate cancer is one of the most common malignancies in men worldwide, with surgery, radiation therapy, and androgen deprivation therapy (ADT) as the principal treatment options [1]. In photon-based radiotherapy, dose escalation has been shown to improve biochemical control; for example, 78 Gy achieved superior long-term outcomes compared to 70 Gy [2]. More recently, carbon ion radiotherapy (CIRT) has gained attention for its enhanced physical and biological properties, offering improved tumor control with reduced normal tissue toxicity [3, 4].

Unlike proton therapy, where the relative biological effectiveness (RBE) is assumed constant, CIRT requires accounting for RBE variations influenced by dose per fraction, tissue type, and linear energy transfer (LET) [5]. This complicates the conversion from physical dose to biologically effective dose (BED), resulting in inter-institutional variability. To address this, several modeling approaches have been developed. European centers often employ the local effect model (LEM), whereas Japanese institutions predominantly use the modified microdosimetric kinetic model (mMKM). The mMKM is based on microdosimetry, theory of dual radiation action and correction of non-Poisson distribution of lethal lesions per nucleus. The mMKM provides a mechanistic framework that links microscopic energy deposition events to macroscopic biological outcomes, accounting for stochastic variations in energy clustering and its impact on DNA damage, particularly double-strand breaks [6]. This model has been successfully integrated into clinical treatment planning systems (TPS) for scanned carbon ion beams, improving the biological accuracy of dose distributions. At the National Institute of Radiological Sciences (NIRS), and likewise at our institution, the updated clinical dose system has been implemented in TPS, incorporating the mMKM together with radiobiological parameters derived from human salivary gland (HSG) cell data [7].

In Japan, the NIRS has conducted extensive research to optimize dosing regimens for prostate cancer using CIRT. Initially, a regimen of 66 Gy(RBE) delivered in 20 fractions (66G20) was employed. Subsequent studies explored hypofractionated schedules, including 63 Gy(RBE) in 20 fractions (63G20), 57.6 Gy(RBE) in 16 fractions (57.6G16), and 51.6 Gy(RBE) in 12 fractions (51.6G12) [8]. The BED is derived using the linear-quadratic (LQ) model with photon ratio (photon LQ model) based on the clinical dose, ensuring that each dose regimen maintains equivalent BED to preserve treatment efficacy while optimizing fractionation to shorten overall treatment duration. However, a study from NIRS using clinical data from low-risk prostate cancer patients without ADT demonstrated that the clinical RBE of carbon-ion radiotherapy decreases with increasing dose per fraction [9]. Ultra-hypofractionated CIRT delivered as 40 Gy(RBE) in 4 fractions (40G4) has shown favorable safety and encouraging 3-year biochemical control outcomes in patients with low- and intermediate-risk localized prostate cancer. However, applying the LQ model directly to the clinical dose yields a higher tumor BED for the 4-fraction regimen than for 12-, 16-, or 20-fraction regimens [10].

The Ion Prostate Irradiation (IPI) trial represents the first randomized controlled trial directly comparing proton and CIRT for prostate cancer. This landmark study reported significantly inferior 5-year biochemical failure free survival (bFFS) for CIRT (50%) compared to proton therapy (85%), despite identical prescribed doses [11]. This finding challenges the assumption that CIRT, with its higher LET and RBE, is inherently superior. One plausible explanation is that the prescribed carbon dose—derived using the LEM—may have been biologically inadequate for effective tumor control [12].

The present study aims to evaluate and compare the BED delivered to the prostate tumor using different BED modeling approaches with variable parameters. Through this analysis, we seek to estimate the potential variation in BED across different fractionation schemes and evaluate its impact on the predicted tumor control probability (TCP).

Method

BED

According to the LQ model, BEDs are calculated as:

Where , and represent the total dose, the dose per fraction and the number of fractions, respectively. BEDs for six dose-fractionation schemes—66G20, 63G20, 57.6G16, 54G12, 51.6G12, and 40G4—were calculated using the four methods described below.

Modified microdosimetric kinetic model (mMKM)

In mMKM, cell survival follows the LQ model, with the term dependent on the microdosimetric distribution: [6, 13]

where:

Here, is the initial slope of the survival curve as LET approaches 0, is fixed across radiation types, is the dose-averaged saturation-corrected specific energy, the dose-averaged saturation-corrected lineal energy, the domain density (≈ water), and the domain radius.

with as the probability density of and the nuclear radius.

Simplified mMKM

Here, we propose a novel method—a simplified version of the mMKM—that requires only the ratio, which is readily available from photon radiotherapy data, and the parameter ​, to estimate BED. It should be noted that the simplified mMKM is not a new radiobiological model, but a reduced formulation of the established mMKM derived through algebraic simplification, rather than an approximate or empirically modified model.

According to mMKM, the for photon, denoted as , can be expressed as: [14]

,

Thus:

Then the for particles, denoted as , can be expressed as:

The photon-equivalent BED for particle is:

Because is largely insensitive to and other parameter variations across most of the spread-out Bragg peak (SOBP) [14, 15], its value using reasonable parameter sets is nearly identical to that from HSG parameters. The spatial distribution of for HSG can be derived from physical and clinical dose data in the TPS, from which and are also obtained. With ≈ 2.36 keV/µm [16], calculation of photon-equivalent BED for different radiobiological settings requires only the specified and . Methods 3 and 4 in this study described below are based on this simplified mMKM approach.

For comparison, the photon-equivalent BED calculated using the full mMKM model is:

This formulation requires more parameters (, , and ) than the simplified method, even though the resulting BED values are essentially the same. Validation of this equivalence is shown in Supplementary Figure S1.

For simplicity, the term “BED” hereafter refers to photon-equivalent BED, unless otherwise specified.

Method 1: photon LQ model

The NIRS clinical dose system is based on HSG cell survival and fast neutron therapy data. Both the original and updated systems based on mMKM (HSG) produce similar results, with differences in the physical dose within the target region being within 1.5% [6, 17]. Assuming the clinical dose equals the photon dose, BED is calculated by the LQ model:

This approach is widely used [18], especially when transitioning from conventional to hypofractionated CIRT for prostate cancer [8]. We evaluated ratios of 1.5, 3.0, 5.0, and 10 Gy to reflect possible variability, and also tested 3.1 Gy for comparison with mMKM3.1.

Method 2: mMKM(HSG)

HSG cells are widely used to represent typical tumor response [17]. In the updated NIRS dose system, the reference radiation is a 350 MeV/u carbon beam at the center of a 60-mm SOBP [6]. Physical dose is converted to carbon-equivalent dose using mMKM with HSG parameters [19] (=0.172 Gy^{− 1}, =0.0615 Gy^{− 2}, =0.32 μm, =3.9 μm). This is then scaled by a clinical RBE factor of 2.41 (F_NIRS) to obtain the clinical dose. Our TPS follows this system.

Given the carbon-equivalent dose and reference parameters (=0.764 Gy⁻¹, =0.0615 Gy⁻²), carbon-equivalent BED can be calculated as: [20]

BED is then obtained by multiplying carbon-equivalent BED by F_NIRS.

To improve clarity regarding how clinical dose and BED were derived, the conversion chains for this NIRS-based method is summarized as follows: physical dose → carbon-equivalent dose → carbon-equivalent BED → × 2.41 (F_NIRS)→ clinical BED.

Method 3: mMKM_3.1

The IPI trial comparing proton and CIRT for localized prostate cancer reported inferior outcomes with CIRT. Recalculation of CIRT plans using mMKM ( = 0.15 Gy⁻¹, = 0.0484 Gy⁻², = 0.3 μm, = 3.6 μm) assumed ≈ , giving = 3.1 Gy and defining mMKM3.1 [12]. This yielded BEDs of 96 Gy for CIRT and 133 Gy for protons. Applied to a photon-derived TCP model, predicted 5-year bFFS was 51.5% and 85.8%, respectively, closely matching trial results.

In this study, we used the simplified mMKM ( = 3.1 Gy, = 0.3 μm) as a surrogate for mMKM3.1. Since mMKM3.1 clinical doses at 66G20 align with the updated NIRS system, we compared results from the simplified mMKM and our TPS to confirm equivalence. For TCP estimation, we applied the risk/ADT distribution from the IPI trial (58:17:12:4 across low-intermediate ADT–, low-intermediate ADT+, high-risk ADT–, and high-risk ADT+, respectively) to maintain consistency with the established TCP model framework. TCP estimates for each fractionation scheme were then derived using the IPI-based TCP model parameters and the same risk-stratification weighting [12].

Method 4: sensitivity analysis

To investigate the impact of parameter variations on BED, all 16 combinations of four values (1.5, 3.0, 5.0, and 10 Gy) and four values (0.2, 0.3, 0.4, and 0.5 μm) were evaluated using the mMKM. The selected range of values reasonably encompasses the radiobiological characteristics of most prostate cancers and other malignancies [8, 21].

Bootstrap-based BED uncertainty propagation

We performed patient-level propagation of and variability using a bootstrap framework. For each patient, 10,000 samples were drawn from the Method-4 parameter space, and BED and TCP were recalculated for all regimens. These bootstrap distributions were then used to derive 95% confidence intervals for BED, TCP, and inter-regimen BED contrasts.

Clinical treatment plan data collection and processing

Ten patients with node-negative prostate cancer treated with 54 Gy(RBE) in 12 fractions [9] at our institute (May 2023–January 2024) were randomly selected. Selection criteria required pathologically confirmed prostate adenocarcinoma, node-negative and non-metastatic disease, and definitive-intent CIRT with complete planning data available. The study was approved by the Institutional Review Board of Taipei Veterans General Hospital (No. 2024-07-011AC). The high-risk clinical target volume (CTV_H) included the prostate and proximal one-third of the seminal vesicles, with the entire vesicles added for T3b disease. The planning target volume was defined by expanding the CTV_H by 10 mm anteriorly/laterally and 5 mm superiorly/inferiorly/posteriorly [8]. Single-field-per-day bilateral parallel plans were created using the VQA Plan v6.1.5 TPS (Hitachi, Tokyo, Japan). Structure sets and physical/clinical dose distributions were exported in DICOM format, processed with pydicom (v2.4.4) in Python (v3.10), and rescaled to all fractionation schemes for BED calculation [20]. BEDs across regimens were then compared. Wilcoxon signed-rank test was used for statistical analysis.

Result

Comparison of BEDs estimated with different methods

Using the LQ model (Method 1), 63G20, 57.6G16, and 51.6G12 yielded similar BEDs (~ 200 Gy and ~ 130 Gy for = 1.5 and 3 Gy). 66G20 and 54G12 were slightly higher (~ 220 Gy and ~ 140 Gy), while 40G4 produced the highest (314.89 Gy and 177.74 Gy). For comparison, the BEDs for conventional photon radiotherapy with 78 Gy in 39 fractions are 182 and 130 for = 1.5 and 3 Gy, respectively. For higher ratios (5 and 10 Gy), the BEDs of schemes with larger fraction sizes (and lower total doses) decreased more markedly, indicating stronger dependence on total dose than fraction size.

With mMKM(HSG) (Method 2) and mMKM3.1 (Method 3), BEDs correlated predominantly with total dose rather than fraction size: 66G20 yielded the highest BED, whereas 40G4 produced the lowest values—more than 20% below those of 63G20—demonstrating a quantitatively small fractionation dependence. mMKM(HSG) produced lower absolute values than mMKM3.1, though ranking trends were consistent. Using mMKM3.1, BEDs were 137.97 Gy for 66G20 and 131.12 Gy for 63G20, similar to photon RT (~ 130 Gy for = 3 Gy). Other regimens were 7.34% (57.6G16) to 23.53% (40G4) lower than 63G20, across regimens yielded a consistent p-value of < 0.002.

With mMKM and varying parameters (Method 4), trends were generally consistent with Methods 2 and 3, except for low (1.5–3 Gy) combined with large (0.4–0.5 μm). Across parameters, 66G20 yielded the highest BEDs, while 40G4 was > 20% lower than 63G20 in most cases. Exceptions occurred when was 1.5 Gy with = 0.4–0.5 μm or 3 Gy with = 0.5 μm, where 51.6G12 produced the lowest BEDs. Median BEDs (CTV_H) by method are shown in Table 1, with relative differences to 63G20 in Table 2; Fig. 1.

Table 1.

Median BED for the CTV across different fractionation schemes

Model	α⁄β (Gy)	rd (µm)		Median BED, median [min–max]
				66.0G20		63.0G20		57.6G16		54.0G12		51.6G12		40.0G4
LQ	1.5	-		216.31[213.32–221.11]		200.00[197.25–204.41]		200.63[197.83–205.13]		221.43[218.25–226.52]		204.50[201.59–209.19]		314.89[310.08–322.64]
	3.0	-		141.62[139.86–144.46]		131.94[130.32–134.57]		129.52[127.89–132.16]		138.09[136.29–141.00]		128.42[126.75–131.10]		177.74[175.16–181.87]
	5.0	-		111.75[110.47–113.80]		104.72[103.54–106.63]		101.09[99.91–102.97]		104.77[103.50–106.80]		97.98[96.82–99.87]		122.87[121.19–125.56]
	10.0	-		89.34[88.44–90.81]		84.31[83.46–85.68]		79.75[78.93–81.08]		79.77[78.91–81.14]		75.16[74.36–76.44]		81.72[80.72–83.33]
mMKM(HSG)	5.09	0.32		74.42[73.76–75.49]		70.71[70.09–71.73]		65.55[64.95–66.49]		63.12[62.54–64.05]		59.96[59.41–60.84]		54.31[53.76–55.21]
mMKM3.1	3.1	0.3		137.97[136.73–139.37]		131.12[129.97–132.41]		121.47[120.36–122.78]		116.80[115.72–118.32]		110.99[109.97–112.38]		100.03[98.71–102.20]
mMKM	1.5	0.2		442.40[435.14–451.28]		421.10[414.11–429.63]		388.27[382.04–395.93]		370.31[364.58–377.03]		352.51[347.01–359.03]		302.97[299.15–306.14]
		0.3		228.53[225.68–230.88]		216.92[214.17–219.19]		201.67[199.25–203.68]		195.34[193.22–197.10]		185.32[183.27–187.04]		172.45[170.56–175.79]
		0.4		153.38[151.89–155.23]		145.24[143.82–146.89]		136.00[134.68–137.84]		133.51[132.19–135.87]		126.30[125.06–128.42]		127.09[124.70–130.78]
		0.5		118.39[116.93–120.81]		111.80[110.46–114.03]		105.53[104.15–107.80]		105.17[103.56–107.72]		99.17[97.70–101.52]		105.84[103.43–109.94]
	3.0	0.2		235.69[232.22–239.63]		224.36[221.04–228.17]		206.83[203.81–210.17]		197.17[194.28–199.94]		187.70[184.96–190.44]		160.26[158.46–161.76]
		0.3		128.75[127.51–129.78]		122.29[121.09–123.28]		113.47[112.42–114.40]		109.50[108.51–110.57]		103.97[103.04–104.95]		94.99[93.82–96.94]
		0.4		90.95[90.05–92.54]		86.22[85.38–87.68]		80.49[79.64–81.94]		78.57[77.62–80.14]		74.42[73.54–75.87]		72.22[70.87–74.44]
		0.5		73.65[72.54–75.34]		69.69[68.67–71.27]		65.38[64.35–66.94]		64.37[63.29–66.09]		60.86[59.85–62.44]		61.64[60.17–64.01]
	5.0	0.2		153.13[151.06–154.95]		145.81[143.82–147.57]		134.31[132.52–135.86]		127.80[126.17–129.14]		121.72[120.15–123.01]		103.18[102.18–104.03]
		0.3		88.76[87.96–89.77]		84.37[83.61–85.30]		78.10[77.40–79.05]		75.02[74.35–76.11]		71.31[70.66–72.30]		64.00[63.12–65.40]
		0.4		66.13[65.29–67.46]		62.76[61.98–64.01]		58.39[57.61–59.59]		56.61[55.79–57.86]		53.69[52.93–54.87]		50.30[49.34–51.90]
		0.5		55.67[54.77–57.15]		52.78[51.94–54.17]		49.25[48.43–50.59]		48.02[47.17–49.42]		45.49[44.70–46.80]		43.97[42.86–45.64]
	10.0	0.2		91.06[90.20–91.76]		86.73[85.92–87.41]		79.80[79.07–80.41]		75.75[75.08–76.34]		72.18[71.54–72.73]		60.28[59.73–61.06]
		0.3		58.73[58.07–59.81]		55.88[55.26–56.89]		51.59[51.00–52.55]		49.30[48.70–50.26]		46.91[46.35–47.82]		40.73[40.08–41.76]
		0.4		47.45[46.71–48.66]		45.11[44.41–46.26]		41.74[41.08–42.83]		40.06[39.40–41.15]		38.08[37.46–39.11]		33.86[33.15–34.99]
		0.5		42.23[41.38–43.50]		40.12[39.33–41.33]		37.19[36.42–38.32]		35.79[35.02–36.92]		34.01[33.28–35.07]		30.72[29.89–31.86]

BED: biologically effective dose; CTV: clinical target volume; rd: radius of domain; min: minimum; max: maximum; xGy: x Gy(RBE) in y fractions; LQ: linear quadric model; mMKM: modified microdosimetric kinetic model; HSG: human salivary gland

Table 2.

Relative differences in median BED of CTV for various regimens, referenced to 63G20, calculated using different models and parameter sets

Model	α⁄β (Gy)	rd (µm)		Relative Difference of median BED of CTV (%), median [min–max]
				66.0G20		63.0G20		57.6G16		54.0G12		51.6G12		40.0G4
LQ	1.5	-		8.16[8.15–8.17]		Ref		0.32[0.29–0.35]		10.71[10.65–10.82]		2.25[2.20–2.34]		57.45[57.20–57.84]
	3.0	-		7.33[7.32–7.35]		-		-1.83[-1.86– -1.79]		4.66[4.58–4.78]		-2.67[-2.74– -2.58]		34.70[34.41–35.15]
	5.0	-		6.71[6.69–6.72]		-		-3.48[-3.51– -3.43]		0.04[-0.04–0.16]		-6.44[-6.49– -6.34]		17.33[17.05–17.75]
	10.0	-		5.97[5.96–5.99]		-		-5.41[-5.43– -5.37]		-5.39[-5.45– -5.30]		-10.86[-10.90– -10.78]		-3.08[-3.28– -2.74]
mMKM(HSG)	5.09	0.32		5.24[5.23–5.26]		-		-7.31[-7.33– -7.29]		-10.74[-10.77– -10.71]		-15.21[-15.24– -15.18]		-23.19[-23.30– -23.03]
mMKM3.1	3.1	0.3		5.23[5.20–5.26]		-		-7.34[-7.42– -7.27]		-10.83[-11.04– -10.64]		-15.28[-15.45– -15.13]		-23.53[-24.34– -22.82]
mMKM	1.5	0.2		5.07[5.04–5.09]		-		-7.78[-7.84– -7.72]		-12.06[-12.24– -11.90]		-16.28[-16.43– -16.15]		-28.15[-28.85– -27.56]
		0.3		5.35[5.31–5.38]		-		-7.04[-7.15– -6.94]		-9.96[-10.28– -9.70]		-14.58[-14.83– -14.36]		-20.29[-21.45– -19.30]
		0.4†		5.64[5.59–5.68]		-		-6.28[-6.42– -6.16]		-7.85[-8.25– -7.50]		-12.85[-13.18– -12.57]		-12.31[-13.84– -10.97]
		0.5†		5.89[5.83–5.95]		-		-5.61[-5.76– -5.46]		-5.95[-6.40– -5.53]		-11.31[-11.67– -10.97]		-5.19[-7.01– -3.59]
	3.0	0.2		5.04[5.02–5.07]		-		-7.83[-7.89– -7.78]		-12.20[-12.37– -12.05]		-16.40[-16.54– -16.28]		-28.67[-29.29– -28.13]
		0.3		5.28[5.25–5.31]		-		-7.21[-7.29– -7.13]		-10.46[-10.70– -10.24]		-14.98[-15.18– -14.80]		-22.13[-23.08– -21.30]
		0.4		5.49[5.45–5.54]		-		-6.64[-6.77– -6.55]		-8.88[-9.18– -8.60]		-13.68[-13.94– -13.47]		-16.19[-17.39– -15.10]
		0.5†		5.67[5.63–5.71]		-		-6.20[-6.31– -6.08]		-7.60[-7.96– -7.27]		-12.66[-12.95– -12.39]		-11.39[-12.76– -10.19]
	5.0	0.2		5.02[5.00–5.04]		-		-7.89[-7.94– -7.84]		-12.36[-12.51– -12.24]		-16.53[-16.65– -16.43]		-29.26[-29.80– -28.80]
		0.3		5.21[5.19–5.24]		-		-7.40[-7.46– -7.33]		-10.97[-11.17– -10.77]		-15.40[-15.56– -15.24]		-24.01[-24.77– -23.33]
		0.4		5.37[5.34–5.39]		-		-6.98[-7.08– -6.91]		-9.83[-10.08– -9.61]		-14.47[-14.67– -14.28]		-19.77[-20.72– -18.92]
		0.5		5.48[5.44–5.50]		-		-6.70[-6.78– -6.61]		-9.00[-9.26– -8.77]		-13.79[-14.01– -13.61]		-16.64[-17.64– -15.75]
	10.0	0.2		4.98[4.96–4.99]		-		-8.00[-8.03– -7.96]		-12.66[-12.77– -12.56]		-16.78[-16.87– -16.70]		-30.42[-30.82– -30.07]
		0.3		5.10[5.09–5.13]		-		-7.68[-7.72– -7.63]		-11.79[-11.91– -11.65]		-16.06[-16.16– -15.94]		-27.12[-27.64– -26.60]
		0.4		5.19[5.15–5.19]		-		-7.47[-7.53– -7.41]		-11.19[-11.35– -11.05]		-15.58[-15.71– -15.46]		-24.89[-25.47– -24.36]
		0.5		5.23[5.21–5.26]		-		-7.33[-7.40– -7.28]		-10.81[-10.96– -10.67]		-15.25[-15.38– -15.15]		-23.44[-24.00– -22.91]

BED: biologically effective dose; CTV: clinical target volume; rd: radius of domain; min: minimum; max: maximum; xGy: x Gy(RBE) in y fractions; LQ: linear quadric model; mMKM: modified microdosimetric kinetic model; HSG: human salivary gland

^†indicate parameter combinations under which BED ranking is inverted (51.6G12 becomes lowest)

Fig. 1.

Forest plot of the relative differences in median BED across fractionation schemes, calculated using various methods (LQ, mMKM(HSG), mMKM3.1, mMKM) and parameter settings ( = 0.2–0.5 μm, = 1.5–10 Gy). The BED of the 63.0 Gy/20 fractions scheme (63G20) was used as the reference. Schemes are color-coded as follows: 66.0 Gy/20 fractions (66G20, red circles), 57.6 Gy/16 fractions (57.6G16, green hexagons), 54.0 Gy/12 fractions (54G12, light green diamonds), 51.6 Gy/12 fractions (51.6G12, blue squares), and 40.0 Gy/4 fractions (40G4, purple triangles). Error bars indicate the minimum and maximum relative differences. Abbreviations: BED: biologically effective dose; HSG: human salivary gland; LQ: linear-quadratic; mMKM: modified microdosimetric kinetic model

Bootstrap uncertainty analysis demonstrated stable BED ranking across all – combinations. From the bootstrap distributions, 95% confidence intervals were obtained for BED values, inter-regimen BED differences, and TCP estimates (Supplementary Tables 1–2 and Supplementary Figure S2).

α_particle / β

Because the difference between and depends on (with ∝ ), smaller values yielded larger differences. In mMKM(HSG) and mMKM3.1, was relatively low (5.09 and 3.1 Gy), whereas was much higher (12.42 and 14.05 Gy). With varying parameters, differences between and were 21.63, 9.61, 5.41, and 3.46 Gy for = 0.2, 0.3, 0.4, and 0.5 μm, respectively. Table 3 summarizes at the SOBP center using mMKM (Methods 2–4) across parameter settings.

Table 3.

The estimated α_particle⁄β ratio at the center of a standard SOBP

Model	αphoton⁄β (Gy)	rd (µm)	y* (keV/µm)	z* (Gy)	y*photon (keV/µm)	z*photon (Gy)	αparticle⁄β (Gy)
mMKM(HSG)	5.09	0.32	19.33a	9.63	4.60b	2.29	12.42
mMKM3.1	3.1	0.3	19.33	10.95	0.00c	0.00	14.05
mMKM	1.5	0.2	19.33	24.64	2.36d	3.01	23.13
	1.5	0.3	19.33	10.95	2.36	1.34	11.11
	1.5	0.4	19.33	6.16	2.36	0.75	6.91
	1.5	0.5	19.33	3.94	2.36	0.48	4.96
	3.0	0.2	19.33	24.64	2.36	3.01	24.63
	3.0	0.3	19.33	10.95	2.36	1.34	12.61
	3.0	0.4	19.33	6.16	2.36	0.75	8.41
	3.0	0.5	19.33	3.94	2.36	0.48	6.46
	5.0	0.2	19.33	24.64	2.36	3.01	26.63
	5.0	0.3	19.33	10.95	2.36	1.34	14.61
	5.0	0.4	19.33	6.16	2.36	0.75	10.41
	5.0	0.5	19.33	3.94	2.36	0.48	8.46
	10.0	0.2	19.33	24.64	2.36	3.01	31.63
	10.0	0.3	19.33	10.95	2.36	1.34	19.61
	10.0	0.4	19.33	6.16	2.36	0.75	15.41
	10.0	0.5	19.33	3.94	2.36	0.48	13.46

^a: The value of y* at the center of a 60 mm SOBP carbon ion beam with an energy of 350 MeV u⁻¹.⁶

^b: Calculated using standard HSG parameters for 200 KVP X-ray and carbon ion radiotherapy^6.19 

^c: The assumption made in mMKM3.1¹²

^d: The value of y* for 6 MV X-ray¹⁶

Abbreviations: SOBP: spread-out bragg peak; mMKM: modified microdosimetric kinetic model; HSG: human salivary gland

TCP

Estimated TCPs (five-year bFFS) are shown in Fig. 2. Using the LQ model ( = 3.1 Gy), TCPs were 89.7%, 85.2%, 83.8%, 88.1%, 83.0%, and 97.3% for 66G20, 63G20, 57.6G16, 54G12, 51.6G12, and 40G4, respectively. With mMKM3.1, they were 89.2%, 86.0%, 79.9%, 76.1%, 70.5%, and 57.3%. Differences were minimal for 66G20 and 63G20 but widened with greater hypofractionation, paralleling BED disparities.

Fig. 2.

TCPs for six dose-fractionation schemes plotted against BED values calculated using two different models: (left) the LQ model with = 3.1 Gy, and (right) the mMKM3.1 model. Each data point represents the TCP corresponding to a given fractionation scheme: 66.0 Gy/20 fractions (66G20, red circles), 63.0 Gy/20 fractions (63G20, orange hexagons), 57.6 Gy/16 fractions (57.6G16, green pentagons), 54.0 Gy/12 fractions (54G12, light green diamonds), 51.6 Gy/12 fractions (51.6G12, blue squares), and 40.0 Gy/4 fractions (40G4, purple triangles). TCP curves represent fits derived from clinical photon radiotherapy outcomes, stratified by risk group and ADT status: low-intermediate risk with ADT (dotted), low-intermediate risk without ADT (dashed), high risk with ADT (dash-dotted), high risk without ADT (long dash-dotted), and an average curve (solid). Abbreviations: ADT: androgen deprivation therapy; BED: biologically effective dose; HSG: human salivary gland; LQ: linear-quadratic; mMKM: modified microdosimetric kinetic model; TCP: tumor control probability

Flatness

Figure 3 shows BED profiles from the LQ model, mMKM(HSG), and mMKM3.1 (Methods 1–3), all demonstrating preserved target uniformity, with Homogeneity Index (HI) values ranging from 1.0 to 2.1%. Even with Method 4, under the extreme 40G4 regimen and broad parameter variations ( = 0.2–0.5 μm, = 1.5–10 Gy), BED flatness remained consistent (Supplementary Fig. S3).

Fig. 3.

Top left panel (a): The BED distribution overlaid on the computed tomography image, calculated using the LQ model (=3.1 Gy) for the 66G20 plan. The red dashed line indicates the location of the BED profile sampling line. Top right panel (b): BED profiles based on the LQ model (=3.1 Gy) and six dose-fractionation schemes: 66.0 Gy/20 fractions (66G20, red), 63.0 Gy/20 fractions (63G20, orange), 57.6 Gy/16 fractions (57.6G16, green), 54.0 Gy/12 fractions (54G12, light green), 51.6 Gy/12 fractions (51.6G12, blue), and 40.0 Gy/4 fractions (40G4, purple). Bottom left panel (c): BED profiles using the mMKM(HSG) model. Bottom right panel (d): BED profiles using the mMKM3.1 model. Homogeneity Index (HI) was calculated as (D₂% – D₉₈%) / D₅₀%. Falloff distance (FO) was defined as the distance from the prescribed-dose level to the 50% isodose. Abbreviations: BED: biologically effective dose; HSG: human salivary gland; LQ: linear-quadratic; mMKM: modified microdosimetric kinetic mode

mMKM3.1 vs. mMKM(HSG) for 66G20

The BEDVHs for the 66G20 scheme using mMKM(HSG) and mMKM3.1 were highly similar (Fig. 4), validating that our implementation of mMKM3.1 is consistent with the originally published model.

Fig. 4.

The dose volume histograms for the 66 Gy in 20 fractions scheme using the modified microdosimetric kinetic model (mMKM) with human salivary gland (HSG) parameters and mMKM3.1

Discussion

The most critical finding of this study is the progressive reduction in estimated tumor BED with increasing degrees of hypofractionation in CIRT for prostate cancer, observed across different implementations of the mMKM and a wide range of parameter sets. The BED for 51.6G12 is 11.31–16.78% lower than that for 63G20— arising from the parameter sets of = 0.5 μm with = 1.5 Gy and = 0.2 μm with = 10 Gy, respectively—corresponding to a 15.5% reduction in predicted TCP. In contrast, photon LQ-based BED estimates with = 1.5 and 3 Gy remained nearly identical across regimens—except for 40G4—which is expected, as these schemes were designed to be biologically equivalent using this model. Our findings underscore potential risks of tumor underdosage in more aggressively hypofractionated regimens, emphasizing the critical importance of thoroughly evaluating BED equivalence when adopting novel fractionation strategies in clinical practice.

In our Method 1, BED values were calculated based on the photon LQ model, which has been referenced for designing hypofractionated CIRT protocols in prostate cancer [8, 18]. Assuming ratios of 1.5 Gy, the estimated BEDs for 63G20, 57.6G16, and 51.6G12 were 195.3, 195.8, and 199.5 Gy, respectively [8]. However, in a study which analyzed outcomes from the NIRS in a cohort of low-risk prostate cancer patients, TCP analysis demonstrated that the 10-year estimated bFFS were 86%, 73%, 66%, and 61% for 66G20, 63G20, 57.6G16, and 51.6G12, respectively [9]. This discrepancy suggests that BED estimates based solely on the photon LQ model may not fully reflect the actual clinical efficacy of different CIRT regimens.

The ultrahypofractionated regimen has recently emerged in CIRT, with protocols delivering 36G4, 40G4, and 44G4 [10]. For instance, 40G4 yields a BED of 306.7 Gy (α⁄β = 1.5 Gy), markedly exceeding the BEDs of moderate hypofractionation regimens (e.g., 195.3–199.5 Gy for 63G20, 57.6G16, and 51.6G12), revealing a notable inconsistency between the theoretical biological effectiveness of ultrahypofractionation and more hypofractionated schedules. To address this, a study from NIRS applied HSG-based carbon-equivalent BED calculations (equivalent to our Method 2) [20]. Their analysis showed that 51.6G12 (24.49 Gy carbon-equivalent BED) is biologically equivalent to 43.32G4; however, both remain substantially lower than the BED of 63G20.

Our method 3 was based on the IPI trial, which employed a validated TCP model using the mMKM3.1 framework. A key strength of this approach is its ability to provide biologically grounded comparisons across photon, proton, and carbon ion therapies, thereby supporting cross-modality evaluation. However, when applied to 51.6G12, the model predicted a 5-year TCP of 70.5%, notably lower than the > 90% 5-year bFFS reported by the Japanese CIRT center [22, 23]. This discrepancy likely reflects the parameter used in mMKM3.1, which was derived from non–prostate cancer cell data and happened to fit well for 66G20 in the IPI trial. Consequently, this parameter may be suboptimal for predicting TCP in more hypofractionated regimens such as 51.6G12, particularly without prostate cancer–specific clinical calibration. It is noteworthy that the original IPI-1 trial used LEM-I rather than mMKM. LEM-I systematically deviates from MKM/mMKM—overestimating RBE in low-LET regions and underestimating it in mid-LET regions. In contrast, mMKM has demonstrated superior agreement across in vitro, in vivo, and patient-plan analyses. Their study also showed that model choice alone can introduce ± 20–30% variability in clinical RBE-weighted dose [24].

Method 4, which employed the mMKM framework with a broad range of radiobiological parameters, was designed to explore the uncertainty inherent in model-based BED estimations. Despite this wide parameter variation, our analysis consistently demonstrated a progressive decline in BED with increasing hypofractionation, underscoring the robustness of these findings.

Our proposed simplified mMKM-based method provides a practical bridge between advanced microdosimetric modeling and conventional radiobiological paradigms. It requires only two key parameters—photon ratios and —the former of which is well-established in the photon radiotherapy literature. Using these parameters, the framework enables biologically meaningful cross-modality comparisons and supports the translational application of photon-based radiobiological insights to CIRT.

Although our study suggests that the estimated BED of 51.6G12 may be insufficient, recent reports from multiple CIRT centers in Japan have shown favorable short-term outcomes, with 5-year bFFS rates of 87.5% and 95.1% in low-risk patients [22, 23]. However, long-term data indicate that early biochemical control may not always predict durable outcomes. For example, MD Anderson data showed the difference in bFFS between 78 Gy and 70 Gy photon arms widened over time—85% vs. 78% at 5 years, 78% vs. 59% at 8 years, and 73% vs. 50% at 10 years [2]. Similar temporal trends were seen in RTOG 0415 [25] and in NIRS data [9]. Thus, while differences may be minimal at five years, significant disparities can emerge with longer follow-up, underscoring the need for extended monitoring to capture late biochemical recurrences that may not be evident at 5 years.

To improve the accuracy of BED estimation, we employed a range of parameter variations within the mMKM framework. While these variations primarily affect the absolute magnitude of the calculated BED, their influence on dose uniformity in clinical treatment planning is minimal—particularly in bilateral or opposing-field beam configurations, where the symmetric contribution of each field maintains dose flatness across the target.

The were derived by systematic modulation of microdosimetric parameters within the mMKM framework and were notably higher than . The physical dose per fraction at the center was approximately the clinical dose fraction divided by 2.41, remaining below 2 Gy for all schemes except 40G4, which had a physical dose of 4.15 Gy per fraction. Both the high ratios and the low physical dose per fraction explains the limited fractionation effect across schemes. Using these parameters, radiation oncologists and medical physicists can interpret CIRT fractionation effects within an LQ-like framework, thereby enhancing clarity and clinical applicability in dose assessment and fractionation strategy design.

Several limitations warrant discussion. First, while discrepancies in BED estimates were observed across radiobiological models, it remains uncertain whether lower BED values translate into meaningful reductions in bFFS. Longer follow-up, ideally ≥ 10 years, is needed to establish their clinical significance. Second, uncertainties exist in parameters used for BED calculations. To address this, we applied multiple models with a broad parameter range, thereby delineating plausible BED intervals and improving robustness. In summary, this study demonstrates the feasibility of applying the mMKM framework to evaluate biological dose distributions and associated error margins in prostate CIRT. While we recognize that extrapolating photon/IPI-based TCP parameters to hypofractionated CIRT introduces inherent uncertainty, the present work provides a necessary foundational analysis of relative BED and TCP trends. A crucial next step—beyond the scope of this study—will be a fully data-anchored bridging analysis, in which key radiobiological parameters (including and ) are fitted directly to contemporary prostate CIRT outcomes. Such recalibration will be essential for achieving clinically validated BED/TCP predictions in future work.

Conclusion

This study shows a progressive decline in tumor BED with increasing hypofractionation in carbon ion radiotherapy for prostate cancer using the mMKM framework, while photon LQ-based estimates remained largely unchanged. These findings may suggest a risk of tumor underdosage with hypofractionation and underscore the need for biologically robust models like mMKM to guide dose equivalence.

Supplementary Information

Below is the link to the electronic supplementary material.

Abbreviations

ADT

Androgen deprivation therapy

CIRT

Carbon ion radiotherapy

RBE

Relative biological effectiveness

LET

Linear energy transfer

BED

Biologically effective dose

LEM

Local effect model

mMKM

Modified microdosimetric kinetic model

TPS

Treatment planning system

NIRS

National institute of radiological sciences

HSG

Human salivary gland

LQ

Linear-quadratic

IPI

Ion prostate irradiation

bFFS

Biochemical failure free survival

TCP

Tumor control probability

SOBP

Spread-out bragg peak

CTV_H

High-risk clinical target volume

Author contributions

The conception and design of the study were led by Yu-Wen Hu and Chi-Chuan Chiou. Data analysis and interpretation were collaboratively conducted by Chi-Chuan Chiou and Yu-Wen Hu, while Jui-Pin Chen was responsible for radiation treatment planning. Manuscript preparation, including writing and editing, was primarily undertaken by Chi-Chuan Chiou and Yu-Wen Hu. All authors provided critical feedback and contributed to the revision of the manuscript. Furthermore, all authors have read and approved the final version of the manuscript submitted for publication.

Funding

Not applicable.

Data availability

Due to institutional IRB restrictions and patient privacy regulations, the individual-level clinical and treatment planning datasets analyzed in this study cannot be publicly shared.

Declarations

Ethical approval and consent to participate

This study was conducted in accordance with the Declaration of Helsinki and was approved by the Institutional Review Board of Taipei Veterans General Hospital (No. 2024-07-011AC), with a waiver of informed consent because all data were de-identified prior to analysis.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.