MI_VMAT: A New Comprehensive Modulation Index for Volumetric Modulated Arc Therapy Based on Three Key Variables

Abstract

Objective:

This study introduces a novel modulation index for volumetric modulated arc therapy (MI_VMAT), which simultaneously accounts for the combined effects of three key delivery variables. The index is validated for its ability to predict gamma pass rates (GPRs) across two different linear accelerator platforms.

Materials and Methods:

MI_VMAT is derived from beam modulation at the control points (CPs) of VMAT plans. Modulation due to variations in gantry speed and dose rate is represented by changes in meter set weight across CPs and combined with the corresponding aperture area at each CP. In addition, the Field Weight (FW=MU/cGy), obtained from plan optimization and inherently dependent on these variables, is incorporated into the MI_VMAT formulation. The predictive performance of MI_VMAT was evaluated by analyzing its correlation with measured GPRs for VMAT plans delivered on two linear accelerators with different characteristics. Its performance was also compared with previously published and commonly used plan complexity metrics (PCMs). Patient plans were prospectively analyzed on both machines.

Results:

MI_VMAT demonstrated strong predictive capability. For the Unique linac, deviations were within 3% for GPR criteria of 3%/3 mm and 2%/3 mm. For the TrueBeam linac, deviations were within 1% for GPR criteria of 3%/3 mm, 3%/2 mm, and 3%/1 mm. Compared with a multivariate prediction model incorporating multiple PCMs, the univariate MI_VMAT model showed comparable or slightly superior performance, with lower mean deviations and reduced variability. MI_VMAT also exhibited strong to moderate correlations with most established metrics.

Conclusion:

The univariate MI_VMAT model shows substantial predictive power for VMAT delivery accuracy and performs comparably or better than multivariate PCM-based models, supporting its clinical utility.

INTRODUCTION

Intensity-modulated Radiation Therapy (IMRT), delivered either using fixed-field IMRT (FF-IMRT) or Volumetric Modulated Arc Therapy (VMAT), introduces additional complexity due to deliberate modulation of the radiation fluence.[1,2,3] Increased modulation of machine parameters leads to greater plan complexity and is associated with higher uncertainty in dose calculation and delivery when compared with unmodulated treatments.[2,4] Nevertheless, an appropriate level of modulation is essential to achieve the desired dose conformity and normal-tissue sparing.[5,6] In contemporary clinical practice, VMAT has become the routine modality among modulated treatment techniques, largely because it enables highly conformal dose delivery within substantially reduced treatment times.[6,7,8]

Plan complexity in VMAT is commonly quantified using modulation indices, which serve as surrogate predictors for Patient-Specific Quality Assurance (PSQA) outcomes.[9] The previously published Plan Complexity Metrics (PCMs) evaluated in this study included the Modulation Complexity Score for VMAT (MCSv),[10,11,12] Leaf Travel Modulation Complexity Score (LTMCS),[11,12] Modulation Index for VMAT (MIVMAT) proposed by Li and Xing (MISPORT),[11,13] Normalized Planned Monitor Units (PMU),[11,14] MLC speed and MLC acceleration.[11,15] These indices have been widely investigated for their correlation with gamma pass rates (GPRs) obtained from PSQA measurements.[16,17] However, many of the earlier complexity metrics were originally developed for fixed-gantry IMRT, where plan modulation is governed primarily by MLC motion alone.[9,11,15,18]

In VMAT delivery, plan modulation arises from the combined and simultaneous variation of three machine parameters: MLC motion, dose rate, and gantry speed.[19] To better reflect this intrinsic multi-parameter modulation, the present study introduces a new MI_VMAT that jointly accounts for the influence of all three variables. The objective of this work is to investigate the correlation between GPRs and MI_VMAT and to evaluate its potential as a predictive metric for PSQA outcomes. By estimating expected GPR values prior to measurement-based QA, MI_VMAT aims to support informed decisions on whether a VMAT plan requires reoptimization or may be exempted from routine PSQA, thereby enhancing clinical efficiency without compromising treatment safety.

MATERIALS AND METHODS

Formalism of the modulation index for volumetric modulated arc therapy index

Demonstrating the measure of modulation index on control points

VMAT plans typically consist of single or multiple arcs, each comprising a large number of control points (CPs). Beam modulation in VMAT arises from the simultaneous variation of three machine parameters across CPs:

1. MLC aperture geometry

2. Dose rate

3. Gantry speed.

The formalism modulation index on CP (MI_CP) quantifies modulation along CPs by characterizing variations in dose rate, gantry speed, and aperture area introduced along the VMAT beam delivery. Modulation arising from gantry speed and dose rate variations is represented through the meter set weight (MSW) differences across adjacent CPs. In contrast, modulation due to MLC motion is quantified using the aperture area variability at each CP (AAVcp), corresponding to the second component of the MCSv formulation described by Masi et al.[12]

The modulation index MI_CP is derived from the formalism introduced by Webb (2003).[20] Wp, P = 1, 2, 3…n representing the numerical value contributing to the three variables. Its mean and standard deviation (SD), σ_W, is calculated. Let ΔWp = ∣Wp − Wp − 1∣| for P = 1, 2, 3,…, n − 1 be the magnitude of the difference between the CP values Wp and Wp − 1, where Wp is the value at the p-th CP and Wp − 1 is the value at the (P − 1) th CP. Counting N, is the changes for which ΔWp > f σ_W, where f = 0.01, 0.02,… 2, and hence the fraction z (f), is the significant modulations among all modulations along the CPs, i.e. z (f) = (f; ΔWp > f σ_W). The function z (f) always falls monotonically as f increases. The area under the spectrum (z (f) vs. 100fσ_W) is an indication of the degree of modulation.

MI_CP =

Upper limit selected as 1.5 as z (f) approaches zero well beyond the selected upper limit of 1.5. Alternatively, it can be set to extend f until z (f) = 0. It is explained in Supplementary File 1.

Field weight-a function of the three variables (I_FW)

The optimization process is explained in Varian documents as follows: the MU is calculated for each voxel from each beamlet according to the dose map generated during the optimization stage. The MU delivered to each voxel may differ from that of other voxels depending on the optimized dose distribution. In this context, the concept of dose rate in MU calculation is applicable only when it is considered at the voxel level; that is, the dose rate can be determined only when a specific point is taken as the reference for treatment using dose-calculation parameters. Therefore, the field weight value (MU/cGy) extracted from the plan data is only a factor related to the optimization process and not a parameter used directly for MU calculation as the dose rate at a specific treatment point. It is assumed to be a numerical factor representing MU per cGy. Thus, it can be inferred that the field weight is a function of three variables and is also considered a factor in the complexity measure.

Therefore, MI_CP is corrected using an optimization index (I_FW) to obtain the more apt MI, which can be calculated using the formula,

MI_VMAT = MI_CP I_FW

The index on field weight is determined by the equation I_FW = e^-2FW in order to transform the MI_CP into a nonlinear form, as it exhibits nonlinear correlation with other complexity matrices and GPR values.

MI_VMAT = MI_CP I_FW

where, MI_CP, , I_FW = e^-2FW

Volumetric modulated arc therapy treatment plans and patient-specific quality assurance

A total of 100 VMAT treatment plans for oropharyngeal cancer were prospectively generated using the Eclipse treatment planning system (v16.1, Varian Medical Systems). All plans were created for a Varian Unique linear accelerator equipped with a 120-leaf Millennium™ MLC, using 6 MV photon energy and a nominal maximum dose rate of 600 MU/min.

As this study focuses on plan complexity measurements, the dataset comprised 60 single-arc and 40 double-arc VMAT plans intentionally designed to span a wide range of modulation complexities by varying user-controlled optimization parameters such as constraint number, weighting, and priority.[21,22] The plans were generated for planning target volumes (PTVs) prescribed doses of 54 Gy, 60 Gy, and 66 Gy in 30 fractions, with organs at risk (OARs) including the parotid glands, spinal cord, brainstem, mandible, and a normal-tissue ring surrounding the PTV.

PSQA measurements were performed using an Amorphous Silicon Electronic Portal Imaging Device (EPID, aS1000). Gamma analysis was conducted using global criteria of 3%/3 mm, 3%/2 mm, 2%/3 mm, 2%/2 mm, 1%2 mm, and 1% 1 mm. The corresponding GPRs were recorded for further analysis.

To study MI_VMAT, the correlation between GPRs and MI_VMAT was analyzed to evaluate its potential as a predictive metric for PSQA outcomes. In addition, its predictive capability was compared with other established PCMs. The correlations between MI_VMAT and other PCMs were also examined. The previously reported PCMs selected for this study include MCSv, LTMCS, MI_SPORT, PMU, mean MLC speed, and mean MLC acceleration.

Computation of modulation index for volumetric modulated arc therapy and plan complexity metrics

To calculate the proposed index and the other selected PCMs, an in-house Python-based software tool, Compliance Checker, was developed. The software extracts MI_VMAT, PCMs, and their constituent parameters from the DICOM RT Plan file. The accuracy of the code was validated using data from four patients through manual calculations performed using raw optimization data and control-point information copied to Microsoft Excel.

The proposed modulation index (MI_VMAT) was initially evaluated for VMAT plans delivered on the Varian Unique linear accelerator. The analysis was performed in a step-wise manner, ensuring homogeneity in delivery configuration: first, MI_VMAT was assessed for single-arc VMAT plans; subsequently, its behavior was examined for multi-arc delivery by comparing single-arc and double-arc plans.

The applicability of MI_VMAT to double-arc VMAT plans was investigated using three different arc-combination strategies. All approaches yielded nearly identical index values and correlations, demonstrating the internal consistency of the formulation. The final approach based on total field weight and weighted MI_CP across arcs was adopted due to its stronger physical basis.

Statistical correlation analysis

MI_VMAT and PCM result datasets are continuous, and hence each matrices and the corresponding GPRs across all gamma criteria was effectively assessed using Karl Pearson’s correlation coefficient (ρ). P < 0.10 was considered statistically significant to increase sensitivity for detecting potentially meaningful predictors.

Predictive modeling

To assess the capability of MI_VMAT to predict PSQA outcomes prior to measurement-based verification, both univariate and multivariate linear regression models were developed.

Univariate regression model (MI_VMAT only):

Y = A + B1X1 + ε

Multivariate regression model (multiple PCMs):

Y = A + B1X1 + B2X2 + ··· + BnXn + ε

where:

Y = predicted GPR,

X₁…Xₙ = selected PCMs,

A, B₁…Bₙ = regression coefficients, and

ε = residual error term.

Multivariate models were developed using predictors with statistically significant correlations, with predictors included or excluded to formulate equations that maximized overall correlation strength.

Model validation

The relationship between MI_VMAT and GPRs was analyzed to develop a univariate predictive model, and a multivariate model based on selected PCMs was also constructed. The predictive performance of the MI_VMAT-based model was assessed by comparing the deviation between the measured and calculated GPRs, and the results were further compared with those obtained from the PCM-based multivariate model.

The prediction models were validated using 28 additional VMAT plans, comprising 16 head and neck cases and 12 clinically treated multi-site cases, generated on the Varian Unique system. For each validation plan, PCMs were extracted using the Compliance Checker, and predicted GPRs were compared against measured EPID-based GPRs. The agreement between predicted and measured values (<3%) served as the validation criterion.

Extension to a high-precision linac (TrueBeam)

To evaluate the generalizability and machine dependence of the proposed index, the study was extended to a Varian TrueBeam linear accelerator using Eclipse v16.1. A cohort of 50 clinically delivered double-arc VMAT plans from multiple anatomical sites was randomly selected to investigate whether the models could be applied more generally. Statistical analyses were performed to assess the performance of the univariate model (based on MI_VMAT) and the multivariate model (based on selected PCMs). The models were further validated and compared using an additional independent cohort of 14 randomly selected clinically treated patients (from multiple cities) on the TrueBeam system. MI_VMAT exhibited a different correlation pattern with GPRs for the TrueBeam data. Therefore, the index MI_CP was evaluated by excluding the field weight (MU) component from MI_VMAT. The modified index demonstrated behavior similar to other PCMs in the TrueBeam dataset and was subsequently considered more suitable for machines with high positional accuracy.

RESULTS

Gamma pass rate evaluation using Amorphous Silicon Electronic Portal Imaging Device system

To evaluate the performance of the proposed modulation index (MI_VMAT), analyses were performed separately for single-arc and double-arc VMAT plans, as well as for plans delivered on a high-precision Linac (TrueBeam). GPRs were analyzed for 60 single-arc VMAT plans and 40 double-arc VMAT plans (out of a total of 100 VMAT plans) delivered on the Varian Unique linear accelerator, as well as for 50 double-arc VMAT plans delivered on the Varian TrueBeam system. The corresponding GPR results are summarized in Figure 1.

Figure 1.

Mean gamma pass rate ± standard deviation for volumetric modulated arc therapy plans delivered on Unique single arc, Total Unique Plans, and TrueBeam Plans across all gamma criteria. X-axis: gamma criteria; Y-axis: gamma pass rate (%). GPR: Gamma pass rate

For the Unique cohort, an institutional action level of 95% was applied for the 3%/3 mm gamma criterion, and 90% for all other criteria. For the TrueBeam cohort, a uniform action level of 95% was adopted across all evaluated gamma criteria, reflecting the higher delivery precision of the system. This approach is in accordance with the recommendations of Crowe et al.,[23] which suggest that the selection of thresholds may be tailored based on institutional requirements. In this context, the TrueBeam system was predominantly used for advanced techniques such as SBRT, which are typically verified using tighter criteria. Therefore, a higher action level was maintained to prevent achieving a full (100%) pass rate, thereby enabling a more rigorous and meaningful evaluation of plan quality. Due to the observed isocentric precision of approximately 2 mm in diameter for the Unique system, adopting the TG-218 recommended criterion (3%/2 mm) as the primary evaluation standard was not practically feasible for statistical analysis; therefore, the institutional protocol was followed.

For the Varian Unique single arc plans, the mean GPR for the 3%/3 mm criterion was 98.0% ± 1.24% (range: 94.70%–99.80%), with 98.3% of plans meeting the institutional action level of 95%. Similarly, the 3%/2 mm and 2%/3 mm criteria demonstrated high levels of agreement, yielding mean GPRs of 96.44% ± 2.07%and 95.78% ± 1.65%, respectively, with 100% of plans satisfying the corresponding 90% action level. In contrast, further tightening of the gamma evaluation criteria resulted in a pronounced decline in plan performance.

When considering the entire cohort of 100 VMAT plans (including both single-arc and double-arc deliveries) on the Unique system, the mean GPR for the 3%/3 mm criterion was 97.62% ± 1.38% (range: 94.70%–99.80%), with 97% of plans satisfying the institutional action level of 95%. The 3%/2 mm and 2%/3 mm criteria also showed strong agreement, yielding mean GPRs of 94.70% ± 3.14% and 94.86% ± 2.24%, with corresponding pass rates of 91% and 97%, respectively. For the 2%/2 mm, 2%/1 mm, 1%/2 mm, and 1%/1 mm criteria, the mean GPRs decreased to 89.58% ± 4.64%, 77.31% ± 9.08%, 79.03% ± 7.24%, and 58.82% ± 11.43%, respectively. GPRs beyond the clinically relevant criteria, up to 1%/1 mm, were also analyzed to see their correlation behavior with the new Index and with PCMs.

For the Varian TrueBeam Linear Accelerator, the mean GPRs for the 3%/3 mm, 3%/2 mm, and 3%/1 mm criteria were 99.95% ± 0.17%, 99.92% ± 0.22%, and 99.76% ± 0.44%, respectively, with 100% of plans meeting the institutional action level of 95% for each criterion. Similarly, the 2%/3 mm, 2%/2 mm, and 2%/1 mm criteria demonstrated excellent agreement, yielding mean GPRs of 99.37% ± 1.47%, 99.22% ± 1.55%, and 98.58% ± 1.79%, with corresponding pass rates of 98%, 98%, and 96%, respectively. The 1%/2 mm and 1%/1 mm criteria produced substantially lower mean GPRs of 64.09% ± 19.54% and 59.41% ± 19.13%, with only 8% and 2% of plans meeting the predefined action levels, respectively.

Results-complexity indices

The new index and the selected previously published PCMs were extracted for VMAT plans delivered on the Varian Unique and Varian TrueBeam linear accelerators using the in-house Compliance Checker software.

Modulation index for volumetric modulated arc therapy_-unique

Table 1 summarizes the distribution of MI_VMAT values used here as a relative metric to characterize VMAT plan modulation for three delivery configurations: single-arc VMAT on Unique, double-arc VMAT on Unique, and VMAT on TrueBeam. For each configuration, the mean, SD, minimum, and maximum values are reported.

Table 1.

The mean values of index modulation index for volumetric modulated arc therapy and ranges analyzed for volumetric modulated arc therapy plans in true beam machine, and single arc and double arc plans in unique machine

MIVMAT	Mean±SD	Minimum	Maximum
Unique single arc	0.0907±0.0391	0.0246	0.1950
Unique double arc	0.0744±0.0371	0.0295	0.1500
True beam	0.1544±0.1467	0.0122	0.9535

MI_VMAT: Modulation index for volumetric modulated arc therapy, SD: Standard deviation

Table 1 summarizes the distribution of MI_VMAT values for the analyzed VMAT plans. For the Varian Unique system, single-arc plans (0.0907 ± 0.0391) and double-arc plans (0.0744 ± 0.0371) showed comparable results. In contrast, plans generated on the Varian TrueBeam system exhibited a higher mean value (0.1544 ± 0.1467) and a wider range (0.0122–0.9535), indicating greater variability in modulation complexity.

Modulation index on control point-truebeam

The descriptive statistics of the MI_CP index for VMAT plans delivered on both the Varian TrueBeam and Varian Unique linear accelerators, encompassing single-arc and double-arc delivery techniques, are summarized in Table 2. For each group, the table reports the mean, SD, and minimum–maximum values of the MI_CP index.

Table 2.

The mean values of index modulation index on control points and ranges analyzed for volumetric modulated arc therapy plans in TrueBeam machine, and single-arc and double-arc plans in unique machine

Index	Mean±SD	Minimum	Maximum
MICP single arc (Unique)	16.93±2.71	12.24	22.4
MICP double arc (Unique)	16.28±1.77	13.5	19.81
MICP TrueBeam	18.91±3.24	13.39	28.55

MI_CP: Modulation index on control points, SD: Standard deviation

The TrueBeam plans exhibit the highest modulation complexity, with a mean MI_CP value of 18.91, ranging from 13.39 to 28.55, and the largest variability (SD = 3.24). In comparison, the Unique single-arc plans demonstrate a lower mean MI_CP value of 16.93 and a moderate spread (SD = 2.71), with values ranging from 12.24 to 22.40. The Unique double-arc plans show the lowest modulation complexity among the three groups, with a mean of 16.28, minimal variability (SD = 1.77), and a narrower range of 13.50–19.81.

Commonly used plan complexity metrics

The selected previously published PCMs extracted for all VMAT plans delivered on Unique and TrueBeam are summarized in Table 3. The Table reports the mean, SD, and minimum–maximum range for each metric, based on 100 VMAT plans for the Unique system and the corresponding cohort for the TrueBeam system. In addition, the table details the average proportion of MLC speed and acceleration occurrences within predefined magnitude ranges, along with the mean MLC speed and mean MLC acceleration.

Table 3.

Mean values, standard deviations, and ranges of commonly used plan complexity metrics for volumetric modulated arc therapy plans generated with unique and TrueBeam systems

PCMs	Unique			True Beam
	Mean±SD	Minimum	Maximum	Mean±SD	Minimum	Maximum
MCSv	0.2613±0.1003	0.0858	0.4563	0.5286±0.1229	0.3596	0.9283
LSVcp	0.7473±0.0151	0.7117	0.7881	0.818±0.0261	0.7035	0.8769
AAVcp	0.2529±0.0335	0.1175	0.3103	0.3283±0.0754	0.2263	0.5476
LSVcpAAVcp	0.19±0.025	0.0933	0.2344	0.2682±0.0605	0.1828	0.4628
LTmean	726.35±50.08	597.49	820.46	636.17±99.38	343.92	824.21
LT	0.2736±0.0501	0.1795	0.4124	0.3638±0.0994	0.1758	0.6561
LTMCS	0.072±0.0322	0.0303	0.1401	0.1983±0.0919	0.0711	0.575
PMU	538.97±57.15	447.54	670.25	514.65±81.68	311.07	762.32
MISPORT-max	50,078.27±18,848.18	19,491.7	96,734.97	47,282±23,730.3	9570.07	111,824
MISPORT-avg	26,692.36±8901.28	12,904.84	47,334.2	19,040.8±7112.66	5139.72	34,008.9
MISPORT-avg/max	0.5449±0.0627	0.3678	0.6782	0.4446±0.107	0.2381	0.6649
S(0–4)	0.2536±0.0217	0.2074	0.3313	0.2948±0.0454	0.2132	0.3984
S(4–8)	0.1809±0.01	0.1623	0.2171	0.1885±0.0146	0.1554	0.2201
S(8–12)	0.1038±0.0066	0.0915	0.1322	0.0938±0.0077	0.0732	0.1078
S(12–16)	0.0723±0.0075	0.0601	0.0967	0.0607±0.0094	0.0371	0.0795
S(16–20)	0.0888±0.0109	0.0717	0.1251	0.0708±0.0107	0.0482	0.1036
S>20	0.3006±0.039	0.1529	0.3777	0.2914±0.033	0.2196	0.3738
Savg (mm s−1)	9.6642±0.6509	7.7487	11.0089	7.4303±1.9462	2.8712	10.4956
A(0–40)	0.8851±0.0129	0.8541	0.921	0.8843±0.0167	0.8156	0.9083
A(40–80)	0.1149±0.0129	0.079	0.1459	0.1157±0.0167	0.0917	0.1844
Aavg (mm s−2)	14.5565±0.5213	13.2404	16.0043	11.837±2.685	5.7422	15.1551

where, LTmean in mm, PMU in monitor units, MI_SPORTavg in arbitrary units, MCSv dimensionless, S_avg mm s⁻¹⁰, and A_avg mm s⁻². SD: Standard deviation, PCMs: Plan complexity metrics, MCSv: Modulation Complexity Score for VMAT, AAVcp: Aperture area variability at each control point, LTMCS: Leaf travel modulation complexity score, PMU: Planned monitor units, VMAT: Volumetric modulated arc therapy

Results of statistical analysis: Correlation coefficients and regression models

Modulation index for volumetric modulated arc therapy_-unique

The significance of the MI_VMAT index on GPRs was evaluated using Karl Pearson’s correlation coefficient (ρ), and a univariate regression model was developed to predict GPRs. To compare the predictive performance of the MI_VMAT index with that of selected previously reported PCMs, univariate regression models were developed based on individual PCMs. Correlation analysis revealed weak to moderate associations between individual PCMs and GPR criteria, with no single PCM showing consistently strong correlations across all gamma criteria. Therefore, multivariate regression models were subsequently developed to improve the prediction of GPRs for the 3%/3 mm, 3%/2 mm, 2%/3 mm, and 2%/2 mm criteria. Model performance was assessed using an independent validation cohort of 28 additional patients and compared. The results of the validation analysis are summarized in Tables 4 and 5. The predictive models created are provided in the Supplementary File 2 and Supplementary File 3, Table A and B.

Table 4.

Validation of fitted models for 60 single arc volumetric modulated arc therapy plans, using modulation index for volumetric modulated arc therapy for each gamma pass rate criterion, comparing estimated and measured, and their deviations for additional 28 plans

GPRs Comparisons	3%/3 mm			3%/2 mm			2%/3 mm			2%/2 mm
	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation
Mean±SD	97.36±0.63	96.08±1.67	2.14±1.23	95.37±1.04	91.82±3.19	4.94±2.13	95.19±0.57	93.86±2.39	2.76±1.55	91.48±0.89	87.34±4.39	6.56±2.71
Maximum	96.32	94.00	0.15	93.66	87.80	1.53	94.25	91.00	0.40	90.01	81.60	2.13
Minimum	98.60	99.70	4.19	97.41	99.70	8.46	96.31	99.60	5.00	93.23	99.20	11.78

GPRS: Gamma pass rates, SD: Standard deviation

Table 5.

Validation of fitted models for 100 volumetric modulated arc therapy using modulation index for volumetric modulated arc therapy for each gamma pass rate criterion, comparing estimated and measured values and their deviations for additional 28 plans

GPRs Comparisons	3%/3 mm			3%/2 mm			2%/3 mm			2%/2 mm
	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation
Mean±SD	97.36±0.63	96.08±1.67	2.02±1.15	88.20±1.64	91.82±3.19	4.00±3.49	94.12±0.88	93.86±2.39	2.54±1.56	93.51±1.43	87.34±4.39	8.22±3.81
Maximum	96.32	94.00	0.08	85.50	87.80	0.10	92.67	91.00	0.04	91.16	81.60	0.68
Minimum	98.60	99.70	3.95	91.44	99.70	12.59	95.85	99.60	6.07	96.32	99.20	14.43

GPRS: Gamma pass rates, SD: Standard deviation

Supplementary File 2.

Mean values, standard deviations, and ranges of commonly used plan complexity metrics for volumetric-modulated arc therapy plans generated with Unique (Single Arc and Double Arc) and TrueBeam systems

PCMs	Unique single arc			Unique double arc			True Beam
	Mean±SD	Minimum	Maximum	Mean±SD	Minimum	Maximum	Mean±SD	Minimum	Maximum
MCSv	0.1848±0.0380	0.0858	0.2317	0.3761±0.0414	0.2995	0.4563	0.5286±0.1229	0.3596	0.9283
LSV mean	0.7493±0.0175	0.7117	0.7881	0.7443±0.0102	0.7228	0.7722	0.8180±0.0261	0.7035	0.8769
AAVp mean	0.2496±0.0371	0.1175	0.3059	0.2578±0.0264	0.2026	0.3103	0.3283±0.0754	0.2263	0.5476
AAV×LSV Mean	0.1881±0.0278	0.0933	0.2344	0.1929±0.0198	0.1534	0.2299	0.2682±0.0605	0.1828	0.4628
LTmean×103	0.7372±0.0532	0.5876	0.8205	0.7101±0.0416	0.6268	0.7768	0.6362±0.0994	0.3439	0.8242
LT	0.2628±0.0532	0.1795	0.4124	0.2899±0.0416	0.2232	0.3732	0.3638±0.0994	0.1758	0.6561
LTMCS	0.0478±0.0141	0.0303	0.0730	0.1084±0.0152	0.0765	0.1401	0.1983±0.0919	0.0711	0.5750
PMU×1000	0.5313±0.0576	0.4475	0.6703	0.5505±0.0551	0.4685	0.6271	0.5147±0.0817	0.3111	0.7623
MISPORTmax×105	0.6143±0.1614	0.2804	0.9673	0.3305±0.0707	0.1949	0.5068	0.4728±0.2373	0.0957	1.1182
MISPORTavg×105	0.3266±0.0669	0.1820	0.4733	0.1774±0.0227	0.1290	0.2139	0.1904±0.0711	0.0514	0.3401
MISPORTAvg/Max	0.5432±0.0677	0.3678	0.6782	0.5475±0.0572	0.4221	0.6621	0.4446±0.1070	0.2381	0.6649
S(0–4)	0.2538±0.0253	0.2074	0.3313	0.2533±0.0154	0.2328	0.2941	0.2948±0.0454	0.2132	0.3984
S(4–8)	0.1809±0.0114	0.1623	0.2171	0.1810±0.0077	0.1689	0.1989	0.1885±0.0146	0.1554	0.2201
S(8–12)	0.1045±0.0072	0.0915	0.1322	0.1028±0.0052	0.0924	0.1193	0.0938±0.0077	0.0732	0.1078
S(12–16)	0.0745±0.0084	0.0626	0.0967	0.0690±0.0040	0.0601	0.0796	0.0607±0.0094	0.0371	0.0795
S(16–20)	0.0924±0.0124	0.0735	0.1251	0.0833±0.0040	0.0717	0.0897	0.0708±0.0107	0.0482	0.1036
S>20	0.2939±0.0455	0.1529	0.3777	0.3107±0.0235	0.2527	0.3446	0.2914±0.0330	0.2196	0.3738
Savg (mm s−1) ×10	0.9752±0.0702	0.7749	1.1009	0.9532±0.0557	0.8414	1.0425	0.7430±0.1946	0.2871	1.0496
A(0–40)	0.8925±0.0103	0.8795	0.9210	0.8741±0.0071	0.8541	0.8909	0.8843±0.0167	0.8156	0.9083
A(40–80)	0.1075±0.0103	0.0790	0.1205	0.1259±0.0071	0.1091	0.1459	0.1157±0.0167	0.0917	0.1844
Aavg (mm s−2) ×100	0.1437±0.0042	0.1324	0.1527	0.1483±0.0054	0.1377	0.1600	0.1184±0.0269	0.0574	0.1516

PCMs: Plan complexity metrics, VMAT: Volumetric-modulated arc therapy, SD: Standard deviation, MCSv: Modulation Complexity Score for VMAT, AAV: Aperture area variability, LTMCS: Leaf travel modulation complexity score, PMU: Planned monitor units

Table A.

Univariate predictive models estimated for gamma pass rates at 3%/3 mm, 3%/2 mm, 2%/3 mm, and 2%/2 mm in Unique

GPRs	Predicted model	ρ	ρ2	P
3%/3 mm	Y = 96.289 - 18.963MIvmat	0.598	0.358	<0.001
3%/2 mm	Y = 93.601 - 31.322MIvmat	0.590	0.348	<0.001
2%/3 mm	Y = 94.222 + 17.127MIvmat	0.404	0.164	<0.001
2%/2 mm	Y = 89.961 - 26.856MIvmat	0.399	0.159	0.002

GPRs: Gamma pass rates

Table B.

Multivariate predictive models estimated for gamma pass rates at 3%/3 mm, 3%/2 mm, 2%/3 mm, and 2%/2 mm in Unique

GPRs	Predicted model	ρ	ρ2	P
3%/3 mm	Y = 65.194 - 0.005PMU-54.767LTMCS + 43.589 S0–4 + 21.097LSVcp×AAVcp + 27.384 LSVcp + 12.858MCSv + 0	0.683	0.466	<0.001
3%/2 mm	Y = 112.852 - 19.081MCSv - 0.024PMU + 0	0.729	0.531	<0.001
2%/3 mm	Y = 78.078 + 24.429MCSv - 0.019PMU + 85.017 S0–4 - 69.695LTMCS + 1.483MISPORT-avg + 0	0.702	0.493	<0.001
2%/2 mm	Y = 18.18 - 21.791MCSv - 0.03 PMU + 105.555A0–40 + 0	0.764	0.584	<0.001

GPRs: Gamma pass rates

Both models for the 3%/3 mm and 2%/3 mm criteria demonstrated good agreement between the estimated and measured GPRs, with deviations of <3% in the study with a single arc and a total of 100 plans. The mean percentage deviations for the 100 VMAT plan cohort were 2.02% ± 1.15% for the 3%/3 mm criterion and 2.54% ± 1.56% for the 2%/3 mm criterion, indicating good agreement and demonstrating the robust predictive performance of the MI_VMAT-based univariate model. The multivariate PCM model [Table 6] produced deviations of 2.50% ± 1.56% and 2.91 ± 1.60% for the 3%/3 mm and 2%/3 mm criteria, respectively, showing comparable but slightly better performance for the MI_VMAT model. Larger deviations were observed for the more stringent criteria (2 mm criteria), consistent with the increased sensitivity to delivery uncertainties.

Table 6.

Validation of multivariate fitted model for 100 volumetric modulated arc therapy using selected plan complexity metrics for each gamma pass rate criterion, comparing estimated and measured values and their deviations for additional 28 plans

GPRs Comparisons	3%/3 mm			3%/2 mm			2%/3 mm			2%/2 mm
	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation
Mean±SD	97.74±1.22	95.91±1.74	2.50±1.56	94.98±1.82	91.61±3.19	5.07±2.69	94.73±2.02	93.62±2.48	2.91±1.60	91.30±2.20	87.03±4.40	7.05±3.23
Maximum	100.87	99.70	5.59	97.38	99.70	8.83	101.51	99.60	5.83	97.89	99.20	14.29
Minimum	95.81	93.30	0.08	88.98	87.50	0.24	91.15	89.90	0.13	85.03	81.60	0.14

GPRS: Gamma pass rates, SD: Standard deviation

This comparison is visualized in Figure 2, which highlights the superior consistency and predictive accuracy of the MI_VMAT-based model relative to multivariate PCM-based models for the Unique validation cohort.

Figure 2.

Comparison of prediction error of gamma pass rate prediction model based on new Index modulation index for volumetric modulated arc therapy and multivariate model using selected plan complexity metrics for the Unique validation cohort (28 plans) across four gamma criteria. X-axis: gamma criteria; Y-axis: mean deviation (%). MI_VMAT: Modulation index for volumetric modulated arc therapy, PCMs: Plan complexity metrics

Modulation index on control point_-TrueBeam

In the analysis performed for the TrueBeam system, no significant correlation was observed for MI_VMAT. However, the primary index formulated in this study, MI_CP, demonstrated a significant correlation (P = 0.001). Univariate regression models [Table C: Supplementary File 3] were developed to predict the GPRs and were compared with models derived from selected PCMs. Significant correlations were observed for the GPR criteria of 3%/3 mm, 3%/2 mm, and 3%/1 mm. The models were further validated using an additional 14 clinically used VMAT plans from different treatment sites, created on the TrueBeam system. The validation results are summarized in Tables 7 and 8. The model demonstrated excellent predictive performance for the 3%/3 mm, 3%/2 mm, and 3%/1 mm criteria, with mean percentage deviations of 0.07% ± 0.07%, 0.11% ± 0.22%, and 0.20% ± 0.18%, respectively, well within a 1% deviation.

Table C.

Univariate predictive models estimated for gamma pass rates at 3%/3 mm, 3%/2 mm, 3%/1 mm using modulation index on control points in TrueBeam

GPRs	Predicted model	ρ	ρ2	P
3%/3 mm	Y = 100.417 - 0.025MIcp	0.460	0.212	0.001
3%/2 mm	Y = 100.539 - 0.033MIcp	0.475	0.226	<0.001
3%/1 mm	Y = 100.892 - 0.06MIcp	0.441	0.194	0.001

GPRs: Gamma pass rates

Table 7.

Validation of modulation index on control points univariate models for each gamma pass rate criterion, comparing estimated and measured gamma pass rates, showing its deviations for additional 14 plans (True Beam)

GPRs Comparisons	3%/3 mm			3%/2 mm			3%/1 mm
	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation
Mean±SD	99.95±0.09	99.94±0.13	0.07±0.07	99.92±0.12	99.91±0.16	0.11±0.08	99.77±0.22	99.66±0.26	0.20±0.18
Maximum	100.14	100	0.27	100.18	100	0.28	100.24	99.9	0.74
Minimum	99.77	99.5	0.00	99.68	99.4	0.03	99.33	99	0.04

GPRS: Gamma pass rates, SD: Standard deviation

Table 8.

Validation of the multivariate models of other plan complexity metrics using mean deviation between estimated and measured values based on additional 14 plans

GPRs Comparisons	3%/3 mm			3%/2 mm			3%/1 mm
	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation	Estimated	Measured	Percentage deviation
Mean±SD	99.87±0.18	99.94±0.13	0.07±0.10	99.82±0.24	99.91±0.16	0.12±0.13	99.56±0.51	99.66±0.26	0.30±0.32
Minimum	99.42	99.50	0.00	99.22	99.40	0.00	98.26	99.00	0.00
Maximum	100.02	100.00	0.36	100.01	100.00	0.52	99.92	99.90	1.10

GPRS: Gamma pass rates, SD: Standard deviation

The results were compared with those obtained from the multivariate prediction model using significant PCMs for an additional 14 plans [Table 8], showing comparable or lower deviations. A direct comparison between the univariate predictive model based on the new index and the multivariate predictive model using established PCMs is presented in Figure 3. The results demonstrate comparable predictive performance.

Figure 3.

Comparison of prediction error of gamma pass rate prediction model based on new Index modulation index on control points and multivariate model using selected plan complexity metrics for the TrueBeam validation cohort (14 plans) across four gamma criteria. X-axis: gamma criteria; Y-axis: mean deviation (%). PCMs: Plan complexity metrics, MICP: Modulation index on control points

Correlation of modulation index for volumetric modulated arc therapy with other plan complexity metrics

For single-arc VMAT plans on the Unique system [Table 9], MI_VMAT demonstrated statistically significant correlations with several established PCMs.

Table 9.

The correlation coefficient (ρ-value) and P values observed for the index modulation index for volumetric modulated arc therapy of 60 single arc plans (Unique) with other plan complexity metrics

PCMs	ρ	P	Relation
MCSv	0.448	<0.001	Moderate
LSVcp	0.361	0.005	Weak
AAVcp	0.353	0.006	Weak
LSVcpAAVcp	0.421	0.001	Moderate
LTmean	0.353	0.006	Moderate
LT	0.353	0.006	Moderate
LTMCS	0.70	<0.001	Strong
PMU	0.886	<0.001	Very strong
MISPORTmax	0.622	<0.001	Moderate strong
MISPORTavg	0.628	<0.001	Moderate strong
MISPORTavg/max	0.301	0.02	Weak
S(8–12)	0.241	0.063	Weak
S(12–16)	0.564	<0.001	Moderate
S(16–20)	0.715	<0.001	Strong
S>20	0.308	0.017	Weak
A(0–40)	0.607	<0.001	Moderate strong
A(40–80)	0.607	<0.001	Moderate strong

ρ: Spearman’s rank correlation coefficient, PCMs: Plan complexity metrics¸PCMs: Plan complexity metrics, MCSv: Modulation Complexity Score for VMAT, AAVcp: Aperture area variability at each control point, LTMCS: Leaf travel modulation complexity score, PMU: Planned monitor units, VMAT: Volumetric modulated arc therapy

Moderate correlations were observed with MCSv (ρ = 0.448) and LSVcp AAVcp (ρ = 0.421), while strong to very strong correlations were found with LTMCS (ρ = 0.700) and PMU (ρ = 0.886). Metrics related to modulation intensity, such as MI_SPORTmax and MI_SPORTavg, also showed moderately strong correlations (ρ ≈ 0.62). In addition, MI_VMAT exhibited meaningful correlations with higher MLC speed bins and MLC acceleration, indicating sensitivity to mechanical delivery characteristics.

For double-arc VMAT plans on the unique system [Table 10], correlations were generally stronger. MI_VMAT showed strong to very strong correlations with PMU (ρ = 0.977), MI_SPORTavg (ρ = 0.850), MI_SPORTmax (ρ = 0.793), and MCSv (ρ = 0.718). Moderate correlations were also observed with aperture-related metrics, while correlations with lower MLC speed and acceleration bins were comparatively weaker.

Table 10.

The correlation coefficient (ρ-value) and P values observed for the index modulation index for volumetric modulated arc therapy of double arc plans (Unique) with other plan complexity metrics

PCMs	ρ	P	Relation
MCSv	0.718	<0.001	Strong
AAVcp	0.632	<0.001	Strong
LSVcpAAVcp	0.672	<0.001	Strong
LTMCS	0.378	0.016	Weak
PMU	0.977	<0.001	Very strong
MISPORTmax	0.793	<0.001	Strong
MISPORTavg	0.850	<0.001	Very strong
MISPORTavg/max	0.614	<0.001	Strong
S(8–12)	0.281	0.079	Weak
S(12–16)	0.324	0.043	Weak
S(16–20)	0.370	0.019	Weak
A(0–40)	0.293	0.066	Weak
A(40–80)	0.293	0.066	Weak

PCMs: Plan complexity metrics, MCSv: Modulation Complexity Score for VMAT, AAVcp: Aperture area variability at each control point, LTMCS: Leaf travel modulation complexity score, PMU: Planned monitor units, VMAT: Volumetric modulated arc therapy

For TrueBeam plans, the behavior of MI_VMAT differed from that observed on the Unique system. Consequently, the MI_CP index was used for correlation analysis. Separate detailed PCM tables for single-arc, double-arc plans, and TrueBeam plans are provided as Supplementary File 2.

DISCUSSION

Plans of intentionally varied complexity were generated by manipulating user-controlled optimization priorities to ensure sufficient modulation diversity for modeling. Limiting the dataset to oropharyngeal cases reduced anatomical variability, although natural differences in contour geometry, PTV-OAR relationships, and overlap volumes still contributed to inherent complexity variation.

This study evaluated the newly proposed VMAT modulation index, MI_VMAT, to assess its capability to PSQA outcomes across different delivery configurations. The predictive performance of the univariate MI_VMAT model was compared with a multivariate regression model developed using other established PCMs. In addition, the correlations between MI_VMAT and selected PCMs were also investigated. The analysis was subsequently extended to a high-precision Varian TrueBeam system to investigate the influence of machine characteristics on Modulation Index-GPR relationships.

GPRs were further evaluated under stricter criteria, up to 1%/1 mm, to perform sensitivity analysis with the new index and other PCMs and to assess the influence of machine characteristics. However, no GPRs stricter than 2%/3 mm for the unique system and 3%/1 mm for the TrueBeam system showed a significant correlation with predictive power. This observation may reflect differences in the isocentric positional accuracy of the two machines, indicating that delivery-related inaccuracies may contribute to the observed variations beyond plan complexity alone and may not necessarily correlate with the PCMs investigated in this study.

Nguyen and Chan et al.[16] used the method of deciding the threshold for each PCMs using the ROC curve, considering a false positive rate below 10% with a true positive rate ranging from 27% to 53%. In the study by Park et al. (2015a) and by Park et al. (2015b), they were not able to set a threshold value on the indices of the data they studied, and commented on this as a limitation. Hence, the present study adopts a more practical approach by developing a prediction model that demonstrates predictive potential and consistency with the distance criteria of GPRs. In addition, the model was evaluated and validated using both single-site and multi-site plans to assess its applicability for clinical use.

Statistical analysis and evaluation

Modulation index for volumetric modulated arc therapy in unique (single-arc and double-arc plan evaluation)

To study the behavior of the new index MI_VMAT, single-arc and multiple-arc plans were analyzed separately, and the computation method for multiple-arc plans was developed. Finally, the analysis was extended to VMAT plans delivered on the Varian TrueBeam system to investigate machine-specific effects.

Correlation analysis using Karl Pearson’s correlation coefficient revealed a statistically significant association (P < 0.001) between MI_VMAT and measured GPRs. In single arc and double arc plans, the correlation shows a significant correlation in the same pattern. Moderate correlations were observed for the 3%/3 mm and 2%/3 mm criteria (ρ ≈ 0.598 and 0.590), while weaker correlations were seen for more stringent criteria (ρ ≈ 0.404 and 0.399). These findings suggest that MI_VMAT captures the dominant modulation features influencing delivery accuracy.

A univariate regression model based on MI_VMAT was developed and validated using an independent cohort of 28 additional plans [Tables 4 and 5] for both single-arc and double-arc cases. For the 3%/3 mm criterion, the prediction errors showed mean deviations of 2.14% and 2.02% for single-arc and total 100 plans, respectively. For the 2%/3 mm criterion, the corresponding mean deviations were 2.76% and 2.54%, respectively. These results demonstrate that MI_VMAT provides reliable PSQA prediction for the 3 mm gamma criteria (3%/3 mm and 2%/3 mm), reinforcing its suitability for routine VMAT complexity assessment and PSQA prediction.

Importantly, when compared with the multivariate prediction model incorporating multiple PCMs, it is 2.50% and 2.91%, respectively, the univariate MI_VMAT-based model showed slightly superior performance, with lower mean deviations and reduced variability [Table 6]. This finding highlights a key advantage of MI_VMAT: its ability to encapsulate relevant modulation information in a single, interpretable metric, avoiding the complexity and potential overfitting associated with multivariate models.

In contrast, tighter criteria (3%/2 mm and 2%/2 mm) exhibited larger deviations, frequently exceeding 5%, reflecting the increased sensitivity of PCMs to geometric and dosimetric uncertainties. Prediction accuracy deteriorated for the 2 mm distance criteria, largely due to the mechanical limitations of the Unique linac, whose observed isocentric accuracy (~2 mm diameter) approaches the spatial tolerance of the gamma calculation, which is consistent with the increased sensitivity to delivery uncertainties. A similar behavior was observed for the multivariate prediction model incorporating multiple PCMs [Tables 5 and 6].

Relationship with established plan complexity metrics

The robustness of MI_VMAT was further evaluated through correlation analysis with established PCMs for single-arc and double-arc plans [Tables 9 and 10], including MCSv, LTMCS, PMU, MI_SPORT, and MLC kinematic parameters. Strong correlations (ρ >0.6, P < 0.001) were observed with LTMCS, PMU, MI_SPORT, MLC speed, and MLC acceleration, while MCSv exhibited a moderate correlation. These findings confirm that MI_VMAT is sensitive to both fluence modulation and mechanical delivery characteristics, bridging the gap between purely fluence-based metrics (e.g., MCSv) and kinematic descriptors. This dual sensitivity likely underpins its strong association with GPRs.

Modulation index on control point-evaluation on a high-precision TrueBeam system

The behavior of MI_VMAT was markedly different when applied to the TrueBeam system. For this platform, all evaluated plans achieved near-perfect GPRs (100% pass rate) for the 3%/3 mm, 3%/2 mm, 3%/1 mm, 3%/2 mm, 2%2 mm, and 2%1 mm criteria. Despite substantially higher and more variable MI_VMAT values, no significant correlation was observed between MI_VMAT and GPRs for plans delivered on the Varian TrueBeam system. This suggests that increased modulation complexity, as quantified by MI_VMAT, does not necessarily translate into delivery uncertainty in high-precision systems.

Consequently, the MI_CP, in which field weight (MU) is excluded, was further investigated. MI_CP demonstrated statistically significant correlations with GPRs for the 3%/3 mm, 3%/2 mm, and 3%/1 mm criteria (P < 0.001). Similar behavior has been reported for other PCMs on the TrueBeam system, where PMU was not identified as a significant predictor in the multivariate prediction model. In contrast, PMU was a major contributing predictor in all models for the Varian Unique system [Table D: Supplementary File 3]. It is also noted that, in the unique system, PMU showed strong correlations with MI_VMAT, with correlation coefficients of 0.886 and 0.977 for single-arc and double-arc plans, respectively [Tables 9 and 10].

Table D.

Fitted multivariate model for 3%/3 mm, 3%/2 mm, and 3%/1 mm in TrueBeam

GPRs	Predicted model	ρ	ρ2	P
3%/3 mm	Y = 95.238 + 1.596 LSV mean + 6.239 S (8 - 12) + 3.191 A (0 - 40) + 0	0.622	0.387	<0.001
3%/2 mm	Y = 97.7434 + 2.304 LSV mean + 8.060 S (8 - 12) - 3.978 A (40 - 80) + 0	0.646	0.417	<0.001
3%/1 mm	Y = 90.710 + 9.184 LSV mean + 16.432 S (8 - 12) + 0	0.686	0.471	<0.001

GPRs: Gamma pass rates

Modulation index on control point and relationship with established plan complexity metrics

Statistically significant correlations were observed between MI_CP and MLC displacement, speed, and acceleration, indicating that kinematic delivery parameters dominate modulation behavior on the TrueBeam platform [Table 11]. Consistently, the same predictors were identified in the multivariate regression model for this cohort [Table D: Supplementary File 3]. These findings suggest that delivery accuracy in TrueBeam is less sensitive to field weight (PMU). Consequently, MI_CP, which excludes field weight, demonstrates stronger predictive capability for high positional accuracy systems such as TrueBeam.

Table 11.

Comparison of the index modulation index on control points from true beam data with previously suggested plan complexity metrics with significant P values

PCMs	ρ	P	Relation
LSVcp	0.306	0.031	Weak
S(16–20)	0.345	0.014	Weak
Savg	0.317	0.025	Weak
A (0–40)	0.459	0.001	Moderate
A(40–80)	0.459	0.001	Moderate
Aavg	0.309	0.029	Weak

PCMs: Plan complexity metrics

A univariate prediction model based on MI_CP demonstrated excellent agreement with the measured GPRs in an independent validation cohort of 14 patients, with mean deviations well below 1% [Table 7]. When compared with a multivariate prediction model incorporating multiple PCMs, the univariate MI_CP-based model showed equal or slightly superior performance, exhibiting lower mean deviations and reduced variability [Table 8].

These findings confirm that MI_CP is sensitive to both fluence modulation and mechanical delivery characteristics, bridging the gap between purely fluence-based metrics (e.g. MCSv) and kinematic descriptors. This dual sensitivity likely underpins its strong association with GPRs.

Influence of machine characteristics and physical interpretation

The divergent behavior observed between Unique and TrueBeam systems underscores the machine-dependent nature of modulation-GPR relationships.

Machine mechanical characteristics are sensitive to studied PCMs and also sensitive for new index. In the TrueBeam, delivery accuracy is largely decoupled from MU modulation and field weight, as evidenced by the lack of correlation between PMU (field weight factor) and GPRs. Instead, geometric and kinematic descriptors such as LSV_CP, MLC speed, and MLC acceleration emerged as dominant predictors. This finding suggests that, in high-end systems where gantry and positional uncertainties are minimal, residual GPR variations are driven primarily by leaf motion dynamics rather than fluence intensity modulation. MI_CP, by emphasizing control-point–level aperture variation and delivery kinematics, therefore becomes a more appropriate predictor than MI_VMAT in such environments.

Furthermore, it is clear that the PCMs and MI_VMAT/MI_CP are sensitive and predictors of spatial accuracy, and hence the models are limited to 3 mm criteria in Unique and do up to 1 mm criteria for TrueBeam. The PCMs studied are based on mechanical parameters of the machine, and hence it is correlated with special accuracy and directly capable of pataial criteria prediction up to 1 mm accuracy.

Clinical implications and summary

Overall, this study demonstrates that MI_VMAT/MI_CP, when applied appropriately, provides powerful, machine-specific predictors of PSQA performance. For conventional systems such as the Varian Unique, MI_VMAT offers a compact, univariate predictor capable of matching or exceeding multivariate PCM-based models. For high-precision systems such as the TrueBeam, MI_CP better reflects the dominant physical drivers of delivery accuracy. When interpreted within the appropriate geometric and mechanical context, MI_VMAT/MI_CP can serve as reliable tools for PSQA decision-making, potentially enabling selective QA, improved workflow efficiency, and enhanced clinical confidence in VMAT delivery.

The results suggest that the MI_CP index for high-end machines can be further optimized in future work by incorporating MLC speed and acceleration at CPs. Its application to SRS/SBRT plans is yet to be evaluated and is outlined as future work.

CONCLUSION

This study demonstrates that the proposed modulation index, MI_VMAT, is sufficiently robust to function as a stand-alone PCM for VMAT and as an effective predictor of GPRs. Unlike previously reported indices, MI_VMAT explicitly incorporates the three fundamental components of VMAT modulation – MLC motion, dose-rate variation, and gantry speed – within a unified framework. The new index showed strong and consistent correlations with PSQA outcomes and achieved predictive performance comparable to, or better than, multivariate models constructed from multiple established PCMs, including MCSv, LTMCS, PMU, MI_SPORT, and MLC speed and acceleration. It was also observed that both the proposed index and the evaluated PCMs are primarily based on machine mechanical parameters and are therefore more sensitive to spatial delivery accuracy. And hence, in the Varian Unique system, predictive capability was reliable mainly for the 3 mm gamma criteria, whereas for the TrueBeam system, prediction remained reliable up to the 1 mm criterion. In the high-precision TrueBeam platform, the predictive capability of MI_VMAT and other PCMs appears to be predominantly governed by fluence-editing parameters such as MLC positions, speed, and acceleration. Since MU and field weight contributions are effectively minimized, MI_CP emerged as a more appropriate predictor for PSQA performance in high-precision delivery systems.

Conflicts of interest

There are no conflicts of interest.

SUPPLEMENTARY FILES

Supplementary File 1

Method: Demonstrating the Measure of MICP

The modulated beam (Wp) is represented by varying intervals of meter set weight (MSW) multiplied by corresponding relative AAVcp. This explanation adopts Webb’s method for measuring the degree of modulation. For simplicity, consider a treatment plan using an arc beam, which consists of a series of control points (CPs) in 2-degree arc lets.

Let the arc beam have (n + 1) =180 control points with corresponding sequential cumulative MSW values. From this series, the difference between the cumulative MSW values at each CP is Xp, where P = 1, 2, 3,…, n. where n = 179 in this case. Aperture area AAVp for n control points taking corresponding to MSW Xp. Where, AAV_p = (AAV_p + AAV _P+1)/2

Calculate the mean and standard deviation (σ_W) of these WP values. Let ΔWp = ∣Wp − Wp − 1∣| for P = 1, 2, 3,…, n − 1 be the magnitude of the difference between the CP values Wp and Wp − 1, where Wp is the value at the p-th control point and Wp − 1 is the value at the (P − 1) th control point.

Now, count the number N (f;ΔWp > f σ_W) of changes for which ΔWp > f σ_W, where f = 0.01, 0.02,…, 2 and hence the fraction z (f) of significant changes among all changes in the CPs, i.e., z (f) = (f; ΔWp > f σ_W). The function z (f) is a spectrum of the number of adjacent-element changes that exceed a certain fraction of the standard deviation (σ_W) in the beam. It always falls monotonically as f increases. The area under the spectrum [Figure 1 ^{(558.6KB, tif)} ] is an indication of the degree of modulation and

MI = has been chosen as a single numerical measure of the modulation, termed the ‘modulation index on control points’ (MI_cp). i.e., The upper limit chosen as 1.5 after that z (f)=zero.

Calculation Steps for MI _cp :

1. Extract Data: The modulated MSW values are extracted from the plan data.

2. Accounting AAV: Multiply AAVp and Wp

3. Compute Differences: Calculate the difference between adjacent indexed control points (CPs) to obtain the WP values. These values are multiplied by 100 to simplify calculations since ΔWp accounts for very small values.

4. Calculate Differences: Compute ΔWp= ∣Wp − Wp − 1∣, the absolute difference between the MSWs of adjacent control points.

5. Calculate mean and standard deviation,σ_W of Wp

6. Compute z (f): Calculate z (f) = (f; ΔWp > fσ_W), where N (f; ΔWp > fσ_W) is the number of ΔWp values exceeding fσ_W. Here, z (f) is the fraction of ΔWp values that are greater than a certain fraction f of the standard deviation σ_W.

7. Plot and Integrate: Plot z (f) against 100f σ_W. Calculate the area under the curve to determine MI.

8. Monotonic Decrease: Observe that z (f) decreases monotonically as f increases and reaches zero before f = 1.5 in all cases. Hence, the upper limit of integration is set to 1.5.

9. Tabulate Values: For each f value, compute N (f; ΔWp > fσ_W) and tabulate z (f)= (f; ΔWp > fσ_W)

10. Perform Integration: Approximate the integral as the summation of fine intervals with f = 0.01. The area under the curve, representing MI_cp, is

where Δf_w = 0.01

It is also evident from the direct plot of z(f) versus f that z(f) approaches zero well beyond the selected upper limit of 1.5. Alternatively, the Python code can be set to extend f until z (f)=0

Field weight-a function of the three variables

Methods adopted in modulation in the planning level are explained as per the Varian documents, which are given below

Steps involved in the plan optimization and calculations are explained as follows:

1. The volume representation is a voxel model and is converted to voxel matrix

2. During optimization, a required fluence map is generated at the Iso Centre level (or at any level to get 3D pattern), and dose is calculated using the Multi Resolution Dose Computation (MRDC) algorithm [Figure 2 ^{(463.8KB, tif)} ]

3. The optimized and calculated dose pattern returns to the leaf motion calculator (“calculate MLC motions” to MLC control points) to obtain modulated MLC opening (fluence) and MU/deg [Figure 3 ^{(503.8KB, tif)} ]

4. Re-optimization up to the required level is done, again and again, and converting this optimized dose pattern inversely to MLC motions and MSW modulation in the optimization stage [Figure 4 ^{(420.8KB, tif)} ].

5. After optimization, the 3D dose distribution is then calculated using AAA or Acuros XB.

6. The MUs calculated are in such a way that they deliver the optimized dose to each voxel (to each point) three-dimensionally and provide adequate dose coverage to the target volumes.

7. MU is differentially distributed from different gantry angles with respect to MSW and differential beam opening.

Treatment monitor unit (MU) calculation primarily depends on parameters such as field size and depth. From the description of the optimization procedure, it is evident that

• Intensity modulation in VMAT or FF-IMRT is achieved through volume optimization, and differences in the depth of each voxel from different angles are considered or accounted for as an objective during optimization. (Considering not as a factor of dependence in MU calculation, dose calculation parameters.)

• The varied Aperture area is the result of optimization. (Considering not as a factor of dependence in MU calculation, dose calculation parameters.)

• The MU is calculated for each voxel from each beamlet as per the dose map generated in the optimization part, and also, the MU to each voxel may be different from any other voxel, depending on the optimized dose map. Here, the concept of dose rate in MU calculation is working only if it is considering each voxel, i.e. dose rate can be calculated only if considering a point to be treated using dose calculation parameters.

Hence, the field weight value (MU/cGy) extracted from plan data is a factor of optimization only, not the MU calculation parameter, as the dose rate at a point to be treated. It has been merely taken as a numerical value, and therefore, it is clear that the field weight is a function of three variables. And also, field weight is a factor to be considered in the complexity measure.

Therefore, Modulation Index (MI_CP) is corrected using an optimization index (I_FW) to obtain the more apt Modulation Index MI, which can be calculated using the formula,

MI_VMAT = MI_CPI_FW.

The index on field weight is determined by the equation I_FW = e^-2FWin order to transform the MI_CP into a nonlinear form, as it exhibits nonlinear correlation with other complexity matrices and gamma pass rate values.

MI_VMAT = MI_CPI_FW

where,

********^{end of theory part} *********

Figure 1

z (f) vs 100fσ_W

Figure 2

Varian document 1

Figure 3

Varian document 2

Figure 4

Varian document 3

Funding Statement

Nil.