Extracting the gradient component of the gamma index using the Lie derivative method

Objective. The gamma index (γ) has been extensively investigated in the medical physics and applied in clinical practice. However, γ has a significant limitation when used to evaluate the dose-gradient region, leading to inconveniences, particularly in stereotactic radiotherapy (SRT). This study proposes a novel evaluation method combined with γ to extract clinically problematic dose-gradient regions caused by irradiation including certain errors. Approach. A flow-vector field in the dose distribution is obtained when the dose is considered a scalar potential. Using the Lie derivative from differential geometry, we defined L, S, and U to evaluate the intensity, vorticity, and flow amount of deviation between two dose distributions, respectively. These metrics multiplied by γ (γL, γS, γU), along with the threshold value σ, were verified in the ideal SRT case and in a clinical case of irradiation near the brainstem region using radiochromic films. Moreover, Moran’s gradient index (MGI), Bakai’s χ factor, and the structural similarity index (SSIM) were investigated for comparisons. Main results. A high L-metric value mainly extracted high-dose-gradient induced deviations, which was supported by high S and U metrics observed as a robust deviation and an influence of the dose-gradient, respectively. The S-metric also denotes the measured similarity between the compared dose distributions. In the γ distribution, γL sensitively detected the dose-gradient region in the film measurement, despite the presence of noise. The threshold σ successfully extracted the gradient-error region where γ > 1 analysis underestimated, and σ = 0.1 (plan) and σ = 0.001 (film measurement) were obtained according to the compared resolutions. However, the MGI, χ, and SSIM failed to detect the clinically interested region. Significance. Although further studies are required to clarify the error details, this study demonstrated that the Lie derivative method provided a novel perspective for the identifying gradient-induced error regions and enabled enhanced and clinically significant evaluations of γ.


Introduction
Stereotactic radiotherapy (SRT) and intensity-modulated radiation therapy (IMRT), including volumetric modulated arc therapy, generate complex dose distributions.Particularly, SRT creates a steep dose gradient to concentrate high doses on target tumors and to minimize dose exposure to organs at risk (OARs) or to maximally reduce doses to tumor-adjacent OARs.Typically, in an actual treatment, an unintended higher peripheral dose than the prescribed dose may cause harm (Hsu et al 2017).Thus, the high-dose gradient region adjacent to OARs should be carefully evaluated as it may cause adverse clinical outcomes (Benedict et al 2010, Videtic et al 2015, Videtic et al 2019).Previously, only a 2 mm expansion of a high-dose area was reported to increase cerebral necrosis in brain SRT (Nataf et al 2008, Kirkpatrick et al 2015).Thus, the steep dose-gradient region should be appropriately evaluated through actual measurements, i.e. patient-specific quality assurance (PSQA) (Bender 2014, Wen et al 2016, Wootton et al 2018, Peng et al 2020).Moreover, an agreement of the gradient region between the planned and actual dose distributions indicates an accurate implementation of the expected treatment.
When comparing two dose distributions for a patient, a typical method used in radiotherapy for evaluating deviations is the gamma index (g ) analysis (Low et al 1998, Low 2010).The g -index represents a value that combines the dose difference (DD) and distance-to-agreement (DTA) with their tolerance limits for the evaluation of the appropriateness and accuracy of the measured and calculated dose distributions (see appendix equation (C1)).Recently, a guideline for γ analysis has been established, i.e. task group (TG) 218 by the American Association of Physicists in Medicine (Miften et al 2018).Consequently, an increase in appropriate use following the guideline recommendations by experts has been noted, despite the prevalent facility-specific techniques and clinical objectives in Japan (Anetai et al 2022b).Although the method was initially proposed by Low et al (Low et al 1998, Low et al 2013), the γ-analysis exhibits radically problematic characteristics during evaluation (Heilemann et al 2013, Stojadinovic et al 2015).First, the ellipsoidal interpretation for evaluation points does not readily identify errors caused by the DD, DTA, or both (Park et al 2018).Second, an arbitrary dose threshold is required, as extremely low doses are unsuitable for evaluation.This threshold is typically set to 10% of an evaluation base dose, for instance, a prescribed dose per fraction, and is simultaneously ambiguous with respect to the undetermined base dose.Third, the obtained γ-distribution has various parameters hidden within its facility-specific background.Thus, the resolution of the dose calculation and interpolation, dose calculation method depending on the treatment planning system (TPS), measurement device or phantom, analysis software, and global or local comparison, among other factors, should be considered (Hussein et al 2017).Therefore, steep dose gradient regions cannot be accurately evaluated using the γ index, and the TG-218 guideline primarily provides documentation for γ use in general dose distributions.
In several studies, the dose gradient has been measured using the dose-volume index (DVI), (Ohtakara et al 2011, Cao et al 2019), gradient index (GI) (Paddick and Lippitz 2006), dose-gradient index (DGI) (Wagner et al 2003, Reynolds et al 2020), and the dose-gradient curve represented by the dose-volume surface and isodose levels (Sung and Choi 2018).Specifically, the GI value gained popularity in clinical environments owing to its simplicity and convenience, as it can evaluate the dose-fall amount outside the treatment volume of irradiation as the ratio of the 50% prescribed dose-volume against the 100% prescribed dose-volume or as the effective radius defined by a sphere (Ohtakara et al 2011, Reynolds et al 2020).Although the GI and DGI provide the degree of dose fall-off outside the planning target volume (PTV), these criteria do not identify heterogeneously distributed complex strong dose gradients aimed at a dose reduction for critical OARs as shown in figure 1.Moreover, no significant gradient differences have been observed between CyberKnife and TrueBeam Edge from the GI values defined by the both of indices using the effective radius and treatment volume (Cao et al 2019).The GI and DGI conveniently evaluate the dose gradient based on a simple criterion.However, these indices are insufficient for evaluating the gradients of complex dose distributions.An accurate evaluation of the dose gradient requires the location information of errors.
A combined analysis for the dose gradient with γ can serve as a powerful tool, provided that the γdistribution locationally specifies errors.Dose-gradient compensation (Moran et al 2005) has been proposed for the dual evaluation of the DD and gradient (Moran's GI: MGI); however, this method is unsuitable for SRT PSQA, which considers the averaging of gradients in different directions.In addition, the MGI calculates the gradient solely with respect to the reference dose distribution (see appendix equation (C2)).An alternative method is Bakai's χ factor (Bakai et al 2003), which provides the DD considering the error propagation term of the DD and DTA criteria (see appendix equation (C3)).Although the gradient term provides a certain correction for the DTA term in the χ factor and  c g, as demonstrated by Bakai, the gradient-induced region remains unclear, as the larger norm of the gradient term suppresses χ-value growth.Moreover, χ also considers only the gradient of the reference dose distribution.The structural similarity index (SSIM) is an alternative solution for the comparison method from a new perspective (see appendix equation (C4)).This index can directly compare two image qualities based on luminance, contrast, and structural comparison factors.Peng et al applied the SSIM to extract dose distribution errors by comparing g distributions (Peng et al 2020).Peripheral irradiation errors were primarily extracted by the luminance term; however, the relationship between extracted errors and the gradient region were not sufficiently extracted.These values were considered more comprehensively, as detailed further in this study.A more direct representation method for comparison for the discrepancy between gradients in the two dose distributions is required.
From the differential geometric approach, discrepancies in the dose distribution are generally considered as deviations of the two dose-flow fields.In a previous study, radial and angular evaluations of the discrepancies using the Lie derivative between the calculated and irradiated simple rectangular fields were reported (Anetai et al 2022a), and it was found that the Lie derivative facilitates the useful detection of errors in the penumbra region based on the flow deviation intensity.The Lie derivative is radically related to diffeomorphism, which is familiar in the radiotherapy field as a technique of deformable image registration (Ashburner 2007, Avants et al 2008, Mok and Chung 2020).In this study, the Lie derivative method concept was extended by applying it to the discrepancy in the general dose distribution to enhance the evaluation of the dose-gradient region of the γ-distribution in a PSQA process.Then, we aimed to extract the gradient-induced γ to assess the PTV-to-OAR region that should be evaluated more accurately for SRT.

Material and methods:
Lie derivative metrics and their applying to the deviation between dose distributions The Lie derivative naturally defines the discrepancy between vector fields X and Y as a commutator X Y , .

[
] For example, the discrepancy between the planned dose-flow X and delivered dose-flow Y is represented as = Y X Y L , .
X [ ] This deviation can be applied to the orthogonal three-dimensional x y z , , -coordinate system (see appendix B); however, for simplicity in this study, we considered the two-dimensional comparison in an axial plane.Here, a dose distribution deemed a velocity potential, which is natural because the dimension of dose is represented as | }denotes coordinate suffixes, x and y represent right-to-left and anterior-to-posterior, respectively, and the direction from a higher dose to a lower dose of the velocity vector is positive.Then, the Lie derivative ) ) can be specifically calculated as follows: where a partial derivative operator ¶ m also denotes a basis vector.We defined the following three metrics, L, S, and U, according to the deviation vector fields yielded by the Lie derivative.Thus, as shown in the case, a dose-gradient evaluation using GI or R50 assuming sphere is insufficient.
where L is the intensity of the deviation represented by 2-norm Y L , X | | S denotes the strength of the deviation vector field (rotational axis rigidity as the vorticity term, leading to robust deviation), and U represents the flow amount of the deviation vector field (the influence of the gradient-induced deviation).Detailed calculations of L, S, and U are presented in appendix B.

Verification in the ideal and clinical case dose distributions
We obtained the calculated dose distributions using an Eclipse TPS (Varian, Palo Alto, CA, USA).First, we verified the basic characteristics of L, S, and U with respect to the various complex gradient regions in an ideal SRT study case.Hence, a PTV was set as a sphere with a diameter of 4 cm covered by a 95% prescribed dose of 5 Gy per fraction, to include various understandable dose gradients; this was developed using a static-field IMRT technique with a 2 mm grid calculation of the analytical anisotropic algorithm (AAA).The details of the beam planning for the dose distribution are depicted in figure 2. The gradient difference in the complex dose distributions between the original and laterally 2, 4, 6, 8, and 10 mm-shifted dose distributions was compared (plan versus plan study).Second, the film measurement-base dose distribution was verified with a +2 mmshifted isocenter case using a phantom (film measurement study).The analytic resolution of the film was 72 dpi (0.353 mm pixel −1 ), and the TPS calculation was expanded via bilinear interpolation for comparison in the measurement resolution.The g distribution was computed using VeriSoft (PTW, Freiburg, Germany).Gafchromic EBT3 (Ashland Inc, Wayne, NJ, US) was used in the film measurement.
The differential dose distribution was calculated according to the second-order finite difference (see appendix D) using Python-based in-house software, where the numerical infinitesimal intervals (D D , j i ) were adopted as 1.0 in the TPS calculation and the film measurement to prevent significant magnifications in the differential value due to an influence up to the third power (equations (1)-(2).The distributions of metrics L, S, and U were compared with the g distribution with respect to the 3%/2 mm, 2%2 mm, and 3%/1 mm criteria of DD/DTA.We then defined the threshold values for g filtered by L, S, and U as s and determined the effective value in the film study using ideal and clinical cases requiring high dose-gradients.For comparisons, we also calculated MGI, c, and SSIM in this study (see appendix C).The CTV region in this study was defined as a sphere with a diameter of 4 cm at the center of the phantom, which was anteriorly and posteriorly adjacent to OARs; here, posterior OAR1 defines a sphere with a diameter of 3 cm, and anterior OAR2 denotes a sphere with a diameter of 5 cm.The PTV was added as a 5 mm threedimensional (3D) margin to the CTV (approximately equal to a sphere with a diameter of 4 cm).The dose distribution was designed to satisfy the 100% isodose (= 5 Gy) line and PTV (D95 = 5 Gy) and to maximally reduce the high-dose to the OAR using the static-field IMRT technique (nine fields per 15°of the gantry angles from 300°to 60°were used to exclude scatter effects by the treatment couch).Abbreviations: CTV, clinical target volume; PTV, planning target volume; OAR, organ at risk.

Results
Gamma and Chi indices, MGI, and SSIM calculation for the plan versus plan study We obtained two high-intensity-modulated dose distributions based on the TPS calculations in an ideal case (figures 3(a)-(b) and conducted their comparisons.The +2 mm laterally (patient-left side) shifted isocenter case D Y was recalculated in the TPS, and it was compared with the non-shifted dose distribution D .
X The DD, -D D , Y X and γ and c distributions for the DD/DTA criteria of 3%/2 mm, 2%/2 mm, and 3%/1 mm were obtained, respectively, as shown in figures 3(c)-(i).Although the γ distributions could be more sensitive with respect to the gradient region when the DTA was subjected to a more stringent criterion, the nearest strong DDs were also affected, as shown in figure 3(c).Moreover, -D D Y X does not radically indicate the intensity of the gradient.By contrast, Bakai's c detected more gradient regions owing to its gradient term; however, the gradient-induced error still remained unclear, since the dose difference term also radically affected.
The SSIM and its components (luminance, contrast, and structural similarity) were obtained as shown in figures 3(j)-(l).Notably, the SSIM provided the dose error between two dose planes and similar γ distributions.The luminance factor barely extracted strong DTA; however, the contrast and similarity factors sensitively detected small dose differences.Overall, as shown in figure 3(f), the SSIM was unclear for the gradient-induced error region, indicating that the SSIM is unsuitable for comparing similar dose distributions.The MGI expects the gradient region to conform to the total derivative; therefore, the gradient averaging effect was observed (figure 3(l)).

Lie derivative metrics for the plan versus plan study
Each dose distribution had a flow field as a gradient when the dose was considered as a potential (figure 4).Highdose gradients, such as those in the peripheral region of the PTV, were observed in the area of high-intensity modulation.The flow-vector fields of D X (dose for reference) and D Y (dose for comparison) were obtained as ) represent the flow deviation between D X and D Y for the x and y-direction base vectors ¶ x and ¶ , y respectively.Figure 5(i) presents the Lie derivative deviation Y L X as a vector field.In general, the deviation vector field is not uniform because of the non-uniform dose distribution; thus, the vector field becomes a locally characterized turbulent flow.
The Lie derivative deviation facilitated the extraction of a high gradient of the dose distribution with respect to the surroundings, which explicitly indicated that D Y required a change in the flow of the dose distribution with a clockwise rotation with respect to D .
X Figures 5(j)-(l) depict L, S, and U with respect to the Lie derivative deviation.These values detected each distinctive point in the deviation vector fields: the L-value was used to extract particular points where the flow changed rapidly (figure 5(j)), S-value demonstrated a significant vorticity (i.e. the robust deviation against the surroundings) with coordinate-invariant quantities (figure 5(k)), and U -value indicated the amount of movement during the flow change (figure 5(l)).In particular, metrics L, S, and U -metrics accurately detected the deviation of the dose gradient owing to different dose distributions.Among these metrics, L demonstrated point-oriented deviations, whereas S and U performed areal detections.This could be attributed to the relationship L with respect to the tangential vector and the relationship S, U with respect to the normal vector.A higher L was observed in the strong different gradient region caused by the isocenter shift such as the peripheral curved surface of the phantom, tangential path of the beam owing to the strong intensity modulation, and peripheral boundaries of the PTV.
Sensitivity and characteristic of the proposed L, S, U-metrics with or without the γ index Figure 6(a) illustrates the deterioration in the three metrics under different dose distributions.As can be observed from the figure, we obtained D Y as recalculated dose distributions with +2-10 mm isocentric shifts.The results revealed the sensitivity of the metrics: L and U reflected any gradients between two dose distributions.By contrast, S indicated specific gradients receiving the difference from the original dose distribution D .
X Given that the S-value was radically related to the vortex in the turbulent flow, the S-value decreased corresponded to the dissipation of deviation vector when the disorganized flow effect in turbulence increased.This result indicated that the L, S, U-metrics should be evaluated in the range of each valid metric in a complementary manner.
In the comparisons of the g -distribution values under different DTA/DD conditions (figure 6(b)), high Lvalues had g > 1 under the tighter DTA condition.The discrepancy of the dose gradient in the distribution of g was appropriately extracted using > L 1.However, L was insensitive to the tighter DD constitution.When S > 0.20, g was approximately greater than 0.5 (figures 5(b), (e), and (h)).A high S-value was related to g > 1 under the tighter DTA condition.By contrast, U was more appropriate for the gradient-induced DD than for DTA.A higher U was related to a tighter DD condition with respect to the same point receiving different flows.Low L, S, and U values ignored the g -value unrelated to the gradient region.
Lie derivative analysis and metrics for ideal plan versus film measurement study Similar to the plan versus plan study, the dose distribution of film measurement (+2 mm shifted isocenter) was compared with that of the TPS calculation.The film dose (figure 7(b)) was compared with the reference dose (TPS), and the g, SSIM, MGI, c, and L, S, U distributions were obtained as shown in the figures 7(c)-(l), where g detected an intentional 2 mm shift under a tighter 1 mm DTA, whereas the 2 mm DTA yielded a lenient evaluation (figures 7(d)-(f)).However, g uniformly evaluated the dose distributions, which is irrelevant to directional errors (figure 7(f)).The SSIM was found to be susceptible to noise; therefore, it detected multiple non-significant points with respect to the dose gradient (figure 7(g)).The MGI detected strong dose-gradient lines, whereas it simultaneously detected the noise gradient (figure 7(h)).Moreover, c extracted similar g points; however, the weighted DTA factor influenced the irrelevant points in the dose gradient (figure 7(i)).By contrast, metrics L, S, and U performed adequately with respect to the gradient-induced deviations, although several noise effects were observed (figures 7(j)-(l)).
γ distribution filtered by L, S, U-values in the plan versus plan and plan versus film measurement and the determination of threshold σ The abovementioned characteristics were applied to the gamma distribution of the plan versus plan or film measurement.The L, S, U-value filtering, denoted as gL, gS, and gU, respectively, demonstrated a significant detection of the gradient-induced g region according to their characteristics.In this study, we defined the same filtering threshold s value for the three metrics.As indicated for s = 0.1 in figure 8(a) with respect to the 3%/1 mm DD/DTA g criterion, gL successfully extracted g -fail points defined by g > 1.0 under the shift in the high-dose gradient.Moreover, gS extracted g -fail points where the flow difference caused an intensive change.The gU extracted inflows and outflows yielded by the high-dose gradient shift.This was primarily in the surroundings of points where the high-dose gradient shifted owing to an indirect relation with the rigor DTA g -fail points.
For s = 1.0, gL extracted a significant gradient-induced error region, which was associated with peripheral irradiation or tangential beams; however, gS and gU disregarded numerous g -fail points that should be considered.Inversely, when s = 0.01, the filtering was excessive and detected meaningless g -values.To balance these, we determined s = 0.1 in this plan versus plan case, such that each metric contributed to the flow deviation.Figure 8(c) depicts the number of detected g -value points with respect to s.When s = 0.1, each   Figure 8(b) depicts comparisons between the expected and measured dose distributions with respect to the 3%/1 mm DD/DTA g criterion.In the actual measurement using a film, the measurement noise accompanied by a sub-millimeter resolution for both axes increased; therefore, the filter threshold was down to an order of approximately - 10 .
2 Similarly, we determined s = 0.001 for the film measurement based on the following rationales: (1) gS remains valid for different DD/DTA criteria, (2) gU exhibits a high DD sensitivity, and (3) gL detects gradient regions with a noise increase, resulting in the maximum L-value at the - 10 2 level in the gradient area when compared with the plan versus plan (figures 7(j)-(l)).As shown in figure 8(d), the detection ratios g g S L / and g g U L / in the film measurement were 0.294 and 1.025 for s = 0.001, respectively; these ratios were similar to the plan versus plan ratios for s = 0.1, which were 0.268 and 0.881, respectively.
Applying L, S, U-values to the clinical evaluation of the PSQA using the film measurement We applied the proposed metric approach to the clinical evaluation based on the above results.Figure 9(a) presents a clinical example of brain SRT prescribed by 42 Gy per 10 fractions.As shown in the figure, the tumor region (PTV) was significantly close to the brainstem.We provided a treatment plan to satisfy the condition that the maximum received dose decreased by 15% with respect to the prescribed dose (35.7 Gy/42 Gy), considering that the biologically equivalent dose when the fractional dose was 2 Gy, i.e.EQD2, with an α/β value of 2.0, equaled to 49.71 Gy, which was less than the brainstem-tolerable dose of 54 Gy.Therefore, an accurate strong dose gradient had to be achieved for PTV-to-OAR.Figures 9(b)-(c) depict dose distributions in the phantom from the TPS calculation and film measurement.The γ criterion with 3%/1 mm DD/DTA achieved a mean γ of 0.34 with a favorable pass rate 98.7%, as shown in figure 9(d).The γ-fail points when g 0.5  included multiple noise events during evaluation (figure 9(e)).By contrast, g > 1.0 extracted specific points; however, the causes of the error and its clinical significance could not be readily discerned.
The red circles in figures 9(e) and (f) indicates the region wherein clinical focus is required, which directly relates the strong dose gradient of PTV-to-OAR.The original g analysis neglected the region despite the DTA deviation of more than mean-γ.Filtered by the L, S, and U-values, we re-mapped the dose-gradient-related γ distribution (figures 9(g)-(i)) and appropriately accounted for the PTV-to-OAR region for evaluation.Higher s values in the gL map detected extreme gradient-induced flow deviation points, whereas gS indicated that the intensity of deviation was weak for the PTV-to-OAR region.The detection ratio of g g S L / at 0.002 against 0.001 of the s value, 34/588 versus 351/2476, was 0.82%.Therefore, we determined that the PTV-to-OAR dosegradient region was realized in the PSQA process.Conversely, MGI detected the strong gradient of PTV-to-OAR; however, the detected gradient region was influenced more by the tangential beam path than the gradient difference of dose distributions (figure 10(i)).With respect to the SSIM, based on the statistical similarity between the distributions, the points at which the dose was less correlative were extracted regardless of the gradient region (figure 10(j)).The pass rate was 98.7% with the mean-g value 0.392. (e)-(f) Illustrations of the g 0.5  and g > 1.0 distributions with the number of detections.The red circle indicates the clinically significant region for evaluating the PTV-to-OAR dose gradient.(g)-(i) L, S, and U filters with respect to the g -value (3%/1 mm, DD/DTA) in the clinical case with the number of detections via metric thresholds of 0.001, 0.002, and 0.005, respectively.

Discussion
The g index analysis is expressed in a mathematically rigorous form with the clinical tolerance terms and can be conveniently used for clinical evaluations.However, the differences in dose distributions are complex in terms of the dose modulation or high convergence to the cancer target, resulting in an insufficient evaluation based on the pass-fail g > 1 distribution.Moreover, the DD is highly sensitive to the gradient, whereas the DTA presents a significant problem associated with the evaluation of the dose-flat region.These unfavorable characteristics of g remain unchanged respect to different DD/DTA criteria.Moreover, the facility-or patient-specific considerations regarding the criteria of the DD, DTA, and dose thresholding may be used.Bakai's c modifies the term of the DTA criterion according to the gradient term and ignores of the DTA error due to the dose-flat region.Although innovative, this factor still underestimates the gradient-induced error region, as shown in figures 3(g)-(i), owing to the sole consideration of the reference dose-gradient.Apart from DD/DTA terms, the MGI reconsiders the dose gradient from the distribution.However, this metric also only considers the reference dose distribution.Moreover, the term requires the averaging of all directional gradients, leading to deterioration in the gradient intensity (figure 3(n)).By contrast, the SSIM statistically detects the difference between distributions; however, this factor does not readily focus on the gradient region in similar dose distributions (figures 3(j)-(m)).
The use of traditional methods to evaluate specific gradient regions is difficult.However, g has been extensively investigated in clinical practice.A novel perspective combined with g can serve as a powerful tool for clinical evaluations.Therefore, we approached this issue based on differential geometry using the Lie derivative.The Lie derivative can quantify the deviation between different vector fields, which is a mathematically rigorous description, as indicated by equations (A1)-(A4) in appendix A. The proposed method using the Lie derivative can be used to directly evaluate gradient points with locational information, considering the deviations between two dose distributions.Thus, the proposed approach can identify gradient-enhanced discrepancies in the g distribution.In this study, we obtained the deviation quantities as L, S, and U from the deviation of the dose- flow vector field Y L ; X the intensities of the flow field deviation L, vorticity S, and dose-flow amount U were multiplied by g.The use of this g -value for SRT is appropriate for 1 mm or tighter DTA conditions, which are required to be more sensitive to the gradient region of dose distribution; therefore, we mainly evaluated 3%/1 mm DD/DTA.
First, L directly represented the intensity of the dose-flow deviation.This provided the point at which a significant flow change occurred within a particular high-dose gradient region.A high L (L 1  ) related with a high gradient-induced g value, as shown in figure 6(b), thus indicating a significant gradient-induced deviation linked with a high g value error.Moreover, the L filtering (gL) process detected gL 1;  the detected points were significant error points with L 1  and g 1.
 If these points were found in the PTV or normal-tissue area with a slight change in the dose distribution, such as a difference equivalent to a one-slice thickness of a CT scan, it would be necessary to determine whether the gradient deviations were acceptable.This approach for the plan versus plan consideration or calculation-based PSQA without measurement (Zhu et al 2021) could be a useful method.
Second, S retrieved the strength of vorticity in the deviation vector field Y L , X which implies rotational axis rigidity against surroundings of the point of interest, denoting deviation robustness.The deviation vector field became weak when the perturbation due to the disorganized vector was dominant, that is, appearance of the stagnation points in the flow, followed by the vorticity term approaching to zero.Therefore, S disappeared when the difference between two dose distributions was significantly large (figure 6(a)).Thus, S can be used to measure similarity with respect to the reference dose distribution.This characteristic is useful, as focus can be directed toward the gradient error between close distributions, although the deviation satisfies the gamma index analysis.
Finally, U extracted the strength of the flow amount, thus indicating that the surroundings of the strong gradient point were extracted by this metric.From the deviation vector field viewpoint, the points at which a large DD occurred near the high-dose gradient points according to the flow change were mainly included (figure 3(n)).Thus, gU was more sensitive to the DD change than the DTA (figure 6(b)).Although the U -filter exhibited weaker and indirect locating characteristics for the gradient-induced g compared with L and S, a high L-value located near a high U -value represented robust deviations; therefore, gL and gU should be comparable in the evaluation.
Hence, gL should be evaluated when gS has a valid value, and this should be comparable to gU.The metric threshold s for gL, gS, and gU was determined to satisfy the condition.As shown in figures 7(c)-(d), s = 0.1 for the plan versus plan and s = 0.001 for the plan versus film measurement were appropriate.Thus, the s value should be determined considering the resolution for comparisons, which is derived from numerical calculations for the differential value of the Lie derivative.In particular, the difference in resolution at the 1-2 mm scale (dose calculation grid level) was small, whereas the difference at the - -10 10 1 2 -scale was large (detector level).According to the results, comparison thresholds of s = 0.1 for the plan versus plan and s = 0.001 for the plan versus film measurement are sufficient.However, alternative calculations or detection conditions may enable a more appropriate evaluation, with the fine-tuning of s according to the resolution.It should be noted that we investigated g using the 3%/2 mm, 2%/2 mm, and 3%/1 mm DD/DTA criteria that closely match the PSQA guideline recommendations (Miften et al 2018).Other criteria such as 5%/0.5 mm (DTA weighted) or 1%/5 mm (DD weighted) influences on the g distribution and not on the L, S, and U distributions.Thus, this manipulation of the g distribution can enhance the g value, whereas the detection also includes increased noise.The dose-gradient region is detected by g multiplied by the Lie derivative metrics with s; however, the adjustable range for the parameter exceeds the scope of this study.
In clinical practice, the gradient region can be assessed for issues using the gL metric supported by gU, enabling the extraction of the PTV-to-OAR dose-gradient errors (approximately higher than mean-g ), despite a favorable gamma analysis pass rate, as shown in figure 9.Moreover, our method can measure the error robustness for the focused region using gS.In this clinical case, high gL and gU were not observed in the clinically relevant region with an increase in s.In other words, we can observe that the deviation of the dose- gradient region between the actual measurement and TPS calculation was not significantly high, and the considered region did not contain excessive issues.Thus, with the clinical reevaluation of the gradient region, the result of PSQA could be more significant.The proposed method demonstrated adequate performance for other actual clinical SRT cases where various prescribed doses corresponded to various lesions including simultaneous integrated boost, which are presented in appendix figure E1.Reproducibility of the method was also verified with the remeasurement of different dpi film analyses using different dose-grid calculations as shown in appendix figure F1.Overall, the computational load was not sufficiently high to calculate metrics L, S, U (assumed to be equivalent to g calculation); their practical use is sufficiently plausible.
However, this study had several limitations.First, the compared dose distributions required an increased resolution to obtain the differential terms of the Lie derivative.The sparse sampling of the dose distribution with simple linear interpolation could lead to inappropriate numerical differential values.Second, although the aim of this study was to appropriately evaluate the dose-gradient region in an SRT dose distribution, the gradientinduced g errors extracted by the three metrics L, S, and U could be further discussed in terms of whether they are clinically problematic deviations.Several clinical cases were reported in this study to demonstrate the concept; however, its usefulness, advantages, limitations, and pitfalls must be studied and discussed in a broader spectrum of clinical applications.Third, the deviation vector field in the dose-flat region was neglected during filtering.The distinction of mechanical irradiation errors caused by the multi-leaf collimator, jaws, or beam stability remains unclear when using our method.In future studies, further investigations will find more clinically focused meaning with regard to the metrics with the abovementioned limitations.

Conclusions
Gamma index g, which provides an inclusive evaluation combining the DD and DTA within a certain tolerance level, was previously used to assess the agreement or deviation between two dose distributions.This criterion has been traditionally evaluated and validated in medical physics.However, an insufficient evaluation of dose gradients is common, particularly in SRT, owing to the characteristics of the DD/DTA criteria.This study proposes a novel method for comparing dose distributions using the Lie derivative from differential geometry.A dose distribution has a flow-field when the dose is considered a scalar potential.The Lie derivative provided a deviation-vector field between the two flow fields yielded by two different dose distributions.We defined metrics L, S, and U, which represented the deviation intensity, deviation robustness, and in/outflow performance of deviation, respectively.These metrics provided a well-enhanced gradient region in a dose distribution with location information, which facilitated the extraction of the region representing the gradientinduced increase in the g distribution during the planning comparison and actual film measurement.Moreover, this enabled clinical reassessment with respect to the highly critical dose-gradient region in SRT, such as PTV-to-OAR, where g performs a lenient evaluation.However, to maximize the use of the concept, further studies on diverse clinical cases are required for sufficient clinical evaluations.

Appendix A. Lie derivative for a deviation between vector fields
From the differential geometry, the discrepancy between two dose distributions can be addressed.are the tangent spaces of M and N. The following operation can then be obtained: The Lie derivative mathematically represents the discrepancy between fields X and Y using smooth mapping to compare the vectors at the same point P. The Lie derivative Y L X is expressed as follows: where t is the time parameter for shifting along flow f.This is interpreted as the deviation of Y p and f Y p ( ) t -time flow point.Equation (A3) can be expanded using equations (A1)-(A2), as follows: Therefore, the Lie derivative for the vector fields is defined as follows: where m and n are suffix notations of the tensor, m X is a contravariant vector, m X is a covariant vector, and the partial derivative m ¶ = ¶ ¶ m : / represents the direction or basis vector.Under the assumption of the Levi-Civita connection (Riemannian manifold), the Lie derivative can be defined using covariant derivatives  m and  n as follows: where G ml n is the connection coefficient, and l denotes the suffix notation of the tensor.The Levi-Civita connection requires a torsion tensor ] which is equal to zero.Therefore, G = G ml lm n n , and this representation explicitly describes the discrepancy between Y moving with the smooth curve X, i.e.  Y , In the Lie derivative image analysis (Anetai et al 2022a), this can be further expressed as follows (see equation (A6)): The The Lie derivative yields a vector field representing the deviation between two different flows received when they follow each other.Here, vector field Y L X is a flow field, and L denotes the intensity of the deviation.This deviation vector field includes vortex ) which can be expressed as follows: The field strength of the Lie derivative vector field can be expressed in terms of the vector field strength tensor mn F : This can be re-written in the orthogonal coordinate system x y z , , ( ) with the contravariant dual tensor mn F as follows: The field strength S 2 defines the inner product as follows: The field strength S 2 relates the vorticity.In general, while the vorticity is regionally evaluated by surface integral as circulation, it can be deemed a point-wise resistance against a flow yielded by a force-couple of vortex, which is the rotational axis rigidity (rotational stability) in the deviation vector field, interpreted as deviation robustness.This relation is described as the following vorticity equation when the local fluid density, local pressure, and viscosity are neglected.
where the flow velocity is v.A change in vorticity is affected by the disorganizing factor of the flow.
The strength of the stream by inflow and outflow, the influence of gradient-induced deviation, is measured by U 2 as follows: In this study, we investigated three metrics, i.e.L, S, and U distributions, by comparing γ distributions in the planning and measurement studies.These values are provided as positive square roots of L S , , 2 2 and U .
2 For simplicity, we considered a two-dimensional comparison in the axial plane.Therefore, m n x y , , x y where x y , ,and z denote patient right-to-left, posterior-to-anterior, and inferior-to-superior orientations, respectively.Therefore, the three metrics are given as equation (3): Appendix C. Applying the modified γ methods (MGI and χ) and SSIM to the deviation between dose distributions  where the gradient term D r r ( ) provides a certain correction for the DTA term of γ.We also considered statistical image analysis using SSIM.The SSIM (Wang et al 2004) is expressed as follows: Coefficients C 1 and C 2 denote regularization constants used to prevent significantly small denominators in the numerical calculations.Specifically, we adopted = ´-C 1.0 10 Appendix E. Examples of clinical case using Lie derivative analysis

Figure 1 .
Figure 1.Example of a lung SRT case for a D95 prescription with 42 Gy per four fractions.(a) Plan developed to fit a 100% (42 Gy) isodose line to the shape of the PTV; the 50% (21 Gy) isodose line was stretched toward anterior-posterior directions to prevent a pulmonary hilum.(b) Comparison of Profile lines A and B, where the lines indicate similar 100% lengths and different 50% lengths.Thus, as shown in the case, a dose-gradient evaluation using GI or R50 assuming sphere is insufficient.

Figure 2 .
Figure2.Beam planning for the ideal SRT dose distribution case.The CTV region in this study was defined as a sphere with a diameter of 4 cm at the center of the phantom, which was anteriorly and posteriorly adjacent to OARs; here, posterior OAR1 defines a sphere with a diameter of 3 cm, and anterior OAR2 denotes a sphere with a diameter of 5 cm.The PTV was added as a 5 mm threedimensional (3D) margin to the CTV (approximately equal to a sphere with a diameter of 4 cm).The dose distribution was designed to satisfy the 100% isodose (= 5 Gy) line and PTV (D95 = 5 Gy) and to maximally reduce the high-dose to the OAR using the static-field IMRT technique (nine fields per 15°of the gantry angles from 300°to 60°were used to exclude scatter effects by the treatment couch).Abbreviations: CTV, clinical target volume; PTV, planning target volume; OAR, organ at risk.

Figure 3 .
Figure 3. (a) Ideal dose distribution for the gradient evaluation based on the prescribed 5 Gy per 1 fraction using a static-field IMRT technique (D X ); (b) re-calculation dose with the 2 mm left-shifted isocenter location of (a) (D Y ); (c) simple DD of (a) and (b); and (d)-(f) γ and (g)-(i) Bakai's c analyses for the deviation in the DD/ DTA criteria of 3%/2 mm, 2%/2 mm, and 3%/1 mm, respectively.These figures depict values greater than 0.5.(j)-(l) SSIM index for D X and D .Y Factors of (g) luminance l, (h) contrast c, and (i) structure comparison s. (m) SSIM and (n) MGI were depicted.(j)-(n) Each value presented as 0-1 on a normalized scale.Both axes of the figures represent image pixels (dose-grid unit, 2 mm).
4(e)-(f)).A slight rotation of the dose flows was observed between D X and D Y due to the isocenter shift, which could not be readily distinguished visually and qualitatively.The Lie derivative method detected these sensitive gradient changes.Differential images were obtained as numerical derivatives, as shown in figures 5(b)-(f).Figures5(g)-(h) present the Lie derivative images according to equations (1) and (2), where Y L X x ( ) and Y L X y (

Figure 4 .
Figure 4. Flow field in the dose distributions.Both axes of the figures represent image pixels (dose-grid unit, 2 mm).(a)-(c) Normalized D X and (d)-(f) normalized D .Y (b)-(c) Vector fields of the flow field and (e)-(f) corresponding magnifications.Figure (c) was compared with (f), and the vector field rotation was demonstrated under the gradient change with respect to the dose distribution shift, as indicated by the red circle in (f).

Figure 5 .
Figure 5. Differential image of the dose distribution in figure 3(a).Both axes of the figures represent image pixels (dose-grid unit, 2 mm).Here, x denotes the column direction, and y denotes the row direction of the images. (a)-(l) Normalized images from 0-1 with the minimum and maximum values of the image, which were obtained as (the original image-min.value) / (max.value-min.value).(a) Dose distribution D , X (b)-(c) first-order derivative of D , X and (d)-(f) second-order derivative of D .X The Lie derivative of the image is obtained using equations (1)-(2).(g) and (h) Intensity of the vector field Y L X with respect to components x and y, respectively, and (i) the vector field Y L .X (j)-(l) Characteristics of (j) L, (k) S, and (l) U between the dose distributions D X and D Y defined in figure 2(a) and (b), respectively.Note that (i) is presented with different color scales for comparison.

Figure 6 .
Figure 6.Sensitivity verification of the proposed metrics L, S, and U with or without the g index.(a) Changes in L, S, and U influenced by shift distances from the original dose distribution in figure 4(a).From this figure, L and U are sensitive to gradients between two dose distributions, and S detects the specific gradients representing the difference from the original dose distributions.Both axes of the figures represent image pixels (dose-grid unit, 2 mm).(b) L, S, and U are compared with respect to the g -value defined by the 2%/ 2 mm, 3%/2 mm, and 3%/1 mm DD/DTA criteria.Each red circle indicates the expansion of sensitivity for detection with high L, S, and U -values compared with the g -value of 3%/2 mm DD/DTA.

Figure 7 .
Figure 7. (a) Ideal dose distribution for the gradient evaluation based on a prescribed 5 Gy per 1 fraction using the static-field IMRT technique (D X ); (b) measured film dose with a 2 mm left-shifted isocenter location of (a) (D Y ); (c) a simple DD of (a) and (b); and (d)-(f) γ analyses for the deviation in the DD/DTA criteria of 3%/2 mm, 2%/2 mm, and 3%/1 mm, respectively.These figures depict values greater than 0.5.(g)-(h) SSIM index and MGI values less than 0.5 are depicted.(i) Bakai's c factor greater than 0.5. (j)-(l) L S U, , -values less than 0.02.Both axes of the figures represent image pixels corresponding to 72 dpi (0.353 mm).

Figure 8 .
Figure 8. Characteristics of the L, S, U filters with respect to the g -value (3%/1 mm, DD/DTA). (a)-(b) Points satisfying the given threshold values (a) (0.01, 0.10, 0.25, 0.50, and 1.0) in the planned dose versus planned dose case and (b) (0.001, 0.002, 0.005, and 0.010) planned dose versus film measurement dose case.(c)-(d) Number of points extracted by the metric thresholds in the g -value distribution using gL, gS, and gU with respect to 2%/2 mm, 3%/2 mm, and 3%/1 mm in (c) the plan versus plan and (d) plan versus film measurement, respectively.The red arrow indicates the threshold value s adopted considering the detection sensitivity against different DD/DTA criteria of g.
metric performed adequately with respect to different g criteria.When the condition s = 0.1 was considered the g = 0.5 and = L 0.2 level, the = L 0.2 condition eliminated meaningless g from the relevance between g and L (figure 6(b)), and demonstrated extracting appropriate gradient regions from the deviation intensity (see figure 5(j): 10% of the maximum L-value 2.09).

Figure 9 .
Figure 9. Clinical dose distribution in the brain-region SRT case prescribed by 42 Gy per 10 fractions.(a) Dose distributions in the patient-and (b) phantom-geometry at one-fractional prescribed dose.(c) Film measurement and (d) g -value (3%/1 mm) analysis results.The pass rate was 98.7% with the mean-g value 0.392. (e)-(f) Illustrations of the g 0.5 and g > 1.0 distributions with the number of detections.The red circle indicates the clinically significant region for evaluating the PTV-to-OAR dose gradient.(g)-(i) L, S, and U filters with respect to the g -value (3%/1 mm, DD/DTA) in the clinical case with the number of detections via metric thresholds of 0.001, 0.002, and 0.005, respectively.

Figure 10 .
Figure 10.Metrics in the clinical brain-region SRT case depicted in figures 8(a).(a)-(b) illustrate the referenced dose distribution (TPS) D X and compared dose distribution (film measurement) D .Y (c) Flow deviation vector fields extracted by the Lie derivative Y L .X (d)-(f) Obtained L, S, U -values within the range of 0-0.02. (g)-(h) Distributions of g and c greater than 0.5; (i)-(j) MGI within the range of 0-0.5, and SSIM with respect to the normalized scale of 0-1.Both axes of the figures represent image pixels corresponding to 72 dpi (0.353 mm).

X
and X moving with the smooth curve Y , i.e.  X. Y Moreover, it is in good agreement with the deviation of the vector fields X Y , .[ ]Appendix B. Applying the Lie derivative to the deviation between dose distributionsIn the orthogonal x y z , , -coordinate system, the two dose distribution = direction from a higher dose to a lower dose is positive.Using the Lie derivative Y L X as the flow deviation, the following is obtained: The g -index(Low et al 1998) is calculated as: their tolerance limits (DD, Dr) are combined.The compared and reference doses at point vectors r c and r r and the position vectors at the same dose are related.For comparisons, modified γ methods, MGI and Bakai's χ, were investigated in this study.The MGI (Moran et al 2005) is obtained as follows: root mean square of the gradients considering the nearest neighboring point i (r i ) of the reference dose with respect to the reference point r .0 Bakai's χ factor(Bakai et al 2003) is obtained as follows: the local mean, standard deviation, and cross-covariance of N- neighbor points, respectively.Each function is represented by the reference D X and compared D Y dose distribution:

6
The components of the SSIM, luminance l, contrast c, and structural comparison s are as follows: Differential imaging using numerical calculation.The numerical derivative of the discretized image I , i j, where Î i  is a row (y), and Î j  is the column (x) of the image, is calculated as follows: D j and D i represent the resolution of the discretization.