Monitor Unit Calculations for External Photon and Electron Beams The TG-71 Report: Monitor Unit Calculations for External Photon and Electron Beams Med Phys 41(3), 2014 Chen-Shou Chui, PhD, FAAPM, DABR Department of Medical Physics Sun Yat-Sen Cancer Center, Taipei, Taiwan

Why Manual MU Calculations? For patients without a computerized treatment plan (such as palliative or emergency cases). Conditions where computer algorithms may not be as accurate or appropriate (such as TBI). As a QA device, TG-40 and TG-114 recommend independent MU check. To understand the basic concepts in dose calculation.

Table of Contents Introduction Calculation Formalism Determination of Dosimetric Quantities Interface with Treatment Planning System MU Calculations for IMRT Fields Quality Assurance Summary Recommendations Examples Appendix A: Derivation of Monitor Unit Equations Appendix B: Calculation of Sc Using a PEV Model

General Concept For a given beam energy, the dose to a point depends on: Beam on time (MU), Field size, Depth, Distance, and if applicable, Beam modifying devices (wedge, tray, spoiler…) Conversely, the MU needed to deliver a given treatment dose can be determined by the treatment dose, field size, depth, distance, and if applicable, the beam modifying device. 4

Photon beams are treated either with the SAD or the SSD technique.

2.A.1.a - Photon calculations Using Tissue Phantom Ratio Appendix A: Derivation of Monitor Unit Equations - 1. TPR (“Isocentric”) Method *Typically, SPD0 = 100 cm, r0 = 1010 cm2, d0 = 10 cm, D0’ = 1 cGy/MU. Dose/MU under normalization conditions* SPD0 SAD Collimator scatter factor (r0  rc), Phantom scatter factor (r0  rc), SPD Distance changed from SPD0 to SPD. d0 Inverse square correction r0 rc d x depth changed from d0 to d, rd Blocking tray factor Off-axis ratio Wedge factor (1)  PDD method  Eq sq  Field size  Inhom  Photon ex 1a 6

2.A.1.b - Photon calculations Using Percent Depth Dose Appendix A: Derivation of Monitor Unit Equations - 2. PDD (“nonisocentric”) Method *Typically, SSD0 = 90 cm or 100 cm, r0 = 1010 cm2, d0 = 10 cm, D0’ = 1 cGy/MU. Dose/MU under normalization conditions* SSD0 SAD Collimator scatter factor (r0  rc), SSD Phantom scatter factor (r0  rc), r0 rc Distance SSD0  SSD. x d0 r Inverse square correction d0  d, d rd0 Blocking tray factor Off-axis ratio Wedge factor (3)  TPR method  Photon ex 1b 7

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.a. Dose per MU under normalization conditions (D0’ ) SPD0 = 100 cm SSD=100 cm d0 r0 r0 D0’ dm Dref’ TG-71 recommended Normalization conditions Existing Reference conditions normalization depth d0=10 cm Field size r0 = 1010 cm2 normalization dose rate D0’ = 1.0 cGy/MU Reference or calibration depth dref (e.g. dm) may not be = d0, then to keep existing Dref’ D0’ = Dref’TMR(10cm,1010)[(100+dm)/100]2 Keep things simple, try to make reference conditions = normalization conditions 8

Example – determination of D0’ from Dref’ = 1 cGy/MU  TPR method

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.e. phantom scatter factor: Sp Flattening filter SPD0 Reference depth Mini-phantom Reference field r0 r r0 r air phantom Scatter factors Sc,p Sc,p: output factor, field size factor Sc,p = Sc Sp Sc: collimator scatter factor Sp: phantom scatter factor Sp = Sc,p / Sc Sc 1.0 reference field (1010) field size 10 10

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.d. collimator scatter factor: Sc  = 4 cm d = 10 cm Flattening filter FIG. 5. Diagram illustrating measurement setup for Sc. The cylindrical mini-phantom is aligned coaxially with the central axis of the beam, with the ion chamber positioned at the source-detector distance corresponding to the chosen normalization conditions. The field size is maintained large enough to ensure coverage of the mini-phantom, and other scattering materials are removed from the treatment field.  TPR method 11

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.c. Tissue Phantom Ratios: TPR(d,rd) SPD0 = 100 cm SPD0 = 100 cm d d0 d0 r r D’(d0,r) D’(d,r) Normalization depth d0 field size r depth d0  d same field size 12

Example – Tissue Phantom Ratios (TPRs) for 6-MV Photons  TPR method

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.g. Tray Factors: (TF) Blocking tray SPD0 = 100 cm SPD0 = 100 cm d0 d0 r0 r0 D0’ D0’tray Normalization depth d0 field size r0 Blocking tray in place Same depth, field size TF is almost independent of field size, depth, and SSD, and a constant value is sufficient in most cases (typically 0.95-0.98). 14

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.f. Off-Axis Ratios: OAR(d,r) OAR is taken as the primary off-axis ratio (POCR). largest field size shallow depth d D’(d,r) D’(d,r=0) Diagonal profiles at various depths for largest field size d large depth … this task group recommends that every attempt be made to keep this calculation on the central axis to avoid the complications associated with off-axis calculations. 15

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.i. Wedge Factors Wedge factor (WF): d WF depends on the distance from the central axis x, the depth d, and weakly on field size rd. rd rd D’wedge D’open ÷ = The wedge factor WF is defined as the ratio of the dose rate at the point of calculation for a wedged field to that for the same field without a wedge modifier. To avoid complications associated with off-axis calculations; this task group recommends that every attempt be made to keep this calculation on the central axis. 16

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.i. Wedge Factors 3.B.1.i.i Physical Wedges – external and internal wedges external wedge For Varian and Siemens machines, a set of external wedges is provided, attached below the lower collimator jaws. The standard wedge angles are 15°, 30°, 45°, and 60°. Wedge angle w is the angle between the isodose curve and the horizontal line at 10 cm depth. d = 10 cm w internal wedge For Elekta machines, a single internal wedge is used, located above the upper collimator jaws, inside the head. The wedge is motor driven and the wedge angle is 60°.

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.i. Wedge Factors 3.B.1.i.i Physical Wedges – external and internal wedges Dependence of Wedge Factor on Depth energy spectrum Co-60 For Co-60, photons are nearly mono-energetic, wedge filter does not have much effect on beam quality. Thus the wedge factor does not appear to change with depth. 1.17-MeV 1.33-MeV open wedge relative fluence linac For linac-produced photon beams, the wedge filter preferentially attenuates low-energy photons. As a result, the wedged-beam is ‘harder’ than open field beam, that is, more penetrating. photon energy depth dose distribution linac dose Thus, the wedge factor (which is the ratio of wedge-field-dose to open-field-dose) increases with the depth. open wedge depth 18

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.i. Wedge Factors 3.B.1.i.i Physical Wedges – external and internal wedges Dependence of Wedge Factor on Field Size Fixed wedge motorized wedge WF WF Center of wedge fixed at beam axis. Wedge factor varies weakly with field size. Thin edge of wedge aligns with the field edge. Wedge factor varies strongly with field size, (greater wedge factor with smaller field size), resulting in more efficient use of beam-on-time 19

Example – Wedge Factors for 6-MV Photons  Dynamic wedge

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.i. Wedge Factors 3.B.1.i.ii Nonphysical Wedges – Enhanced Dynamic wedge (EDW) and Virtual wedge (VW) Open field Physical wedge Dynamic/virtual wedge collimator Fluence or dose Dopen Dwedge(x) Dwedge(y) 21

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.i. Wedge Factors 3.B.1.i.ii Nonphysical Wedges – Enhanced Dynamic wedge (EDW) and Virtual wedge (VW) Example Fixed jaw position 3 4 5 6 7 8 9 For EDW, WF(y) is primarily dependent on the position of the fixed jaw T T(0)/0.84 Y1 T(0)/0.95 Y2 T(0) T(y) T(fixed-jaw-position) -20 -10 3.5 10 Y 1.0  TPR method 22

3.B.1 – Measurements of dosimetric quantities: Photon beams 3.B.1.b. Normalized Percent Depth Dose: PDDN(d,r,SSD) Parallel-plate chamber: measured curve II. 100 curve I Cylindrical chamber: measured curve I, needs to be shifted by 0.6 rcav to get curve II. Curve II is the percent depth ionization curve, can be taken as the percent depth dose curve for photon beams. curve II PDDN(d,r) 100 PDD(d,r) 0.6 rcav 5 10 15 15 20 depth in water PDD is normalized at dm, PDDN is normalized at d0 (10 cm). Conversion of PDDN between two different SSDs. Mayneord F factor 23

Example – PDDN(d,r,SSD=100 cm) for 6-MV Photons  PDD method

= = 2.A.2 – Field-size Determination Sterling’s Method: Area/Perimeter (A/P) method Rectangular field Square field W A rectangular and a square fields are ‘equivalent’ if they have the same ‘area/perimeter’. L S = = (4) rjU: upper jaw setting rjL: lower jaw setting 25

2.A.2 – Field-size Determination Which field size to use? Field size defined by the jaws? Or by the MLC/blocks? Field size defined by Jaw setting at the isocenter Sc(rc) (2.A.2.a) For Elekta machines, MLC = upper jaws, For Siemens machines, MLC = lower jaws SAD SPD Field size defined by Irradiated area (block or MLC) at the point of calculation rd or at surface r r rc d0 d Sp(rd0) (2.A.2.b). TPR(d,rd) or PDDN (d,r,SSD) (2.A.2.c). WF(d,rd,x) for external physical wedges (2.A.2.d.i). rd WF(y) for dynamic wedges is primarily dependent on the position of the fixed Y jaw (2.A.2.d.ii). 26 26

2.A.3 – Radiological Depth Determination To correct for internal heterogeneities within the patient, the calculated homogeneous dose per MU, D’(homogeneous) is multiplied by a correction factor CF: (5) 2.A.3.a. Method 1: TPR method (equivalent pathlength, equivalent depth) (7) d1 re,1 d2 re,2 (6) d3 re,3 rd P Equivalent depth 27

2.A.3 – Radiological Depth Determination To correct for internal heterogeneities within the patient, the calculated homogeneous dose per MU, D’(homogeneous) is multiplied by a correction factor CF: (5) 2.A.3.b. Method 2: Power Law TAR (Batho) Method (8) d1 re=1.0 d2 re d3 re=1.0 rd P  photon TPR method 28

8.A Examples - Photon Calculations These example calculations use a 6-MV beam (Varian Clinac 21EX), 100-cm SAD, with a 0.8 cGy/MU dose rate at the normalization point (SSD0 = 90 cm, rc = 10 × 10 cm2, d0 = 10 cm). The equivalent square of the collimator and effective field sizes has been determined using the A/P approximation. 1. Calculate the MUs required to deliver 250 cGy to the isocenter. Collimator field size of 10.0 × 15.0 cm2, depth = 6 cm and no blocking. SPD0 = 100 cm  photon TPR method (1)  photon equivalent field d0 r0 D0’  scatter table  TPR table (25) 29

8.A Examples - Photon Calculations These example calculations use a 6-MV beam (Varian Clinac 21EX), 100-cm SAD, with a 0.8 cGy/MU dose rate at the normalization point (SSD0 = 90 cm, rc = 10 × 10 cm2, d0 = 10 cm). The equivalent square of the collimator and effective field sizes has been determined using the A/P approximation. 1. Calculate the MUs required to deliver 250 cGy to the isocenter. Collimator field size of 10.0 × 15.0 cm2, depth = 6 cm and no blocking. SSD0 = 90 cm  photon PDD method SSD = 94 cm (3) d0 r0 d D0’  scatter table  PDD table  Mayneord factor (26) 30

8.A Examples - Photon Calculations These example calculations use a 6-MV beam (Varian Clinac 21EX), 100-cm SAD, with a 0.8 cGy/MU dose rate at the normalization point (SSD0 = 90 cm, rc = 10 × 10 cm2, d0 = 10 cm). The equivalent square of the collimator and effective field sizes has been determined using the A/P approximation. 2. Calculate the MUs required to deliver 90 cGy to the isocenter of an AP lung field displayed in Fig. 8. The collimator field size is 18.0 × 12.0 cm2, and the physical depth = 10.0 cm. The field is blocked using a tertiary MLC and a 15◦ physical wedge is added to the field. Estimated rectangular field: 8 cm 14 cm equivalent square: 10.2 cm  photon TPR method (1)  scatter table  TPR table  wedge table (27) 31

8.A Examples - Photon Calculations These example calculations use a 6-MV beam (Varian Clinac 21EX), 100-cm SAD, with a 0.8 cGy/MU dose rate at the normalization point (SSD0 = 90 cm, rc = 10 × 10 cm2, d0 = 10 cm). The equivalent square of the collimator and effective field sizes has been determined using the A/P approximation. 2. Repeat the calculation using a tissue heterogeneity correction using a radiological depth of 6 cm. Estimated rectangular field: 8 cm 14 cm equivalent square: 10.2 cm  equiv. pathlength correction  TPR table 32

8.A Examples - Photon Calculations These example calculations use a 6-MV beam (Varian Clinac 21EX), 100-cm SAD, with a 0.8 cGy/MU dose rate at the normalization point (SSD0 = 90 cm, rc = 10 × 10 cm2, d0 = 10 cm). The equivalent square of the collimator and effective field sizes has been determined using the A/P approximation. 3. Calculate the MUs to deliver 45 cGy to the isocenter (depth = 9.5 cm) of a superior–anterior oblique pituitary field. The collimator field size is asymmetric (X1 = 2.0, X2 = 3.5, Y1 = Y2 = 3.5), with blocking (see Fig. 9) using a tertiary MLC. A 45◦ EDW is added, in the Y1 orientation (Y2 jaw fixed). Collimator setting: 5.5 cm 7 cm equivalent square: 6.2 cm  photon TPR method Irradiated area: 4 cm 5 cm equivalent square: 4.4 cm  scatter table  TPR table  Dynamic wedge (28) 33

8.A Examples - Photon Calculations These example calculations use a 6-MV beam (Varian Clinac 21EX), 100-cm SAD, with a 0.8 cGy/MU dose rate at the normalization point (SSD0 = 90 cm, rc = 10 × 10 cm2, d0 = 10 cm). The equivalent square of the collimator and effective field sizes has been determined using the A/P approximation. 3. Repeat the calculation to an off-axis point at the same depth, located at (1,1) cm in the (X2,Y2) direction. Collimator setting: 5.5 cm 7 cm equivalent square: 6.2 cm  photon TPR method  OAR table OAR(d=9.5, r=1.4) = 1.005 Irradiated area: 4 cm 5 cm equivalent square: 4.4 cm  Dynamic wedge (29) 34

Electron beams are almost always treated with the SSD technique. 35

2.B.1.a - Electron MU calculations at Standard SSDs Dose depends on time, field size, depth, and distance. *Typically, SSD0 = 100 cm, r0 = 1010 cm2 or 1515 cm2 cone/insert dm(r0) = depth of maximum dose for field size r0, D0’ = 1 cGy/MU. Dose/MU under normalization conditions* SSD0 SAD field size changed from r0 to ra, distance changed from dm(ra) to d, ra r0 dm(r0) dm(ra) d (10)  Extended SSD, effective SSD

3.B.2 – Measurements of dosimetric quantities: Electron beams 3.B.2.a. Dose per MU under normalization conditions (D0’ = 1 cGy/MU) SSD0 SSD0 r0 r0 dm(r0) dm(r0) dref D0’ D/MU TG-21 TG-51 calibration depth = dm(r0) calibration dose rate D/MU = normalized dose rate D0’ calibration depth dref = 0.6R50 – 0.1 which may not be the same as dm(r0)  D0’ = (D/MU) /PDD(dref,r0,SSD0)  electron, standard SSD

field size changed from r0 to ra 2.B. - Electrons 3.B.2.c. Electron output factor, Se – includes applicator/insert and treatment distance effect usually SSD = SSD0 Se depends on applicator/insert size SSD0 SSD r0 ra r0 dm(r0) dm(r0) dm(ra) field size changed from r0 to ra reference field size r0 (9) TG-70 notation TG-71 notation

Example - Electron output factor, Se  electron, standard SSD

3.B.2 – Measurements of dosimetric quantities: Electron beams 3.B.2.b. Percent depth dose PDD(d,ra,SSD) Parallel-plate chamber: measured curve II. Cylindrical chamber: measured curve I, needs to be shifted by 0.5 rcav to get curve II. Curve II is the percent ionization curve. Conversion from relative ionization to relative dose (III): Requires water-to-air stopping power ratio and replacement factor(?). Hopefully, conversion from depth-ionization to depth-dose is performed by the water phantom system. (TG-70) Fig.1. Effect of shifting depth-ionization data measured with cylindrical chambers upstream by 0.5rcav for electron beams with rcav=1.0 cm. The raw data are shown by curve I long dashes and the shifted data, which are taken as the depth-ionization curve, are shown by curve II solid line. The value of the percentage ionization at B solid curve, 50% ionization in the electron beam gives I50 from which R50 can be determined. For electron beams, curve II must be further corrected to obtain curve III, the percentage depth-dose curve short dashes. PDD depends on the irradiated field size.

3.B.2 – Measurements of dosimetric quantities: Electron beams 3.B.2.b. Percent depth dose PDD(d,ra,SSD) Field size dependence for large field sizes, PDD nearly constant The PDD increases with field size until it exceeds the lateral range of the electrons, then the PDD is almost constant with field size. The depth of maximum dose, dmax, also increases with field size until the lateral range is reached. For small field sizes, PDD increase significantly with the field size. r=2 cm r=0.1 cm r=1 cm r=0.25 cm r=0.5 cm R ~ 5 cm 1 2 3 4 cm Attix: “introduction to radiological physics and radiation dosimetry.” 41 41

3.B.2 – Measurements of dosimetric quantities: Electron beams 3.B.2.b. Percent depth dose PDD(d,ra,SSD) Field size dependence If PDDs are all normalized to the maximum dose, then the surface dose increases with decreasing field sizes. Attix: “introduction to radiological physics and radiation dosimetry.”  electron, standard SSD 42 42

2.B.1.b - Electron MU calculations at Extended SSDs 2.B.1.b.i. Effective SSD Technique Dose/MU under normalization conditions* SSD0 field size changed from r0 to ra, SSD distance changed from dm(ra) to d, ra r0 r g d dm(r0) dm(ra) PDD at extended SSD ‘inverse square’ correction d0 = dm(ra) (11)  electron, extended SSD air-gap correction 43

PDD at Extended SSD – PDD(d,ra,SSD) TG-71 report did not describe how to obtain PDDs at extended SSDs. Impractical to measure PDD(d,ra,SSD)’s at all possible SSDs. Use Mayneord factor to convert PDD at standard SSD to PDD at another SSD? (This may not be correct.) Mayneord factor The magnitude of ‘F’ may be small for clinical electron beams. For example, for 20-MeV electrons, assume SSDeff=90 cm, dm=2.5 cm, g=20 cm, d=5 cm, then F=1.009 44 44

PDD at Extended SSD – PDD(d,ra,SSD) Some studies showed that in the region beyond dmax, electron PDD is almost independent of SSD. Saw et al. IJROBP 32(1), 159, (1995) 45

PDD at Extended SSD – PDD(d,ra,SSD) But others showed that for high energy electrons, electron PDD is dependent on SSD.  Extended SSD, effective SSD Das et al. Med Phys 22(10), 1667, (1995) 46 46

3.B.2.d. - Electron Virtual SSDs or Effective SSDs Scat. foil virtual source jaws (20) electron beam Example: d0 = 2.0 cm, g = 10 cm slope = (1.14-1.00)/g = 0.014 f = (1/0.014) - 2 = 69.4 cm nominal SSD 100 cm f: effective SSD Slide #4 applicator 1.14 (Khan: Fig. 14.19) d0 d0 I0 I0 g Ig 47

Example - Electron Effective SSD (SSDeff)  Extended SSD, effective SSD 48

2.B.1.b - Electron MU calculations at Extended SSDs 2.B.1.b.ii. Air-gap Technique Dose/MU under normalization conditions* SSD0 field size changed from r0 to ra, SSD distance changed from dm(ra) to d, ra r0 r g dm(r0) d dm(ra) PDD at extended SSD ‘air-gap’ correction some people combine these as the air-gap factor. (12)  field size determination 49

3.B.2.e. Electron Air-gap correction factor (fair) fair depends on SSD and insert size. SSD0 SSD ra ra dm(r0) dm(r0) field size changed from r0 to ra reference field size r0 (21) 50

Example - Electron air-gap correction factor (fair)  electron, extended SSD air-gap correction 51

=  2.B.2 - Electron Field Size Determination Rectangular fields – square root (geometric mean) method =  W L ra: applicator/insert size, WL: rectangular insert size. (13) (22) (note: For PDD, if skin collimation is used, use the irradiated area defined by the skin collimation, rather than the applicator/insert size.) 52

Theoretical Basis for the Square Root Method Based on multiple scattering theory (central limit theorem): The lateral distribution of an electron pencil beam at depth z is approximately Gaussian. www.mathsisfun.com/data/quincunx.html The 2-dimensional pencil beam distribution at z is given by: where dp(0,0,z) is the pencil beam central axis dose at depth z. 53 53

Theoretical Basis for the Square Root Method (cont’d) For a square field of side s, the central axis dose at depth z, Ds(0,0,z), is contributed from all pencil beams: x’,y’ s d or  54 54

Theoretical Basis for the Square Root Method (cont’d) Likewise, for a rectangular field of WL, the central axis dose at depth z, Ds(0,0,z), is then: x’,y’ W L d or 55 55

2.B.2 - Electron Field Size Determination Irregular fields – approximate by a rectangular field Fig.4 When in doubt, measure the output for the irregular field. 56

8.B Examples - Electron Calculations 1. Rectangular fields – standard SSD These example calculations use a 9-MeV beam (Varian Clinac 21EX), with a 1.0-cGy/MU dose rate at the normalization point (SSD0 = 100 cm, rc = 1010 cm2, dm = 2.1 cm). 1. Calculate the MUs required to deliver 200 cGy to a depth of dm at 100-cm SSD for a 6×10-cm2 insert in a 15×15-cm2 applicator.  electron, standard SSD SSD0 (10) ra dm(r0)  determination of field size  output table (30) 57

8.B Examples - Electron Calculations 2. Rectangular fields – extended SSD, effective SSD These example calculations use a 9-MeV beam (Varian Clinac 21EX), with a 1.0-cGy/MU dose rate at the normalization point (SSD0 = 100 cm, rc = 1010 cm2, dm = 2.1 cm). 2. Repeat the calculation for a treatment at 110 SSD.  electron, extended SSD effective SSD SSD = 110 cm (11) SSD0 ra  determination of field size  effective SSD table dm(r0) (31) 58

8.B Examples - Electron Calculations 2. Rectangular fields – extended SSD, air-gap correction factor These example calculations use a 9-MeV beam (Varian Clinac 21EX), with a 1.0-cGy/MU dose rate at the normalization point (SSD0 = 100 cm, rc = 1010 cm2, dm = 2.1 cm). 2. Repeat the calculation for a treatment at 110 SSD.  electron, extended SSD air-gap correction SSD = 110 cm (12) SSD0 ra dm(r0)  determination of field size  air gap table (32) 59