GENERAL EQUATIONS

A System of Dosimetric Calculations

A. GENERAL EQUATIONS

The following general equations cover most of the clinical situations involving MU calculations.

For isocentric fields,

MU D

D_cal.S_c(r_c).S_p(r_d).TPR(d, r_d).WF(d, r_d, x).TF.OAR(d,x).aSCD SPD b²

(10.12) or,

MU D

D_cal.S_c(r_c).S_p(r_d).TMR(d,r_d).WF(d,r_d,x).TF.OAR(d,x).aSCD SPD b²

(10.13) For nonisocentric fields,

MU D

D_cal.S_c(r_c).S_p(r). P

100 (d,r,f).WF(d,r_d, x).TF.OAR(d, x).aSCD ft₀b²

(10.14) or,

MU D

D_cal.S_c(r_c).S_p(r). P_N

100 (d, r, f ).WF(d, r_d, x).TF.OAR(d, x).aSCD fd₀b²

(10.15) where

D = dose to be delivered at the point of interest;

D_cal = Calibration dose per MU at d_ref under reference conditions;

S_c (r_c) = Collimator scatter factor for the collimator-defined field size r_c; S_p (r) = Phantom scatter factor at d_ref for the field size r at the surface;

S_p (r_d) = Phantom scatter factor at d_ref for the field size r_d at depth d;

WF (d,r_d,x) = Wedge factor at depth d, field size r_d, and off-axis distance x;

TF = Tray factor;

OAR (d,x) = Off-axis ratio at depth d and off-axis distance x;

SCD = Source to calibration point distance at which D_cal is specified;

SPD = Source to point of interest distance at which D is delivered;

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ChaPter 10 a System of Dosimetric Calculations 157 d₀= d_ref for TPR and PDD_N;

t₀= d_ref of maximum dose for TMR and PDD.

The above MU equations assume that:

• The calibration dose per MU, D_cal, is specified at the source-calibration point distance, SCD, for the reference field size and at the reference depth.

• d_ref for D_cal and S_p is the same as for the respective dosimetric quantity (TPR, TMR, PDD, or PDD_N) in conjunction with which they are used.

• Tray factor, TF, is a transmission factor for the blocking tray, independent of field size and depth.

• Inverse-square law holds good for change in photon energy fluence in air as a function of distance from the source.

EXAMPLES OF MU CALCULATIONS Example 1

A 4-MV linear accelerator is calibrated to give 1 cGy per MU in water phantom at a refer- ence depth of maximum dose of 1 cm, 100-cm SSD, and 10 × 10 cm field size. Determine the MU values to deliver 200 cGy to a patient at a 100-cm SSD, 10-cm depth, and 15 × 15-cm field size, given S_c(15 × 15) = 1.020, S_p(15 × 15) = 1.010, and P(10,15×15,100) = 65.1%. From Equation 10.14,

MU 200

11.0201.0100.6511111 298

A form for treatment calculations is shown in Figure 10.5 with the above calculations filled in.

Figure 10.5. accelerator calculation sheet.

158 Part II Classical radiation therapy

Example 2

Determine MUs for the treatment conditions given in Example 1 above except that the treatment SSD is 120 cm, given S_c(12.5 × 12.5) = 1.010 and the percent depth dose for the new SSD is 66.7%.

Field size projected at SAD (= 100 cm) = 15100

120 12.5 cm S_c(12.512.5) is given as 1.010 and S_p(1515)1.010

SSD factor = aSCD

ft₀b²a1001

1201b²0.697 From Equation 10.14,

MU 200

11.0101.0100.6671110.697 422 Example 3

A tumor dose of 200 cGy is to be delivered at the isocenter (SAD = 100 cm), which is located at a depth of 8 cm, given a 4-MV x-ray beam, field size at the isocenter = 6 × 6 cm², S_c(6 × 6) = 0.970, S_p(6 × 6) = 0.990, machine calibrated at SCD = 100 cm, and TMR(8, 6 × 6) = 0.787.

Using Equation 10.13,

MU 200

10.9700.9900.7871111 265 Example 4

Calculate MU values for the case in Example 3, if the unit is calibrated nonisocentrically (i.e., SCD = 101 cm).

From Equation 10.13,

MU 200

10.9700.9900.7871111.02 260 COBALT-60 CALCULATIONS

The above calculation system is sufficiently general that it can be applied to any radiation gen- erator, including ⁶⁰Co. In the latter case, the machine can be calibrated either in air or in phantom provided the following information is available: (a) dose rate D₀(t₀, r₀, f₀) in phantom at depth t₀ of maximum dose for a reference field size r₀ and standard SSD f₀; (b) S_c; (c) S_p; (d) percent depth doses; and (e) TMR values. If percent depth dose data for ⁶⁰Co are used, then the S_p and TMRs can be obtained by using Equations 10.1 and 10.5. In addition, the SSD used in these calcula- tions should be confined to a range for which the output in air obeys an inverse square law for a constant collimator opening.

A form for cobalt calculations is presented in Figure 10.6.

Example 5

A tumor dose of 200 cGy is to be delivered at an 8-cm depth, using a 15 × 15-cm field size, 100-cm SSD, and penumbra trimmers up. The unit is calibrated to give 130 cGy/min in phantom at a 0.5-cm depth for a 10- × 10-cm field with trimmers up and SSD = 80 cm. Determine the time of irradiation, given S_c(12 × 12) = 1.012, S_p(15 × 15) = 1.014, and P(8, 15 × 15, 100) = 68.7%.

Field size projected at SAD (= 80 cm)15 80

100 12 cm 12 cm Equation 10.14, when applied to cobalt teletherapy, becomes

Time D

D_cal.S_c(r_c).S_p(r). P

100(d, r, f ). WF(d, r_d, x).TF.OAR(d, x).aSCD ft₀b²² where

D˙_cal is the dose rate under calibration reference conditions = 130 cGy/min; r_c = collimator- defined field = 15 × 80/100 or 12 × 12 cm²; r = field size at surface = 15 × 15 cm²; f = 100 cm;

t₀ = 0.5 cm; SCD = 80.5 cm

Substituting given values in the above equation, we get

˙

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ChaPter 10 a System of Dosimetric Calculations 159

Time 200

1301.0121.0140.687110.642 3.4 min B. IRREGULAR FIELDS

Dosimetry of irregular fields using TMRs and SMRs (or TPRs and SPRs) is analogous to the method using TARs and SARs (Section 9.5). Since the mathematical rationale of the method has been discussed in detail in the literature (5), only a brief outline will be presented here to illustrate the procedure.

An irregular field at depth d may be divided into n elementary sectors with radii emanating from point Q of the calculation (Fig. 9.10). A Clarkson-type integration (Chapter 9) may be performed to give averaged scatter–phantom ratio SPR(d,r_d) of the irregular field r_d:

SPR(d,r_d) 1 n a

n i1

SPR(d,r_i) (10.16)

where r_i is the radius of the ith sector at depth d and n is the total number of sectors (n = 2π/Δθ, where Δu is the sector angle).

The computed SPR(d,r_d) is then converted to TPR(d,r_d) by using Equation 10.9:

TPR(d, r_d)cTPR(d,0) SPR(d,r_d)d S_p(0)

S_p(r_d) (10.17)

where S_p(r_d) is the averaged S_p for the irregular field and S_p(0) is the S_p for the 0 × 0 field.

The above equation is strictly valid only for points along the central axis of a beam that is normally incident on an infinite phantom with flat surface. For off-axis points in a beam with nonuniform primary dose profile, one should write

Figure 10.6. Calculation sheet—cobalt-60.

160 Part II Classical radiation therapy

TPR(d,r_d)cPOAR(d,x)~TPR(d,0)SPR(d,r_d)dS_p(0)

S_p(r_d) (10.18) where POAR(d,x) is the primary off-axis ratio representing primary dose at point Q located at off-axis distance x relative to the primary dose at central axis.

The TMR version of Equation (10.18) is as below:

TMR(d, r_d)cPOAR(d,x). TMR(d,0)SMR(d,r_d)dS_p(0)

S_p(r_d) (10.19) TMR (d,r_d) may be converted into percent depth dose P(d, r, f) by using Equation 10.4:

P (d,r,f )100 cPOAR(d,x)~TMR (d, 0)SMR (d,r_d)dS_p(0) S_p(r_d)

S_p(r_d) S_p(r_t

0

) aft₀

fd b² (10.20) From Equations 10.10 and 10.20, we get the final expression:

P (d,r, f )100 cPOAR(d, x)~ (d, 0)SMR (d,r_d)d 1 1SMR (t₀,r_t

0)aft₀

fd b² (10.21) B.1. Source to Surface Distance Variation Within the Field

The percent depth dose at Q is normalized with respect to the D_max on the central axis at depth t₀. Let f be the SSD along the central axis, g be the vertical gap distance (i.e., “gap”

between skin surface over Q and the SSD plane), and d be the depth of Q from skin surface.

The percent depth dose is then given by

% DD 100 cPOAR(d, x)~ TMR (d, 0)SMR (d, r_d)d 1 1SMR (t₀,r_t

0)a ft₀

fgdb² (10.22) The sign of g should be set positive or negative, depending on whether the SSD over Q is larger or smaller than the central axis SSD.

B.2. Computer Program

A computer algorithm embodying the Clarkson principle and SARs was developed by Cun- ningham et al. (17) at the Princess Margaret Hospital, Toronto, and was published in 1970.

Another program, based on the same principle, was developed by Khan et al. (18) at the Uni- versity of Minnesota. It was originally written for the CDC-3300 computer using SARs and later rewritten for the Artronix PC-12 and PDP 11/34 computers. The latter versions use SMRs instead of SARs.

Although the traditional IRREG programs for 2D computations have been replaced by the current CT-based 3D algorithms, they are briefly reviewed here to illustrate the basic principles of irregular field dosimetry. These methods may be still used to check computer calculations manually, if needed.

In the program described by Khan et al. (18), the following data are permanently stored in this computer: (a) a table of SMRs as functions of radii of circular fields and (b) the primary off-axis ratios, POAR(d,x), extracted from dose profiles at selected depths. These data are then stored in the form of a table of POARs as a function of l/L where l is the lateral distance of a point from the central axis and L is the distance along the same line to the geometric edge of the beam. Usually large fields are used for these measurements.

The following data are provided for a particular patient:

1. Contour points: the outline of the irregular field can be drawn from the port (field) film with actual blocks or markers in place to define the field. The field contour is then digitized and the coordinates stored in the computer.

2. The coordinates (x, y) of the points of calculation are also entered, including the reference point, usually on the central axis, against which the percent depth doses are calculated.

3. Patient measurements: patient thickness at various points of interest, SSDs, and source to film distance are measured and recorded as shown in Figure 10.7 for a mantle field as an example.

Figure 10.8 shows a daily table calculated by the computer for a typical mantle field. Such a table is used in programming treatments so that the dose to various regions of the field can be adjusted. The areas that receive the prescribed dose after a certain number of treatments are shielded for the remaining sessions.

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