1

Chapter 10 A System of Dosimetric Calculations

• Percent depth dose is suitable for SSD treatment technique.

• Tissue-air-ratios (TAR) suitable for SAD treatment technique, but limited to energies no higher than Co-60.

• Tissue-maximum-ratios (TMR) and Tissue-phantom-ratios (TPR) were designed to overcome the limitations of the TAR.

2

10.1 Dose Calculation Parameters

• Dose = Primary dose + Scattered dose

• Primary dose (includes those scattered in the head, so-called collimator scatter) = dose due to unscattered (in phantom) photons. Conceptually, primary dose can be achieved with a beam of zero field size, or a phantom column of zero radius. (In both cases, to eliminate scatter in the phantom.)

• (phantom) Scattered dose = dose due to photons scattered at least once in the phantom.

S S

P

3

10.1 Dose Calculation Parameters (collimator & phantom scatter factor)

air

Buildup cap

Reference field

SA

D

Field size

Reference field

Sc

Reference depth

phantom

Sc,p

Sp = Sc,p / Sc

4

Reference depth

phantomphantom

r r0

D0D

Phantom Scatter Factor (Sp)

keeping same collimator opening

block down to reference

field size

)()()( max rBSFrDdD fs )()()( 0max0 rBSFrDdD fs

)(

)(

)(

)()(

0max0

max

rBSF

rBSF

dD

dDrS p

Dfs unchanged, due to same collimator opening

5

10.1 Dose Calculation Parameters (tissue-phantom & tissue-maximum ratios)

d

rdDdrdDt0

0

),(t

dd D

DrdTPR where t0 is the reference depth

max

),(D

DrdTMR d

d if t0 = dmax

t0

TAR, TMR, and TPR depends on depth and field size, but is SSD independent.

6

10.1 Dose Calculation Parameters (properties of TMR)

Depth in water

TM

R

3030

101000

Like TAR, TMR is independent of SSD, increases with energy and field size.

)( max)0,( tdedTMR

7

Relationship between TMR and TAR

d

rdDdrdDmax

dmax

rdDfs

fs

dd D

DrdTAR ),(

max

),(D

DrdTMR d

d

fsd D

DrBSF max)(

)(

),(),(

d

dd rBSF

rdTARrdTMR

8

Relationship between TMR and PDD

d

rdDd

0100

),,(

D

DfrdPDD drdDmax

dmax

max

),(D

DrdTMR d

d

)(

)( 02

maxmax

0

dp

p

rS

rS

df

df

D

D

dmax

D0

f

r

r0

)(

)(

100

),,(

),(

02

max dp

p

d

rS

rS

df

dffrdPDD

rdTMR

9

Scatter-Maximum Ratio (SMR)

d

rdD1rdD2

dmax

)0,(

)0,(),(),(

max2

11

dD

dDrdDrdSMR dd

)0,()0(

)(),(),( dTMR

S

rSrdTMRrdSMR

p

dpdd

)0,(

)0,(

)0,(

),(

),(

),(

),(

),(),(

max2

1

max2

0max2

0max2

max2

max2

1

dD

dD

dD

rdD

rdD

rdD

rdD

rdDrdSMR d

d

dd

r0 = reference field size

10

SSD

rc

t0

r

SAD

r

SSD

r c

distance changed

10.2 Practical Applications (accelerator calculations)

SSD technique:

r0

K

t0

calibration conditions

SCD

2

0

tSSD

SCD

20)()(),,(

100)dosetarget (

tSSDSCDrSrSSSDrdPDDK

TDMU

pcc

×Sc(rc)×Sp(r)

xSA

D

r’t0

×Sc(rc)×Sp(r’)

xrc

SAD

r

tSCD

r c

0

'

field size changed

100

),,( SSDrdPDD

d

depth changed

11

10.2 Practical Applications (accelerator calculations, example 1)

Machine: 4 MV photons

Calibration conditions: SSD = 100 cm, dmax = 1 cm, field size = 10 10 cm.Calibration dose rate = 1 cGy / MU

Treatment conditions: SSD = 100 cm, d = 10 cm, field size = 15 15 cm, Sc(1515)=1.020, Sp(1515)=1.010, %DD=65.1, TD = 200 cGy.

Dose/MU at prescription point

= 1 1.02 1.01 65.1/100 = 0.6707

MU = 200 / 0.6707 = 298

SSD technique:

12

10.2 Practical Applications (accelerator calculations, example 2)

Machine: 4 MV photons

Calibration conditions: SSD = 100 cm, dmax = 1 cm, field size = 10 10 cm.Calibration dose rate = 1 cGy / MU

Treatment conditions: SSD = 120 cm, d = 10 cm, field size = 15 15 cm, Sc(12.512.5)=1.010, Sp(1515)=1.010, %DD=66.7, TD = 200 cGy.

Dose/MU at prescription point

= 1 1.01 1.01 [(100+1)/(120+1)]2 0.667 = 0.474

MU = 200 / 0.474 = 422

SSD technique:

13

10.2 Practical Applications (accelerator calculations)

SAD technique:

r0

K

t0

calibration conditions

SCD

2)()(),(

)doseisocenter (

SADSCDrSrSrdTMRK

IDMU

dpccd

x

SAD

SAD

r

SCD

r c'

r’t0

×Sc(rc)×Sp(r’)

rcx

field size changed

xt0

rd

2

SAD

SCD

×Sc(rc)×Sp(rd)

distance changed

t0

d

),( drdTMR

rd

depth changed

14

Machine: 4 MV photons

Calibration conditions: SCD = 100 cm, dmax = 1 cm, field size = 10 10 cm.Calibration dose rate = 1 cGy / MU

Treatment conditions: SAD = 100 cm, d = 8 cm, field size = 6 6 cm, Sc(66)=0.970, Sp(66)=0.990, TMR(8, 66)=0.787, TD = 200 cGy.

Dose/MU at prescription point

= 1 0.970 0.990 0.787 = 0.756

MU = 200 / 0.756 = 265

SAD technique:

10.2 Practical Applications (accelerator calculations, example 3)

15

10.2 Practical Applications (accelerator calculations, example 4)

Machine: 4 MV photons

Calibration conditions: SCD = 101 cm, dmax = 1 cm, field size = 10 10 cm.Calibration dose rate = 1 cGy / MU

Treatment conditions: SAD = 100 cm, d = 8 cm, field size = 6 6 cm, Sc(66)=0.970, Sp(66)=0.990, TMR(8, 66)=0.787, TD = 200 cGy.

Dose/MU at prescription point

= 1 0.970 0.990 [(100+1)/(100)]2 0.787 = 0.771

MU = 200 / 0.771 = 259

SAD technique:

16

10.2 Practical Applications (Co-60 unit, example 5)

Machine: Co-60 photons

Calibration conditions: SSD = 80 cm, dmax = 0.5 cm, field size = 10 10 cm.Calibration dose rate = 130 cGy / min

Treatment conditions: SSD = 100 cm, d = 8 cm, field size = 15 15 cm, Sc(1212)=1.012, Sp(1515)=1.014, %DD(8,15 15,100)=68.7, TD = 200 cGy.

Dose/MU at prescription point

= 130 1.012 1.014 [(80+0.5)/(100+0.5)]2 68.7/100 = 58.80

MU = 200 / 58.80 = 3.40 min

SSD technique:

17

)0,()0(

)(),(),( dTMR

S

rSrdTMRrdSMR

p

dpdd

10.2 Practical Applications (irregular fields)

)(

)0(),()0,(),(

dp

pdd rS

SrdSMRdTMRrdTMR

(from slide #9)

For off-axis point:

)(

)0()()0,()(

dp

pp

rS

SdSMRdTMRKdTMR

off-axis ratio

ri

n

iirdSMR

ndSMR

1

),(1

)(

n

iipdp rS

nrS

1

)(1

)(

BEV )(

)0()()0,()(

dp

p

rS

SdSMRdTMRdTMR

18

2

0

)(

)(

)(

)0()()0,(

100

),,(

0

df

tf

rS

rS

rS

SdSMRdTMRK

frdP

tp

dp

dp

pp

d

rd

f

rt0

)(

)(

100

),,(),( 0

2

0 dp

tpd rS

rS

tf

dffrdPDDrdTMR

From slide #8:

2

0

0 ),(1

1)()0,(

100

),,(

0

df

tf

rtSMRdSMRdTMRK

frdP

tp

)0,()0(

)(),(),( dTMR

S

rSrdTMRrdSMR

p

dpdd

(from slide #9)

1)0(

)(1),( 0

p

dpd S

rSrtSMR

(when d = t0)

19

10.2 Practical Applications (asymmetric fields)

SSD technique:

Dose / MU = K

Sc (rc) Sp(r)

(SSD factor)

PDD(d,r)/100 OARd(x)

SAD technique:

Dose / MU = K

Sc (rc) Sp(rd) (SAD factor)

TMR(d,rd)

OARd(x)

MU = TD / (Dose / MU) MU = ID / (Dose / MU)

20

10.3 Other Practical Methods of Calculating Depth Dose Distribution (Irregular fields)

Collimator field vs. effective fields

21

10.3 Other Practical Methods of Calculating Depth Dose Distribution (off-axis points)

baD 224

12 dbD 22

4

11

caD 224

13

d

c

a

b Q

ca

Q

Qa

b

2a

2a

2c

2b

Qd

c2c

2d

Qd b

2d

2b

Px

caD 224

13 cdD 22

4

14

22

10.3 Other Practical Methods of Calculating Depth Dose Distribution (off-axis points)

]100/22%22100/22%22

100/22%22100/22%22[4

)(

cdDDcdBSFcaDDcaBSF

baDDbaBSFdbDDdbBSF

KPDcbdaD Qfs

Q

d

c

a

b Q

Px

Let the dose in free space at P = Dfs(P), then the dose at P(dmax) =Dfs(P)×BSF[(a+d)×(b+c)].

Moreover, let KQ be the off-axis ratio at Q, then the dose at Q is:

]22%2222%22

22%2222%22[

)]()[(4)(

%

max

cdDDcdBSFcaDDcaBSF

baDDbaBSFdbDDdbBSF

cbdaBSF

K

dD

D Q

P

Q

23

d=5

c=10

a=10

b=5

Q

Px

Example 6:

For a = 10cm, b = 5cm, c = 10cm, d = 5cm and a Co-60 beam with KQ = 0.98 and SSD = 80cm, what is %DQ(d=10cm) relative to DP(dmax)?

Given:

Field size 10x10 20x10 20x20 15x15

BSF 1.036 1.043 1.061 1.052

%DD(SSD=80cm,d=10cm) 55.6 56.3 60.2 58.4

8.55

]3.56043.12.60061.13.56043.16.55036.1[052.14

98.0

)(

%

max

dD

D

P

Q

24

10.3 Other Practical Methods of Calculating Depth Dose Distribution (points outside the field)

2

1

bcDbcaDcbaD 2222

1;

a

b ca

b ca

bc

a

b2c a

b2c

25

Example 7:

For a = 15cm, b = 10cm, c = 5cm, and a Co-60 beam with SSD = 80cm, what is %DQ(d=10cm) relative to DP(dmax)?

Given:

Field size 10x10 40x10 15x10

BSF 1.036 1.054 1.043

%DD(SSD=80cm,d=10cm) 55.6 58.8 56.9

1.22

]6.55036.18.58054.1[

043.1

1

2

)]1010(%)1010()1040(%)1040([

)1510(

1

)(

%

max

DDBSFDDBSF

BSFdD

D

P

Q

a=15

b=10 c=5

QPx

26

10.3 Other Practical Methods of Calculating Depth Dose Distribution (points under a block)

a

b

c

tcaDbaDblockaunderD 1

t = block transmission

27

a=15

b=15

c=4

Example 8:

For a = 15cm, b = 15cm, c = 4cm, and Co-60 beam with SSD = 80cm.

Given:

Dfs(15x15,f=80.5) = 120 cGy/min

Block transmission = 5%

Tray transmission = 97%.

QPx

(a) What is the treatment time to deliver 200 cGy to P(d=10cm)? (i.e. DP(d=10) = 200 cGy)

(b) What is the %[DQ(d=10)/DP(d=10)]?

029.1)88(

0.54)80,88,10(%

885.515/

BSF

DD

PA

min09.397.054.0029.1120/200

100/%

time

transDDBSFDtimeDP(a)

28

Method 1: Negative weight PDD

6.3×6.3

a=15

b=15

c=4

QPx

%202003.39

3.39

154%15412009.3154

1515%151512009.31515

05.011541515

PQ

Q

Q

Q

QQQ

DD

cGyD

DDBSFD

DDBSFD

DDD

29

%20694.0

05.01667.0771.0

99,10

177,101717,10

99172.6125.1155.5

77175.4125.1154

1717125.11515

/

/

TAR

transTARTAR

D

D

cmcmcm

cmcmcm

cmcm

P

Q

PA

PA

a=15

b=15

c=4

QPx

Field size magnification at d=10cm: (80+10)/10=1.125

Method 2: Negative weight TAR

30

%20

997.0672.0

05.01989.0651.002.1733.0

9999,10

17777,1017171717,10

p

pp

P

Q

STAR

transSTMRSTMR

D

D

a=15

b=15

c=4

QPx

From slide #7: TAR(d,rd) = TMR(d,rd) × Sp(rd)

Method 3: Negative weight TMR