Time/Monitor Unit calculation
Paweł Kukołowicz
Medical Physics Department, Kielce, Poland
1. Part I – Treatment Time/Monitor Unit (TT/MU) calculation with the
percentage depth doses
How the TT/MU can be calculated for photon beams? First of all the appropriate
dosimetric data should be available. The minimum set of data consists of:
1. output factor for the reference field at the reference calibration depth
2. relative output factors matrix for rectangular fields X;Y ROF(X;Y)
3. a set of the percentage depth doses for some square field
It is widely accepted that the reference field is 10 cm square field and that the reference
calibration depth is 10 cm. In the Part I for simplicity the reference depth will be the depth of
the maximum dose for the photon energy in question. The depth of maximum dose depends
on the size of the field. The larger is the field size the smaller is the depth of maximum dose.
Again for simplicity it is assumed that for all fields the reference depth is the depth for the
smallest field for which the lateral electron equilibrium exists, practically for 5 cm square
field regardless of the energy of the radiation. Let us remember that for cobalt units the output
factor is given in Gy/min and for linear accelerators in Gy/MU.
Symbols (symbols are written in italic and if in the text - in bold)
1) OFref - output factor at a fixed source-skin Fref distance determined in a large water
phantom at depth of maximum dose dm for the reference field (10 cm square field); output
factor is given in Gy/MU for accelerators and in Gy/min for cobalt units
2) OF(X;Y;dm) – output factor for the rectangular field of X;Y at depth of maximum
3) ROF(X;Y) – the ratio of the output factor at the maximum depth for field size X;Y and the
reference field
ROF ( X , Y ) 
OF ( X ; Y ; d m )
OFref
(1)
3) P ( S , d , Fref ) - percentage depth dose at depth d for a square field size S at the source skin
distance Fref
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1
For a rectangular field the percentage depth dose may be calculated by means of the
percentage depth dose for an equivalent square field given by:
E X ,Y  2 
X Y
X Y
P( X ; Y ; d ; F )  P( E X ,Y , d , F )
(2)
(3)
It has to be remembered that all fields sizes are determined at the isocenter distance I. In any
circumstances when we are interested in a field size at any source skin distance F and depth d
the subscripts will be used, i.e. XF,d;YF,d denotes the beam size at source skin distance F from
the source and depth d. One can easily calculate the beam size XF,d from:
X F ,d  X 
F d
I
(4)
1.1 The fixed-reference conditions; actual SSD = reference SSD: Fref
The dose D is to be given for field size X;Y at depth d when the source skin distance is
Fref. From the definition of the percentage depth dose we can calculate the dose at maximum:
D
 100%
P( E; d ; Fref )
Dm 
(5)
Using the relative output factor table OF and the output factor for the reference field one can
calculate the output factor for field size X;Y:
OF ( X ; Y )  OFref  ROF ( X ; Y )
(6)
From (5) and (6) the treatment time or the number monitor units is given by (the formulas will
be given explicitly for accelerator only):
LMu 
LMu 
D
 100%
PDD ( E; d ; Fref )
OFref  ROF ( X ; Y )
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
Dm
OF ( X ;Y )
OFref
D  100%
 ROF ( X ; Y )  P( E; d ; Fref )
(7)
(9)
2
Exercise 1
Calculate the number of monitor unit or the treatment time for dose D=200 cGy, at depth 7
cm, for field size 12;16 and SSD=Fref. Tables enclosed.
The TT/MU can be calculated that way if we have all needed data. In practice it means
that the technique is fixed source skin distance and that the SSD is equal to the distance for
which have made the measurements of the output factors and percentage depth doses. In all
radiotherapy centers equipped in linear accelerators the isocentric technique is very often
used. If this is a case, for each field the source skin distance is usually different so the method
described above has to be slightly changed. We have to know how to recalculate the output
factor and PDD from one SSD to another.
1.2 The isocentric conditions – actual SSD = G
Let us consider two situations shown in figure 1:
Fref
G
d
d
A*Fref,d
AG,d
Figure 1
At isocentre distance the field size is A. Than the field size at distance G and depth d (strictly
speaking at distance G+d from the source) is AG,d. Let us find the field size A* for which the
field size at distance Fref+d conforms equation AG  d  A *Fref  d . Based on the equation (4) we
have:
A* 
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Fref  d
I
 A
Gd
I
(10)
3
F
F
d
d
Dw
Dair
water
air
From (10) A* is given by:
A*  A 
Gd
Fref  d
(11)
Let us define the function called as the Tissue Air Ratio (T). The Tissue Air Ratio is the ratio
of the dose at depth in tissue to the dose in air (dose in air is the dose to small volume of
tissue situated in air and surrounded by just enough tissue equivalent material to generate
electron equilibrium).
T ( AF ,d , d ) 
Dw ( AF ,d , d )
Dair  AF ,d , F  d 
(12)
In principle there is no need to add subscripts on the right side of the formula (12). The
subscripts were put in order to emphasize that whenever we talk about the Tissue Air Ratio
for field A at depth d we keep in mind that A is the field size at the depth d. This is little
confusing in face of what has been said about the definition of the field size. The source of
such „notation” is that the Tissue Air Ratio does not depend on the source skin distance.
If the depth at which the TAR is measured (or calculated) is equal to the depth of the
maximum dose the Tissue Air Ratio is called as the Back Scatter Ratio (BSF) or the Peak
Scatter Ratio (PSF). The PSF or BSF will be denoted with a capital letter B.
From definition the percentage depth dose is given by:
P( A, d , F ) 
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Dw ( A, d , F )
 100%
Dw ( A, d m , F )
(13)
4
This formula may be converted to the one with the Tissue Air Ratios. One need to remember
that 1) for TAR the field size is defined at the depth for which the TAR is defined – not at
isocenter distance, 2) the dose in air depends only on the distance from the source to the
measuring point and on the size of the beam, whenever this size is defined and that is why it is
very convenient to define the beam size at the isocentre distance. The formulae for the dose in
water may be written as:
Dw ( A, d m , F )  T ( AF , d m , d m )  Dair ( A, F  d m )  B( AF , d m , d m )  Dair ( A, F  d m )
Dw ( A, d , F )  T ( AF ,d , d )  Dair ( A, F  d )
(14)
(15)
By substituting equations (14) and (15) in equation (13) and taking into consideration that the
dose in air is inversely proportional to the square of the distance one obtains:
P( A, d , F ) 
( F  dm )2
 100%
B( AF , d m , d m ) ( F  d )2
T ( AF , d , d )
(16)

For situations shown in figure 1 one may write:
P( A, d , G) 
(G  dm )2
100%
B( AG , d m , d m ) (G  d )2
T ( AG , d , d )

Please notice that: T ( AG, d , d )  T ( A*Fref , d , d ) , because
(17)
AG, d  A*Fref , d .
Dividing the left side of the equation (17) by the left side of the equation (18) and right side
T ( A* Fref ,d , d ) ( Fref  d m )

100%
B( A* Fref ,dm , d m ) ( Fref  d ) 2
2
P( A* , d , Fref ) 
(18)
by right side one obtains:
( Fref  d ) 2
B ( A * F ref , d m ) (G  d m ) 2
P ( A, d , G )



B ( AG , d m , d m ) ( Fref  d m ) 2
P ( A * , d , Fref )
(G  d ) 2
P( A, d , G )  P( A* , d , Fref ) 
2
B( A* Fref ,d m , d m ) (G  d m ) 2 ( Fref  d )


B( AG ,d m , d m ) ( Fref  d m ) 2 (G  d ) 2
(19)
(20)
The Back Scatter Factor is a very slow function of the beam size so the ratio of the BSFs in
equation (20) is almost one and if it is neglected one may write:
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(21) 5
2
(G  d m ) 2 ( Fref  d )
P( A, d , G)  P( A , d , Fref ) 

( Fref  d m ) 2 (G  d ) 2
*
(21)
The last two terms in the equation 21 are called as Maynord’s factor. Of course the ratio of the
BSFs may be taken into account. The BSFs may be measured or may be found in the
literature. Searching in the literature one should take the data for the radiation with the quality
index as close as possible to our radiation. The BSFs are the slow function of the energy of
the radiation. By means of the least square method J. van de Geijn and Frass fitted a function
to the BSFs obtained by measurements:
BSF ( A; A, d m )  1  B  [1  exp(  B0  A)]
Here A is the field size at depth of maximum dose, B and B0 are parameters.
(22)
For Co60 B=0,137 and B0=0,032 cm-1, 6 MV X rays B=0,098 a B0=0,029 cm-1. It is
obvious that for the X Rays of our accelerator the BSFs may slightly differ from the
published ones. However one need emphasize that in all equations used for monitor unit or
treatment time calculations the BSFs appear in the form of the ratio of two BSFs which
diminishes the uncertainty. For X Rays energy larger than 9 MV the BSFs almost does not
depend on the beam size so the ratio of BSFs is almost 1. For rectangular fields the BSF may
be calculated by using the idea of the equivalent fields – equation 2.
Exercise 2
Calculate the PDD for Co60 for field size 10;10 cm, at depth 10 cm for SSD = 80 cm and for
6 MV X-Rays for field size 10;10 cm at depth 10 cm and SSD=80 cm. Tables enclosed.
All calculations were performed for a square field. For rectangular fields calculations are done
in the same way. The only difference is that instead of a rectangular field calculations should
be performed for the equivalent square field (equation 2).
For TT/MU calculation one should know how to correct the output factor with the
distance. The actual geometry is shown in figure 1. For PDD calculations the square or
equivalent square field were always used. For output factors all calculations should be
performed for the actual field size, regardless the square or rectangular field is used. The
output factor for fields X;Y and Y;X are different due to different amount of scattered
radiation coming from the X and Y collimators. Therefore it is recommended to measure the
output factors in the whole range of available fields.
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Let us calculate the output factor for the field size L;W. By definition of TAR we may
write:
OF ( L,W , G, d m )  Dair ( L,W , G  d m )  B( LG , d m ,WG , d m )
(23)
, Gmay
)  Dwrite:
air ( AG , d m , G  d m )  B ( AG , d m )
For SSDMD
Fref( A
we
OF ( L,W , Fref , d m )  Dair ( L,W , Fref  d m )  B( LI  d m ,WI  d m )
(24)
Doses in air are related by inverse square law:
Dair ( L,W , G  d m )  Dair ( L,W , Fref  d m ) 
( Fref  d m ) 2
(G  d m ) 2
(25)
By substituting equations (24) and (25) in equation (23) we have:
OF ( L,W , G)  OF ( L,W , Fref ) 
B( LG , d m ,WG , d m ) ( Fref  d m ) 2

B( LI , d m ,WI , d m ) (G  d m ) 2
(26)
Output factor should be corrected for: 1) distance by means of the inverse square low and
2) for back scatter radiation. The second correction factor is very small, usually less than
0,2%.
Exercise 3
Calculate the OF for Co60 for field size 10;10 cm for SSD = 80 cm and for 6 MV X-Rays for
field size 10;10 cm and SSD=80 cm. Tables enclosed.
Part 2 – TT/MU calculations with TAR factors.
For simplicity in the Part 1 it was assumed that the output factors were measured at
depth of maximum dose. It is not recommended to do so because of some dosimetric
problems with electron contaminations. It is recommended to measure the OF at the depth
beyond the electrons range. For most of the clinical accelerators the 10 cm depth is enough
large to fulfill this requirement. From now on the calibration parameters are:
1. calibration field: 10;10 cm
2. calibration depth: 10 cm
3. the source calibration distance (effective point of measurement): the isocentric distance
It is very easy to calculate the output factor at dmax so the recipe of TT/MU calculation
given above is still valid and may be successfully used.
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2.1 Tissue Air Ratio
Let us remind the definition of the Tissue Air Ratio. The tissue air ratio is the ratio of
the dose at depth in tissue to the dose in air. The TAR does not depend on the SSD. Equation
15 informs how to calculate the PDD using the TARs. This equation may be rewritten as:
T ( AF ,d , d )  P( A, d , F )  B( AF ,dm , d m ) 
(F  d )2
1

2
( F  d m ) 100%
(27)
Where AF,d is the field size A at the depth d:
AF ,d  A 
F d
I
(28)
The TAR for Co60 and 6 MV X are printed in the Appendix.
2.2 Dose calculation based on the idea proposed by Clarkson
In 1940 Clarkson proposed a method for calculations of the dose for any regular and
irregular fields. This method was later developed by Cunnigham. All linear accelerators used
in radiotherapy generate a rectangular therapeutic fields. It was shown by Day and other
physicists that a percentage depth dose for a circular field may be approximated by an
equivalent square. According to this rule, a square field is equivalent to a circular one if they
have the same area. Consiquently the TAR for circular field may be calculated from:
TAR( R, d )  TAR( A; A, d )
R
1

A
(29)
The advantage of the circular field is that it has circular symmetry. Based on the symmetry the
dose at any point may be calculated in a very elegant way. Let us consider an irregularly
shaped field as shown in figure 3. Let Q be the point of calculation at a depth dQ in the plane
of the field cross section.
In this part the output factor is measured according to international recommendations
i.e. at 10 cm depth.
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r9 r8 r7
r11 r10
r12


r13


r14

r15

r16
9
8 7
r19
6
Q
1
r17 
r18
5

 



r3
r2
r1






r4
2
4 3
r5
2


r6
r30
r29
r28
r27
r26
r20 r21 r22 r23 r24 r25
Figure 3 - Irregularly shaped field (the field modified by blocks) divided into circular
segments.
The field may be divided by into elementary circular sectors of angles i and a radii r i.
Using TAR table for circular fields, the TAR values for the sectors are calculated and then
summed to give the average TARQ for the field.
30
TARQ  
i 1
i
TAR(ri , d Q )
2
(30)
Exercise 4
Calculate the average TAR for irregular field. The shape of the field is enclosed.
By TAR definition to calculate the dose at Q one need multiply the average TAR by dose in
air at Q (output factor in air). If there is no blocks the dose in air may be calculated in a very
easy way. Let us remind, that from measurements we have the output factors in water
phantom at reference depth. Using the equation (30) we may calculate the TAR for any open
field. By definition the TAR is the ratio of the dose in water and in air at the same point so the
calculation of the dose in air may be done directly from definition:
OFair 
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OFwater
TAR
(31)
9
It is very convenient to calculate once the relative output factor in air (ROFair) for all
rectangular fields for which the relative output factors in water were measured. Then the table
of the ROFair may be treated as the look up table. The dose in air at any distance from the
source (G+d) may be calculated by the equation:
OFair ( X ; Y , G, d )  OFair (10;10, Fref )  ROF air ( X ; Y ) 
OFair (10;10, Fref , d ref ) 
( Fref  d ) 2
(G  d ) 2
(32)
OFwater (10;10, Fref  d ref )
T (10 Fref  d ;10 Fref  d , d ref )
Exercise 5
Calculate the output factor in air for rectangular field 12;16 for Co60.
Irregular fields are usually defined by means of blocks. If this is the situation we should ask a
question: is the output factor in air changed by blocks? In principle the answer is yes, however
in almost all clinical situations this change is so small that may be neglected. If the
uncertainty of less than half percentage is accepted the influence of the blocks on the output
factors in air may be forgotten. Therefore in practice to calculate the output factor for
irregular field in the first step the output factor for open field is calculated – it may be done
with the method described in the Part 1 or according to the method described in Part 2 – and
later the result is corrected for blocks. The correction factor is given by formulae:
CFblocks 
TARopen _ beam
TARblocked _ beam
(33)
The TT/MU calculated for open beam should be multiply by the correction factor. Blocks are
placed at the blocks tray so we must not forget to multiply the TT/MU by the inverse of the
blocks tray attenuation factor - CFTR.
Exercise 6
For the field from the exercise 4calculate the correction factor for blocks.
Summary
1. For open fields the method described in the Part 1 or Part 2 may be used. I prefer to use the
model 2 because this model is more general. Calculations described below are based on the
model 2.
2. The minimum set of data that has to be measured are:
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a. the output factor for a reference field in a reference conditions [symbol OFw(10;10)]
b. the relative output factors for the several square and rectangular fields in a reference
conditions; we usually measure the table with the data for (X;Y) where X and Y are the
smallest clinically used field and 5, 7,10,15,20,30 and the largest one [symbol
ROFw(X;Y)]
c. the Percentage Depth Doses for several square open fields; we usually measure the
smallest clinically used field and 5, 7,10,15,20,30 and the largest one [symbol
PDD(X;X)]
d. the Back Scatter Factor for several square open fields; this data may be taken from
the literature – one need to search for a published data with the Quality Index as close
to our radiation as possible [symbol B(X)]
e. the attenuation tray factors for all trays used in practice [CFTR]
The recommended reference conditions are: 1) depth 10 cm, 2) SSD isocentric distance minus
10 cm, 3) reference field 10 cm square field, 4) measurements in the water phantom.
How to calculate the TT/MU for te dose D to point Q at depth g. The field is A;B
modified by the set of blocks placed at the tray. The shape of the modified field is known. The
SSD = G.
1. Calculate the output factor in water for reference conditions for field A;B
OFw ( A; B, d ref , Fref )  OF w(10;10)  ROF w ( A; B)
2. Calculate the TAR for the A;B open field in reference conditions
n
open
TAR(rdref
, d ref )
n 1 2
N
TAR( AFref ,dref ; BFref ,dref )  
this value may be taken from the look-up table either
3. Calculate the output factor in air for reference conditions for field A;B
OFair ( A; B; Fref , d ref ) 
OFw ( A; B; Fref , d ref )
TAR( AFref ,d ref ; BFref ,d ref )
this value may be taken from the look-up table either
4. Convert the output factor in air from the reference conditions to actual conditions
OFair ( A; B; G, g )  OFair ( A; B; Fref , d ref ) 
( Fref  d ref ) 2
(G  d ) 2
5. Calculate the TAR for the blocked field A;B at SSD = G and depth g
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n
TAR(rgblock , g )
n 1 2
N
TAR( AG ,d ; BG ,d )  
6. Calculate the output factor in water for the blocked field A;B at SSD = G and depth g
OFw ( A; B, g , G )  OFair ( A; B; G, g )  TAR( AG , d ; BG , d )
7. Calculate the output factor in water for the actual conditions corrected for the block
tray
OFwtray ( A; B, g , G) 
OFw ( A; B, g , G)
CFTR
8. Calculate TT/MU
TT / MU 
OF
tray
w
D
( A; B, g , G )
2.3 Correction for inhomogeneities
The human body is composed of water in more than 80% but there are some tissues
that from radiological point of view differs very much from the soft tissue – equivalent to
water. In radiotherapy the most important are lungs. The effective atomic number of lungs
is very similar to the soft tissue, the density is of about four times (0,25 g/cm3) smaller
than for soft tissue. Consequently a radiation attenuation is much smaller than for soft
tissue. The second important tissue which might be taken into consideration is bone. It is
important to understand that in fact only small part of that tissue which considered as bone
differs very much from the soft tissue.
Only outer layer of the bone is very
Figure 4
compact with the density of about 1,6
g/cm3. The thickness of this layer is a
few millimeters only so the influence of
d1
=1
bones on the dose distribution is rather
limited. Therefore regarding the TT/MU
d2
e ≠ 1
calculation only lungs will be accounted
for.
d3
=1
Several correction methods for
inhomogeneities are proposed. The most
accurate and relatively simple is the
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method proposed by Batho. Let us consider the situation shown in figure 4.
We want to correct the output factor at point Q for inhomogenities with density e. Batho
proposed following formula to calculate the correction factor CFinh:
CFinh
 T ( X ; Y , d 2  d 3) 

 
 T ( X ; Y , d 3) 
 e 1
Tissue Air Ratio factor should be calculated for the actual field, i.e. for the field including
the blocks. What shell we do if the d3 is smaller than dmax? If this is a case we should
calculate the T(X;Y,d3) for d3=dmax.
How do we know which is the real value of the lung density? If we have the CT for
our patient we may calculate it according to:
  0,001 * CT  1
  0,0005 * CT  1
for CT<0
for CT>0
2.4 Correction for cobalt source disintegration
Let us assume that for the cobalt source on the date D1 the output factor is OFD1. We
want to calculate the output factor for the date D2. The half time of disintegration in days is
T.=1526 Therefore the searched output factor is given by:
OFD 2  OFD1  e

0.693
( D 2  D1)
T1/ 2
The difference (D2-D1) should be given in days.
2.5 Calculations for physical wedges
For high energy photon wedge beams the so called hardening effect is observed, i.e.
the quality index of the radiation becomes higher. Consequently the Percentage Depth Doses
are little greater than for open beams. The difference may be even larger than 1% so if we
want to have more precise calculations each wedge beam should be treated as a separate
treatment unit. The percentage depth doses should be measured and Tissue Air Ratio factors
should be calculated.
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