XiO_TRN_CLARKSON_B

XiO® Dose Calculation — Clarkson

Contact Information Dose Calculation — Clarkson

ii Document Name: XiO_TRN_CLARKSON_B.pdf

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Revision History

Revision Date Changes XiO_TRN_CLARKSON_A 10/25/2002 Newly baselined. Updated product name from FOCUS to XiO. XiO_TRN_CLARKSON_B 04/18/2007 Update template and logo.

© Copyright 2007 by Computerized Medical Systems, Inc. All rights reserved. Performance characteristics given in this manual are for reference only and are not intended as guaranteed specifications.

XiO® iii

Table of Contents

Overview........................................................1

Physics of the Clarkson Algorithm..........................3

Treatment Aids ................................................9

Wedge................................................................................... 9 Fixed Wedge .................................................................................. 9 Measured Wedge Factors ..................................................................10 Motorized Wedges ..........................................................................11

Enhanced Dynamic Wedge (Varian)..............................................14 Overview .....................................................................................14 Generating a Segmented Treatment Table .............................................15 Dose Calculation ............................................................................15 Effective Wedge Attenuation Factor Calculation......................................17

Virtual Wedge (Siemens)...........................................................18 Overview .....................................................................................18 Generating Virtual Wedge Intensity Profiles ...........................................19 Dose Calculation Algorithms ..............................................................20 Constraints Checking on the Delivery of Virtual Wedge ..............................21 Commissioning of Virtual Wedge .........................................................23

Compensating Filter ................................................................24 Bolus ...................................................................................24 Customized Port and Multileaf Collimators ....................................25

TAR/TPR Calculation ........................................ 26

TAR/TPR for Zero Field Size.......................................................26 Primary Correction Due to Penumbra...........................................26

Collimator Penumbra (Jaw Penumbra) ..................................................27 Cutout and Block Penumbra...............................................................29 Sectorization Process for Cutout and Block Penumbra Calculation.................31 SAR/SPR Integral ............................................................................33 Flattening Filter Correction...............................................................37

Software Implementation .................................. 42


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Source Data Accessed for the Clarkson Calculation ..........................42 OCRs/OCDs................................................................................... 42 TARs/TPRs ................................................................................... 42 Patient Data Sets ........................................................................... 42 Patient Data — CT Input ................................................................... 43 Patient Data - User-Defined Contours and Relative Electron Densities............ 44 Density File Generation.................................................................... 46 Extension of Patient Data ................................................................. 48 Use of Patient Data in Dose Calculation ................................................ 48

Calculation of the Beam Density Matrix ........................................49 Physical and Radiological Depths.................................................50

Normalization Point.................................................................51 Points on Fanline/Depthline Grid ................................................51

Relative Dose.........................................................................51 Patient Dose Matrix .................................................................52 Interest Point Calculations ........................................................52

Clarkson Manual Calculation...............................54

Treatment Plan Setup ..............................................................54 Dose Calculation .....................................................................57

Weight Point................................................................................. 59 “Dose” at Weight Point .................................................................... 63

Calculation Point ....................................................................64 Wedge Transmission........................................................................ 64 TAR for Zero Field Size .................................................................... 68 Primary Correction Due to Penumbra ................................................... 68 SAR Integral.................................................................................. 72 Flattening Filter Correction............................................................... 73

Appendix A: TPR Equation Derivation...................78

Appendix B: Derivation of the Intensity Function ....82

Appendix C: Scatter Calculation Using Day's Rule ....84

Appendix D: Equivalent Radius ...........................88

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Appendix E: Limitations of the Clarkson Algorithm for Small Field Dose Calculations ............................. 90


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XiO® 1

Overview

This Modified Clarkson Sector Integration Algorithm uses patient data, treatment machine data and setup information to simulate dose distributions inside the patient.

The patient data consists of “slices” of relative electron density information that represent a section of the patient. These will have previously been generated by assigning density values to areas enclosed by traced contours or by applying the CT to relative electron density conversion to patient image data.

The patient data is often incomplete because it only defines part of the actual patient when it should define the entire treatment area. If the patient data does not cover the entire treatment volume, certain assumptions must be made during the calculation about the areas where patient data is missing. These are discussed later in this document.

Patient electron density slices may be in the transverse, sagittal or coronal orientations, although for a 3-D plan, only transverse patient data is used in dose calculation.

The treatment machine data required for the Clarkson algorithm consists of a set of Tissue Ratios (TXRs), which are TAR or modified TPR values (see description later in this document), a set of diagonal Off-Center Ratios (OCDs), various machine (energy) specific constants and information about the various beam modifiers.

Setup data consists of information about:

• The position of the beams relative to the patient.

• The various beam modifiers positioned within the beam.

If the beam has a customized port or multileaf collimator, the outlines and transmissions for the blocked areas must be entered. The algorithm takes into account primary dose correction for inhomogeneities in the patient and bolus and the transmission by the wedge, compensator and blocks. Additionally, the Clarkson algorithm takes into account scatter modifications due to field shape. It does not take into account scatter modifications due to differences in field intensity (wedges), patient density or surface curvature. The Clarkson scatter calculation will not accurately model dose for bifurcated structures because it does not account for the scatter reduction from the air space between the separated contours. (Other algorithms may correct for these items in a different manner. See the paper “On methods of inhomogeneity corrections for photon transport”, by Wong and Purdy, for information on this topic. This article is included in the list of dose calculation references under the Bibliographic References, later in this manual.)


2

Each beam is calculated separately and then the beams are summed together. The total dose from all beams represents the dose to the patient.

Dose is calculated on a fanline/depthline grid. It is then interpolated onto a user-defined rectilinear grid. The dose can be viewed in a number of different ways within Teletherapy:

• Interest points only

• Selected planes

• Volume

XiO® 3

Physics of the Clarkson Algorithm

Let D(d,l; fc, fco,ba;r,th,s)MU

be the dose per monitor unit (or per minute)

for an arbitrary point at a physical depth (projected to central axis), d, and radiologic depth (distance from surface to point, scaled by the relative electron density with respect to water and projected to central axis), l, in patient; for a collimator size, fd (not necessarily square field), with its field center off central axis distance, fco; and blocking arrangement, ba, which is defined by customized port or multileaf collimator; the arbitrary point located at a transverse distance, r, from the central axis at angle th from collimator axis along the beam width and an axial distance, s, along the central axis from the radiation source. Parameters fd, fco, and ba are expressed at depth d. In this notation, arguments appearing before the semicolons refer to the depth in patient; those appearing between the semicolons refer to field size, location (with respect to central axis) and shape; those appearing after the semicolons refer to the position of the dose point in the radiation field (see Figure 1).

Figure 1.

F1_00341.DRW

P

ba

fd

fco

Pr

th

fieldcenter

centralaxis

block

S

In the Clarkson method described here, D/Mu (the dose output at calculation point in patient) is given by:

For TAR calculations:

Physics of the Clarkson Algorithm Dose Calculation — Clarkson

4

2( ; ;0; )

( , ; , , ; , , ) *( ; ;0)

* ( ;; ) * (;; , )* (;; , ) * *[ ( ;0;0* (; , , ; , , ) ( ; , , ; , )]

Dc dc ed scD d l fd fco ba r th s scMU

MU TAR dc ed s

INT l r WEGFAC r thCMPFAC r th TRAY TAR lPT fd fco ba r th s SAR l fd fco ba r th

⎡ ⎤⎢ ⎥ ⎛ ⎞= ⎢ ⎥ ⎜ ⎟

⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦

+ (1A)

For TPR calculations:

2

( ; ;0; )( , ; , , ; , , ) (0)*

( ; ;0) ( )

* * ( ;; ) * (;; , )

* (;; , ) * *[ ( ;0;0* (; , , ; , , ) ( ; , , ; , )]

Dc dc ed scD d l fd fco ba r th s SpMU

MU TPR dc ed Sp ed

sc INT l r WEGFAC r ths

CMPFAC r th TRAY TPR lPT fd fco ba r th s SPR l fd fco ba r th

⎡ ⎤⎢ ⎥ ⎡ ⎤

= ⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥

⎢ ⎥⎣ ⎦

⎛ ⎞⎜ ⎟⎝ ⎠

+ (1B)

The equation variables are discussed below:

Dc(dc;ec;0,sc)MU

is the dose output in phantom with tray removed (the dose per monitor unit or dose per minute decayed to treatment date) at the calibration depth, dc, on the central axis at the calibration distance, sc (usually the source-axis distance = sad); for the symmetric open-field equivalent square, ec (at distance sc), for the actual collimator size.

TAR(dc;ec;0) is the central axis tissue-air ratio at calibration depth, dc; for the open-field equivalent square, ec. Therefore,

Dc(dc;ec;0,sc)MU

TAR(dc;ec;0)

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

yields the air dose for

the open-field equivalent square, ec, defined by the actual collimator; on the central axis at distance sc (from radiation source).


XiO® 5

2scs

⎛ ⎞⎜ ⎟⎝ ⎠

corrects the dose from the air dose on the central axis at a distance sc to the air dose on the central axis (for the same collimator setting) at a distance.

INT(l;;r) is the relative beam intensity (relative to photon fluence on central axis) along the beam fanline through the radial distance r, with the effects of the adjustable collimators and phantom scatter removed, since they are being modeled separately. The dependence on l is meant to include in INT the weak effects such as beam hardening, the obliquity of off-axis rays and other effects not modeled explicitly. The argument r without th is used to show that the INT is radially symmetric. The computation of INT is discussed later in this document.

WEGFAC(;;r,th) is the fraction of radiation transmitted through the wedge along the appropriate fanline.

CMPFAC(;;r,th) is the fraction of radiation transmitted through the compensator modifying the beam intensity along the appropriate fanline.

TRAY is the tray factor. If the dose rate (Dc/Mu) is measured with tray in place, this factor defined in machine parameters should be 1.0.

Sp(e) and Sp(0) Sp(e) is the phantom scatter correction factor (PSCF) for a square field of side e (e is the dimension at reference depth). Sp(0) is the PSCF for a square field of side 0. PSCFs are defined at the reference depth where the TPRs are normalized.

TAR(l;0;0) or TPR(l;0;0) is the central axis TAR or TPR at the effective depth, l, extrapolated to zero field size which is performed in the computational TAR or TPR calculation in Source File Maintenance.

PT(;fd,fco,ba;r,th,s) accounts for modifications in the primary intensity due to the field defining elements (collimator jaws and blocking arrangement). It models both radiation transmission and penumbra effects.


6

SAR(l;fd,fco,ba;r,th) or SPR(l;fd,fco,ba;r,th)

is the scatter-air ratio or scatter-phantom ratio at the effective depth l for the blocking arrangement ba at the dose point located at (r,th,s). The determination of this function is a principal objective of this algorithm.

The value proportional to the dose rate (right-hand side of Eq.(1C) or Eq.(1D)) is calculated at each intersection point on the fanline/depthline grid.

For TAR calculations:

2

2

( , ; , , ; , )( ; ;0) 1* * (;; , )

( ; ;0; ) ( * )

* (;; , )*{ ( ;; ) *[ ( ;0;0)* (; , , ; , , )

( ; , , ; , )]}

D d l fd fco ba r thssTAR dc edMU WEGFAC r th

Dc dc ed sc sTRAY scMU

CMPFAC r thINT l r TAR l

PT fd fco ba r th sSAR l fd fco ba r th

⎛ ⎞= ⎜ ⎟⎝ ⎠

+

(1C)

For TPR calculations:

2

2

( )( , ; , , ; , , ) ( ; ;0) *(0)

*( ; ;0, ) ( * )

1 * (;; , ) * (;; , )

*{ ( ;; ) *[ ( ;0;0)* (; , , ; , , )

( ; , , ; , )]}

Sp ecD d l fd fco ba r th s TPR dc edSpMU

Dc dc ed sc TRAY scMU

WEGFAC r th CMPFAC r ths

INT l r TPR lPT fd fco ba r th s

SPR l fd fco ba r th

⎡ ⎤⎢ ⎥⎣ ⎦ =

⎛ ⎞⎜ ⎟⎝ ⎠

+

(1D)

These calculated values are then multiplied by a scale factor to make the value at the weight point equal to a constant (typically 10,000) to retain more accuracy. The scaled values are calculated on the fanline/depthline grid. These values are then interpolated onto the user-defined patient dose matrix. Relative interest point dose is interpolated from the fanline/depthline grid.

The relative dose at the weight point is calculated using conditions from the beam setup, which may not be identical to the conditions for points calculated on the fanline/depthline grid. For example, the relative dose at the weight point for a wedged beam using open normalization is not the same as the relative dose at the weight point for a wedged beam using


XiO® 7

wedged normalization. In Teletherapy, the relative doses will be displayed, which equal the ratio of the “doses” in the patient dose matrix and the “dose” at the weight point multiplied by the beam weight.


8

XiO® 9

Treatment Aids

In the Clarkson calculation the transmission through the treatment aids will be the product of the wedge transmission (WEGFAC) and the compensating filter transmission (CMPFAC). Customized ports and multileaf collimators are calculated in a Clarkson specific manner as described in “Primary Correction due to Penumbra” and “SAR/SPR Integral.” Bolus is treated as part of the patient.

Wedge

A set of wedge transmission values, one for each fan ray, is computed from the geometric intersection of the rays and the wedge. The transmission depends on the distance that the ray “travels” through the wedge and the linear attenuation coefficient of the wedge which in turn depends on the wedge material and the beam energy.

Fixed Wedge

For a wedged beam, the wedge transmission for any point in the fanline/depthline grid is calculated using the equation below. This value is constant along any fanline and need only be computed once for each fanline.

*tT

WFWEGFAC ee

μμ

−−

=

where,

μ is the linear attenuation coefficient (1/cm).

t is the thickness (cm) through the wedge along the ray leading to the point of calculation.

T is the thickness (cm) of the wedge on the central axis.

WF is the central axis wedge factor, linearly interpolated from a stored table of wedge factors versus field size. For a field without a customized port, this wedge factor is looked up for the collimator equivalent square field size. For a field with a customized port or a multileaf collimator, the wedge factor is looked up for the blocked equivalent square field size.

Treatment Aids Dose Calculation — Clarkson

10

Measured Wedge Factors

In Source File Maintenance, wedge factors can be entered for up to 50 square field sizes. The WF and WFW selected in the computation of WEGFAC for any arbitrary matrix point or WEGFACW for the weight pointwill be:

• For the collimator equivalent square field size if the beam does not contain a customized port for a multileaf collimator

• For the blocked equivalent square field size if the beam does contain a customized port or multileaf collimator.

If the field size is outside the range of the field sizes for which wedge factors were entered, then the wedge factor for the nearest field size will be used. If the field size falls between two field sizes for which wedge factors were entered, the wedge factor will be linearly interpolated from the available data. The linear attenuation coefficient for the wedge as well as the physical coordinates of the wedge (relative to the source-to-wedge distance) are also entered in Source File Maintenance.

Open Normalization When using open normalization, the dose matrix is normalized to the open field (without the wedge), i.e. WEGFACw = 1.0. The relative dose is proportional to:

- t- T

w

WF * eWEGFAC e = 1.000WEGFAC

μμ

When the setup is symmetric and the arbitrary dose matrix point is along the central axis, WEGFAC will reduce to the WF because t=T.

When the setup is asymmetric, WEGFAC will not reduce any further because it will not equal T.

When the isodoses for a wedged beam with open normalization are displayed, the doses have already been reduced by the presence of the wedge and therefore need no further correction when calculating monitor units.

Wedged Normalization - Symmetric and Asymmetric Setups For wedged normalization, the WEGFACw at the weight point will be calculated in a similar way to the arbitrary dose matrix point. When normalizing the matrix dose values to the weight point, the relative dose is proportional to:


XiO® 11

w

- t- T

w-w t- T

WF * eWEGFAC e = WFWEGFAC * ee

μμ

μμ

When the arbitrary dose matrix point happens to be the weight point, tw=t

and WF=WFW. Therefore, the ratio 1W

WEGFACWEGFAC

= 12.

NOTE: For wedged fields with customized ports or multileaf collimators, WF=WFw and is the central axis wedge factor for the blocked equivalent square field size. For wedged fields without customized ports or multilear collimators, WF=WFW and is the central axis wedge factor for the collimator equivalent square field size.

When the isodoses are displayed for wedged normalization, the doses have not been reduced by the presence of the wedge and therefore an additional correction is needed when calculating monitor units.

Motorized Wedges

Motorized wedges produce isodose curves for different synthesized wedge angles by varying the portion of the total treatment time that is delivered with the wedge in place. Data for one physical wedge is entered in Source File Maintenance. This data includes:

• The linear attenuation coefficient

• The central axis wedge factors for up to 50 field sizes

• The wedge coordinates

• The type of wedge (motorized or fixed)

• The wedge angle (qn) of this stored wedge when the wedge is left in 100% of the treatment time (“angle”)

In Teletherapy, a “Motorized Wedge Angle” (qs) also is entered. This is the synthesized wedge angle desired in treatment.

Wedged Normalization If wedged normalization was selected for either a symmetric or an asymmetric setup, the beam weight is the desired dose at a point, which is the sum of the (Open Field Dose + Wedged Field Dose). The relationship below is used:

Tan Tans n = F * θ θ


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where, Wedged Field DoseF = (Open Field Dose + Wedged Field Dose)

If for example, qn = 60°, the desired angle qs = 43°, and the Beam Weight = (Open Field Dose + Wedged Field Dose) = 200, the equation becomes:

Tan Tan 43 = F * 60

Wedged Field Dose0.932515 = * 1.7320508200

0.932515Wedged Field Dose = * 200

1.7320508

Wedged Field Dose = 107.68

° °

Open Field Dose + Wedged Field Dose = 200

Open Field Dose + 107.68 = 200

Open Field Dose = 92.32

This means that to achieve a “dose” of 200 with a 60° wedge and to alter the isodose curves as if a 43° wedge is used, a “dose” of 107.68 is given with the 60° wedge in place and a “dose” of 92.32 is given with an open field. Calculations for two beams are actually done, one for an open field with a weight of 92.32 and the second with a 60° wedged field with a weight of 107.68. The resultant doses are added together to give the effect of a 43° wedge.

Open Normalization When using open normalization for a wedged field, the beam weight is the desired dose when the wedge is not in place. The actual dose received at the weight point will be smaller than the beam weight because the wedged field dose = (Beam Weight - Open field dose) * WEGFAC.

Using the same example above, assume the fixed wedge angle is 60° and the wedge factor for this 60° wedge is 0.25. The desired wedge angle is 43° and the desired dose at the weight point in a symmetric field (without the wedge) is 200.

Given that the desired dose at the weight point equals 200, part of this dose will be delivered without the wedge (Open field dose) and part of this dose will be delivered with the 60° wedge (200 - Open field dose).


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Therefore, for a symmetric wedged beam using open normalization, the total dose received at the weight point will be:

Open Field Dose + (200 - Open Field Dose) * WEGFAC

where WEGFAC = WF (the central axis wedge factor defined in Source File Maintenance; 0.25 in this example).

Since the wedged field dose and total dose ratio (to produce the 43° isodose curves) does not change with normalization, the relationship established above is still true:

Tan Tan 43 = F * 60

F = 0.538388

° °

where, Wedged Field DoseF = (Open Field Dose + Wedged Field Dose)

In terms of open normalization with a symmetric setup:

(200 - Open Field Dose) * 0.25F = Open Field Dose + (200 - Open Field Dose) * 0.25

Substituting in the value of F and setting Open field dose equal to (x):

(200 - x)* 0.250.538388 = x +(200 - x)* 0.25

50 - 0.25x0.538388 = x + 50 - 0.25x

0.538388x + 0.538388(50) - 0.538388(0.25x) = 50 - 0.25x

0.403791x + 0.25x = 50 - 26.9194

x = 35.30

Therefore the open field dose is equal to 35.30 and the wedged field dose will equal:


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Wedged Field Dose = (200 - Open Field Dose) * 0.25

Wedged Field Dose = (200 - 35.30) * 0.25

Wedged Field Dose = 41.175

Note that when using open normalization for an asymmetric beam setup, the total dose received at the weight point is slightly different than that described above for symmetric beams:

Open Field Dose + (200 - Open Field Dose) * WEGFAC

where - t- T

WFWEGFAC = * ee

μμ

Enhanced Dynamic Wedge (Varian)

Overview

The XiO Enhanced Dynamic Wedge (EDW) implementation models the dose generated by Varian’s EDW. There are two separate algorithms that are involved---both rely on Segmented Treatment Tables (STTs), which are energy-dependent tables that describe the dose delivery and jaw position “segments” required to deliver EDW dose. In calculating the dose that will be delivered for a particular beam setup, an algorithm generates a “transmission matrix” that will be used to model the modification of fluence emanating from the source. Another algorithm calculates an “Effective Wedge Attenuation Factor” (EWAF), analogous to the wedge factor for a physical wedge, for each beam setup that uses an EDW.

The STTs used for calculating the transmission matrix (which affects the calculation of dose) may be modified by the user. These modifications may help in commissioning photon algorithms, although they are discouraged because of the consistency typically found between measured and calculated doses using the XiO implementation.

The STTs used for calculating EWAFs are not modifiable by the user, as these STTs must be identical to the STTs controlling the delivery of the EDW treatment. If this correspondence is maintained, a first approximation to the EWAF can be derived directly from the STT table. For any treatment setup, this approximate “primary dose” EWAF is the ratio of the relative dose delivered on the axis of the weight point to the total relative dose delivered for the treatment---both are quantities that are found in the STT.


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The XiO implementation goes beyond this simple first approximation, however, with a scatter dose correction for EWAFs. This correction requires the user to only enter measured 60-degree wedge EWAFs at the field center for five fields when commissioning an EDW in Source File Maintenance. The field sizes are 4x4cm, 10x10cm, 15x15cm, 20x20cm, and 30x30cm, which is an asymmetric field size from y = -10 to y = 20cm and x =-15 to x =15. These scatter dose corrections, derived from the measured EWAFs, are interpolated for other wedge angles and field sizes in much the same way that arbitrary wedge angle STT tables are derived from the “Golden STT” as described below.

Generating a Segmented Treatment Table

The Varian Enhanced Dynamic Wedge is based on the concept of an energy-specific “Golden Segmented Treatment Table.” By definition, the Golden STT describes the sequence of jaw positions and doses for each segment of the dose delivered for a 60 degree EDW for the largest possible (full) field size. Generating a segmented treatment table for an arbitrary angle and field size EDW treatment, requires a number of operations on the Golden STT.

The “Golden STT” is first modified for the wedge angle that is to be delivered. The modification process employs the “ratio of tangents” method [Petti and Siddon, 1985], and is performed as follows:

0

0 600 0tan 60 tan tan( ) * (0) * ( )

tan 60 tan 60D y D D yΘ

− Θ Θ= + (1e)

where D0(0) is the “open field” dose, D60

(y) is the cumulative dose delivered at dynamic collimator position “y” in the “Golden STT,” and Θ is the wedge angle to be delivered. Note that open field dose is held constant across the field and is given the Golden STT dose value at position 0.0. The above equation allows the wedge profile implied by the STT to be “flattened” to represent the desired wedge angle.

The final step in producing an STT for a treatment, called “truncation,” is composed of two calculations: The first calculation creates an STT with segments evenly spaced throughout the desired field. The second calculation linearly interpolates the cumulative dose delivered from the STT calculated using the ratio of tangents above to the new evenly spaced set of segments. The result of these manipulations is an STT for the specified field length and wedge angle; it is this modified STT that is used in dose calculation. Note that the STT and primary dose delivery are independent of field width.

Dose Calculation

The delivery of an EDW treatment is a simple one-dimensional intensity modulation. Two parameters are modulated, as specified by the STT,


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during the course of the delivery: the jaw positions and the beam intensity. Together these modulations produce a fluence profile similar to that generated by physical wedges. Both modulations occur in real time when dose is delivered by a Varian linear accelerator. However, since photon dose calculations are time-independent, calculating dose for EDW requires that these modulations be described in a way that is also independent of time.

Segmented treatment tables suit this need well. They specify the relative cumulative dose delivered for various dynamic collimator positions. Varian specifies that the increase in cumulative dose delivered in the interval between two dynamic collimator positions that are specified is linear:

( )( ) b a ay aD y D D Db a

−= − +

− (2e)

where D(y) is the cumulative dose delivered at dynamic collimator position “y”; a, b are adjacent dynamic collimator positions from the treatment STT, a<y<b; and Da, Db are the values of the cumulative dose delivered that are paired with dynamic collimator positions a, b respectively. Thus, it is possible to calculate the cumulative dose delivered to any given dynamic collimator position that is part of an EDW treatment.

The information is written in a format that is independent of time by converting the modified STT to a transmission matrix. Note that the use of this transmission matrix is conceptually different for the Clarkson and Convolution/Superposition algorithms. In the Clarkson algorithm, the transmission matrix is used to modulate dose, whereas in Convolution/Superposition algorithms, it is used to modulate fluence.

Consider a point at the surface of a patient that is going to be treated with EDW. For a portion of the total dose delivery, the point will be directly exposed to the source. The rest of the time, the dynamic collimator jaw will be interposed. Suppose such a point was directly exposed to the source for 85% of the dose delivery, and that for the next 15% of the dose delivery the dynamic collimator jaw was interposed. It follows that for 85% of the dose delivery the transmission of the EDW is ~1 (transmission of air) and for the following 15% the transmission of the EDW is the same as the collimator jaw transmission. Thus, in relative terms, the weight of any element of the transmission matrix is:

( )( )total

D yW yD

= (3e)

where D(y) is determined as described in Eq (2e) and Dtotal

is the total cumulative dose delivered during the treatment. D

total is calculated as the

dose at the “fixed wedge position” - 0.5cm, which is required, to account for the portion of the beam that is always open (See Varian C-Series


XiO® 17

Clinac Enhanced Dynamic Wedge Implementation Guide). Notice that the table is a function only of y, due to the one-dimensional aspect of EDW fluence modulation.

The complete transmission matrix used in the dose calculation algorithms, can be constructed for each fluence (Convolution/Superposition) or primary dose (Clarkson) fan using the Eq (3e) above:

( , ) ( ( , )) (1 ( ( , )))T m n W y m n W y m n= + − (4e)

where the fan under consideration is the m,nth

fan in the matrix, y(m,n) is the position of the dynamic collimator jaw when it is coincident upon the m,nth fan in the matrix, W(y) is the function of dynamic collimator position described in Eq. (3e).

The derived transmission matrix is substituted for the transmission matrix of the collimator jaw in the XiO dose calculation, and dose calculation proceeds as for physical wedges.

Effective Wedge Attenuation Factor Calculation

The Effective Wedge Attenuation Factor (EWAF) is analogous to the wedge factor of a physical wedge. An approximate EWAF, one that only accounts for primary dose, may be computed as the ratio of the cumulative dose delivered to the weight fan (the calculation fan along which the weight point occurs) to the total cumulative dose delivered in the wedged treatment:

,( )( )primarytotal

D yEWAF yDθ = (5e)

where y is the dynamic collimator position coincident with the weight point.

After calculating the primary EWAF, a correction for “scatter dose” is calculated by comparing primary (approximate) EWAFs calculated for calibration setups to user-supplied measured wedge factors for the same setups for a 60-degree EDW. This produces the following correction factor:

,60 ,6060

,60

( ) ( )( )

( )measured measured

centerprimary

total

EWAF w EWAF wCF w

DEWAF wD

= = (6e)

where w is the blocked equivalent square field size of the EDW field to be delivered, ,60 ( )measuredEWAF w is the measured 60 degree EWAF for


18

a square symmetric field of side w cm, and centerD is the cumulative dose value from the STT at the field center.

If we assume that a maximum scatter correction is required for 60-degree wedges, and that the correction for a zero degree wedge is 1, the correction factor for any angle can be obtained using the ratio of tangents method:

0

600 0tan 60 tan tan( ) * ( )

tan 60 tan 60CF w CF wΘ

− Θ Θ= + (7ea)

or, equivalently…

60 0tan( ) ( ( ) 1) * 1

tan 60CF w CF wΘ

Θ= − + (7eb)

where w is the blocked equivalent square field size of the EDW field to be delivered, Θ is the wedge angle to be delivered. The correction factor is used to correct the original approximated, primary EWAF:

,( ) ( ) * ( )primaryEWAF y EWAF y CF wθ Θ= (8e)

The result of (8e) is the final EWAF to be used in the XiO time/monitor unit calculation.

One limitation of this approach is that effective wedge attenuation factors (EWAFs) may be overestimated for weight points away from the central fan of a beam, because scatter decreases with off-axis distance and this decrease is not accounted for in this approach.

Virtual Wedge (Siemens)

Overview

Virtual Wedge is a Siemens treatment modality which generates wedge-shaped dose distributions by moving a collimator jaw from closed to open at a constant speed while varying the dose rate at every 2 mm jaw position. The XiO virtual wedge implementation models the dose generated by Virtual Wedge using the Siemens monitor units (MU) exponential analytic formalism1 that determines the number of MU required to deliver dose at points across the Virtual Wedge beam. For a particular beam setup, the virtual wedge algorithm generates a transmission matrix that will be used to model the modification of fluence emanating from the source. For any treatment setup, the transmission (intensity) map is calculated as the ratio of the MU


XiO® 19

delivered on the axis divided by the monitor units delivered on the central axis MU(0); MU(0) is also the number of monitor units that is entered at the machine console. The MU analytic formalism used for calculating the intensity matrix (which affects the calculation of dose) may be modified by the user through the fine tuning of the effective linear attenuation efficient μ and attenuation correction factor c parameters. These modifications may help in the commissioning of photon algorithms, however, we have found good agreement between calculations and measurements using the default Siemens μ and c values.

In the commissioning process of a virtual wedge in Source File Maintenance (SFM), a key feature is that the user can either select Siemens default machine parameters provided with XiO or define them. These include machine type, collimator head type, wedge orientation (width, length, both), jaw speed, jaw ranges, beam energy, μ, c, dose rate ranges and any desired wedge factors. The virtual wedge implementation models the operational constraints and/or interlocking due to radiation delivery sequence in the different Siemens linear accelerators. Specifically, for a given virtual wedge beam setup, after the dose calculation is complete and a Time/MU function is executed on XiO, the virtual wedge model checks the machine constraints for the delivery of virtual wedge.

Generating Virtual Wedge Intensity Profiles

The delivery of a virtual wedge treatment is a simple one-dimensional (1D) intensity modulation. The jaw motion produces a 1D pattern of intensity g(x) across the field that Siemens expresses by the following analytic formula:

g x c x( ) exp( tan )= μ α (1v)

where x is the distance away from the central axis, c is a Siemens supplied attenuation correction factor (close to unity) used to accurately specify μ for the specific beam, μ is the effective linear attenuation coefficient in water for the nominal beam energy, and alpha is the desired angle to be delivered. The wedge angle can be set to any value when treating a patient and can be delivered if beam constraints are not violated.

In the virtual wedge model, the virtual wedge energy fluence across the field for a given wedge angle calculated using the analytic exponential formula in (1v)

The accelerator works with MU and therefore Siemens assumes proportionality between dose and MU that can be written as:

MU x MU c x( ) ( ) exp( tan )= 0 μ α (2v)


20

where MU(x) is the number of monitor units delivered at a point x and MU(0) is the number of monitor units delivered at central axis (collimator rotation axis) x=0. MU(0) is also the number of monitor units that the user entered at the machine console. This results in central axis cGy/MU calibration factors for a virtual wedge that are identical to those of the open field of the same size since the number of MUs delivered at that position is the same. Therefore, the wedge factor at the central axis is close to unity, for all field sizes. Researchers found that the wedge factors for virtual wedge are close to unity with deviations as large as 4% for large wedge angles and field sizes. The virtual wedge model gives the user the option of using wedge factors of 1 or entering measured wedge factors, for all wedge angles and field sizes.

The total number of monitor units delivered by a Siemens accelerator for a wedged field is equal to MU(xs) the maximum number of monitor units given in the ‘toe’ (static jaw position xs) part of the wedge, and can be calculated as:

MU x MU c xs s( ) ( ) exp( tan )= 0 μ α (3v)

Dose Calculation Algorithms

Two parameters are modulated during the course of the delivery: the jaw positions and the beam intensity. Together these modulations produce an energy fluence profile similar to that generated by physical wedges. Both modulations occur in real time when implemented by a Siemens linear accelerator. However, since photon dose calculations are time-independent, calculating dose for virtual wedge requires that these modulations must be described in a way that is also independent of time.

The MU formula in (2v) fits this situation well. It specifies the dose delivered for any dynamic collimator position provided that the dose at the central axis and the wedge angle are prescribed. Therefore, for any virtual wedge treatment, it is possible to calculate the dose delivered to any given dynamic collimator position. The information in equation (2v) is written in a format that can be used to generate a transmission matrix. Note that the use of this transmission matrix is conceptually different for the Clarkson and Convolution/Superposition algorithms. In the Clarkson algorithm, the transmission matrix is used to modulate primary dose, whereas in the Convolution/Superposition algorithms it is used to modulate energy fluence.

Consider a point at the surface of the patient that is going to be treated with virtual wedge. For a portion of the total MU delivery, the point will be directly exposed to the source. The rest of the time, the dynamic collimator jaw will be interposed. Thus the weight of any element of the transmission matrix is:


XiO® 21

W x MU xMU

c x( ) ( )( )

exp( tan )= =0

μ α (4v)

Notice that the modulation can be a function of width (x direction) or length (y direction) along four wedge orientations (in, out, right, left). The virtual wedge model allows the user to orient virtual wedge along width or length direction. Note that virtual wedge is only available in the length (y direction) for Siemens machines with multileaf collimator (MLC).

The complete transmission matrix used in the dose calculation algorithms, can be constructed for each fluence (Convolution/Superposition) or primary dose (Clarkson) “fan” using the weight ratio described in (4v), above:

T m n W x m n( , ) ( ( , ))= (5v)

where the fan under consideration is the m,nth fan in the matrix, x(m,n) is the position of the dynamic collimator jaw when it is coincident upon the m,nth fan in the matrix. The derived transmission matrix is substituted for the transmission matrix of the collimator jaw in the XiO dose calculation, and dose calculation proceeds as for physical wedges. For any clinical beam setup, the virtual wedge model determines the wedge factor at the weight point as the ratio of the intensity delivered on the axis of the weight point divided by the intensity delivered on central axis. If the virtual wedge beam weight point (weight fan) is at central axis (x=0), then equation (4) indicates a transmission weight factor of 1. For virtual wedge fields with wedge factor at the central axis that deviates from 1, the transmission map is re-scaled to by the measured wedge factor at the central axis (collimator rotation axis).

Constraints Checking on the Delivery of Virtual Wedge

Since the different Siemens linear accelerators equipped with virtual wedge have different dosimetric and mechanical designs, it is critical to know which fields can be delivered during the treatment planning process. The range of combinations of field size, wedge angle, and the number of monitor units at the central axis MU(0) that can be delivered on a Siemens machine is dependent on the maximum and minimum dose rate and jaw speed for that particular machine.

A. The virtual wedge gap Prior to the delivery of a virtual wedge field, the dynamic jaw is moved to the opposing static jaw to leave only a gap of 1 cm width. The 1 cm gap is open for the first few number of monitor units to create a sharper edge at the ‘toe’ part of the wedge field. For any irradiated position in this gap, the monitor units delivered MU(x) (dose) is equal to the maximum monitor units MU(xs) defined in equation (3v). Note that the dose delivered in the gap area, shown in Figure 1, does not follow the exponential formula defined in (2v).


22

In the virtual wedge model, once a virtual wedge is used for a treatment, constraint checking is performed for the field size. The Siemens deliverable field size has limitations along the wedged direction. These include the entire width (or length) of the wedged direction, and the positions of the static and moving jaws. When the minimum gap cannot be maintained because the moving jaw travel across central axis is not far enough to maintain the minimum gap restriction, the virtual wedge model will display a warning message. XiO will still perform the dose calculation, however, a ‘plateau’ or flat region of dose at the ‘toe’ region will be generated.

B. Dose rate and jaw speed limitations Certain combinations of jaw speed, available dose rates and planned MU(0) can not be delivered on the Siemens machines. The XiO virtual wedge model requires the user to perform the dose calculation in order to calculate the monitor units. Consequently, the dose rate and jaw speed checking mechanism is performed in Time/MU calculation part of the virtual wedge model. The idea is to prevent the situation that requires modifying the plan after the isodose distributions and MU calculations are complete.

Once the radiation energy and the wedge angle are selected for planning, the virtual wedge model computes the intensity map for virtual wedge. The intensity at each point in the 2D transmission matrix is calculated by the ratio of monitor units delivered at the calculation point and at the beam central axis. A dose computation and MU calculation then can be performed, respectively.

Following the Time/MU calculation, the virtual wedge model performs the required physics constraint checking. Since Siemens Virtual Wedge uses the field mechanical center as a reference point for the MU setting at the linear accelerator console, all doses are defined from the dose (or the output) at the central axis. Using equation (1) the dose at the static jaw position MU(xs), moving jaw starting position MU(xms) and moving jaw finishing position MU(xmf) shown in Figure 1v can be expressed as:

MU x MU c xs s( ) ( ) exp( tan )= 0 μ α (6va)

MU x MU c xms ms( ) ( ) exp( tan )= 0 μ α (6vb)

MU x MU c xmf mf( ) ( ) exp( tan )= 0 μ α (6vc)

The constraint checking is performed to make sure that the jaw dynamic movement can be accomplished using the available dose rate range assigned in SFM for the linear accelerator.


XiO® 23

The dose variations with respect to position along the moving jaw starting and finishing positions are determined by:

dMUdx

MU c c x

dMUdx

MU c c x

ms

msms

mf

mfmf

=

=

( ) tan exp( tan )

( ) tan exp( tan )

0

0

μ α μ α

μ α μ α

In the delivery of virtual wedge, the dose is highest at the moving jaw start position and the lowest at the moving jaw finishing location as depicted in Figure 2. If we define MUH and MUL as the machine deliverable highest and the lowest dose rate, then the required jaw speeds at each position are:

ν μ α μ α

ν μ α μ α

msH

ms

ms Hms

mfL

mf

mf Lmf

MUdMU

dxMUMU

c c x

MUdMU

dxMUMU

c c x

= =

= =

1 0

1 0

( ) tan exp( tan )

( ) tan exp( tan )

Therefore, in the virtual wedge model, if vms < vmf or vms < vmin, where vmin is minimum jaw speed pre-defined for the machine, virtual wedge beam can not be delivered because the required speed can not be achieved. Therefore, in order to deliver virtual wedge, the maximum dose rate must be lowered, provided that the required dose rate range allows it.

Now the minimum and maximum possible number of monitor units MU(0)min and MU(0)max can be computed:

MU MUv c c x

MU MUv c c x

L

mf mf

H

ms ms

( )tan exp( tan )

( )tan exp( tan )

min

max

0

0

=

=

μ α μ α

μ α μ α

Commissioning of Virtual Wedge

During the validation and verification process of virtual wedge, we have found good agreement between Clarkson and Convolution/Superposition dose calculations and measurements over a large set of test cases for 15, 30, 45 and 60 degree angles, and Siemens Primus 6 MV, 23 MV and Siemens Mevatron MD 15 MV symmetric and asymmetric fields, using the Siemens supplied c and μ values. However, we noticed that the virtual wedge model overestimates the dose in the toe region of the wedge, especially for low beam energy, large wedge angles, and large field sizes.

(7va)

(7vb)

(8va)

(8vb)

(9va)

(9vb)


24

The virtual wedge is delivered initially through a narrow beam strip, e.g., 1x20 cm, for a certain number of monitor units. The jaw starts to move and the field size in the wedge direction gradually increases. There is a lack of electronic equilibrium for very narrow beams. Consequently, the dose delivered is less than the Clarkson or Convolution/Superposition models predict using a broad beam scatter model. Once the moving jaw travels into a more open geometry (more than 1-2 cm along the beam toe), the calculated and measured doses agree better. With increasing beam energy, smaller wedge angles, and smaller field sizes, the difference in dose is smaller to within 1-2%.

Output along the jaw movement for Virtual Wedge is shown:

Compensating Filter

A set of compensating filter corrections, one for each fanline, is computed based on its intersection with the patient surface and the height of the compensator (CMPFAC in the equation above). See Compensating Filter — Technical Considerations for a description of how compensating filters are handled in dose calculation.

Bolus

Bolus is treated as part of the patient. The assigned bolus density will be used to convert the bolus into .ED pixels. These .ED pixels will be merged into the appropriate patient density file. See the section “Dose and Time/MU Calculations for Plans Containing Bolus” in Bolus - Technical Considerations for details on how bolus calculations are performed.


XiO® 25

Customized Port and Multileaf Collimators

Customized ports consist of one or more polygons. These polygons can be defined as either a cutout (where the radiation is blocked outside of this polygon) or as blocks (where the radiation is blocked inside of this polygon). Each polygon is assigned a transmission that is the fraction of the radiation passing through the blocked area. Cutout and block transmissions are used to alter the primary dose calculation (PT, see Primary Correction Due to Penumbra) and they are also used to scale the scatter contributed from the area below the block. See “SAR/SPR Integral” for details.

Multileaf collimators are always defined as cutouts. When MLC is used, the transmission through the cutout is defined as the "MLC transmission" in Source File Maintenance. Multileaf collimators are used exactly the same way in the Clarkson dose calculation as customized ports.

XiO® 26

TAR/TPR Calculation

On the following pages, the calculation of each term in braces { } in equations 1C and 1D is described. Refer to the individual section for each of the terms listed below:

TAR(l;0;0) or TPR(l;0;0) See “TAR/TPR for Zero Field Size”

PT(;fd,fco,ba;r,th,s) See “Primary Correction due to Penumbra”

SAR(d;fd,fco,ba;r,th) See “SAR/SPR Integral”

INT(l;;r) See “Flattening Filter Correction”

TAR/TPR for Zero Field Size

TAR(l;0;0) or TPR(l;0;0) is the TAR or TPR at the radiological depth, l, on the central axis for the 0×0 field size. This value is looked up in a stored data table.

Primary Correction Due to Penumbra

A customized port can be defined as either a cutout (where the radiation is blocked outside of this polygon) or a block (where the radiation is blocked inside of this polygon). The cutout and/or block have diverging edges parallel to the fanlines. A particular customized port can have at most one cutout and multiple blocks. Blocks and cutouts are assumed to be located at the same source to blocking tray distance. Blocks should not overlap cutouts or other blocks, however, the satisfaction of this condition is the user's responsibility. If these polygons do overlap, the penumbra correction in the overlapped region will be considered twice.

Multileaf collimators can be defined only as cutouts. The cutout is assumed to be located at the source-to-MLC distance.

PT(;fd,fco,ba;r,th,s) is the overall penumbra function at the dose point (r,th,s) and is the product of the individual effects arising from each of the upper collimators, lower collimators, cutout, and custom blocks.

TAR/TPR Calculation Dose Calculation — Clarkson

XiO® 27

PT(; fd, fco,ba;r,th,s) = PT(; fu, fco,0;r,th,s)

* PT(; fl, fco,0;r,th,s)

* [PT(;0,0,cutout;r,th,s)

- PT(;0,0,cust.blocks;r,th,s)

- PT(;0,0,hand blocks;r,th,s)]

where,

PT(;fu,fco,0;r,th,s) Is the penumbra for the upper collimators (assumed to control the x size of field)

PT(;fl,fco,0;r,th,s) Is the penumbra for the lower collimators (assumed to control the y size of the field)

PT(;0,0,cutout;r,th,s) Is the penumbra for the cutout

PT(;0,0,cust.blocks;r,th,s) Is the penumbra for the custom blocks

Note that if multileaf collimators are used, the PT (cust.blocks) term drops out of the equation.

Collimator Penumbra (Jaw Penumbra) The collimator penumbra portion of the total penumbra at a calculation point is the product of the penumbra for the upper collimator [PT(;fu,fco,0;r,th,s)] and the penumbra for the lower collimator [PT(;fl,fco,0;r,th,s)] as shown in the following equation:

PT(; fd, fco,0;r,th,s) = PT(; fu, fco,0;r,th,s) * PT(; fl, fco,0;r,th,s)

The values of PT(;fu,fco,0;r,th,s) and PT(;fl,fco,0;r,th,s) are obtained by a procedure similar to that of the Wilkinson extended-source model, in which the fraction of radiation that is transmitted through or past a pair of collimator jaws is estimated as:

PT(; fx, fco,0;r,th,s) = Tx + (1 - Tx)

* Fx[Bx(s); R(fx, fco,0;r,th,s)]

where,


28

x Equals u for the upper collimators, and l for the lower collimators

Tx is the fraction of radiation transmitted through the collimator. One value of collimator transmission is used for both the upper and lower collimators.

Fx is the penumbra function or the fraction of the radiation from the extended source unobscured by the field defining upper or lower collimator jaws, x, (with the other collimator jaws set to infinity) at the dose point (r,th,s).

The penumbra function is comprised of two terms.

Bx(s) Parameterizes the size of the penumbra and is scaled according to the distance, s, the dose point (r,th,s) lies from the field defining element.

R(fx,fco,0;r,th,s) Represents the transverse distances measured from the dose point at (r,th,s) appropriate for the field defining jaw.

To obtain the Fx's of this equation, Wilkinson assumed that the radiation source strength, Sw, was exponentially distributed with R, the radial distance from the center of the source. Suppressing the subscripts and arguments of B and R, Sw and its integral (Fw) are given by:

2-BR

-BR

BSw(B,R) = * e(2* )

Fw(B,R) = 1 - (1 + B * R) * e

π⎡ ⎤⎢ ⎥⎣ ⎦

However, for the Clarkson model used in XiO, it is assumed that the source strength, S, is normally distributed with R, so that:

2

2

2-(BR )

-(BR )

BS(B,R) = * e(2* )

F(B,R) = 1 - e

π⎡ ⎤⎢ ⎥⎣ ⎦

To obtain the penumbra function (Fx) in equation 3 for a pair of collimator jaws, fu or fl, the equation above is integrated and eventually reduced to a form that can be evaluated using error functions, erf. The collimator penumbra for the upper (fu) and lower (fl) collimators can then be rewritten as:


XiO® 29

c c

o

o

PT(; fu, fco,0;r,th,s) = + (1 - )TRAN TRAN

W * 0.5 * {erf[Bc * ( + - x)]x2

W - erf[Bc * ( - - x)]}x2

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

c c

o

o

PT(; fl, fco,0;r,th,s) = + (1 - )TRAN TRAN

L * 0.5 * {erf[Bc * ( + - y)]y2

Y - erf[Bc * ( - - y)]}y2

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

where,

Bc Is equal to c

scd2 * * AL sd * (s - scd)

TRANc Is the collimator transmission.

ALc Is a user-defined source size scaling parameter for the collimator (Collimator edge AL).

scd Is the source to collimator distance which is used for both the upper and lower collimator jaws (cm). (The average of the upper and lower collimator distances from the source).

sd Is the source diameter (cm).

s Is the source to calculation point distance projected to the central axis (cm).

W, L Are the current beam width and beam length projected to the depth of the calculation point (cm).

xo, yo Are the field center coordinates at the depth of the calculation point (cm).

x, y Are the calculation point coordinates with respect to the central axis (cm).

Cutout and Block Penumbra Penumbra calculations are done for:

• Customized port cutouts


30

• Multileaf collimator cutouts

• Customized port blocks

Note that the discussions below center on customized ports (cutouts and blocks). If multileaf collimators are used, only a cutout will exist, which is defined at the source-to-MLC distance. The TRANcutout in this case would be the MLC transmission. The ALcp would be set to the value of MLC edge AL in the case of multileaf collimators.

The cutout and block penumbra portion of the total penumbra at a calculation point is given as the cutout penumbra minus the penumbra from all customized blocks as shown in the following equation:

PT(;0,0,ba;r,th,s) = PT(;0,0,cutout;r,th,s)

- PT(;0,0,cust.blocks;r,th,s)

The cutout penumbra is given by an expression analogous to that for collimators: (If there is no cutout, the cutout penumbra is equal to 1.00).

PT(;0,0,cutout;r,th,s) = Tp + (1 - Tp)

* Fp[Bp(s),R(;0,0,cutout;r,th,s)]

The block penumbra is given by a similar expression:

k

k k k

PT(;0,0,cust.blocks;r,th,s) = (1 - )Tb

* [ (s),R(;0,0,cust. ;r,th,s)]Fb Bb block

∑

Note that since the cutout and blocks are defined with respect to the central axis, the distance r (the transverse distance represented by R in the above equations) is fco independent.

The Fs in the cutout and block penumbra equations are evaluated by sector summations and, like the collimator calculation, are a function of two terms. Bp(s) or Bbk(s) parameterize the size of the cutout or block penumbra and are scaled according to the distance, s, that the dose point (r,th,s) lies from the cutout or block. Because the cutout and blocks are assumed at the same source to blocking tray distance, Bp(s) = Bbk(s), they will be replaced by Bcp(s) in the following write-up. R represents the distances measured from the dose point at (r,th,s) appropriate for the cutout or block.


XiO® 31

Sectorization Process for Cutout and Block Penumbra Calculation

A stored table of average rho (Rn) and theta (θn) values must be calculated for each polygon. First, a list of contour coordinates is read for the customized port. These contour coordinates have been defined relative to the central axis in a counter-clockwise sequence. If the contour points were entered in clockwise sequence, the order of the coordinates is reversed.

The first sector will be the triangle formed by connecting the calculation point, the first contour point of the polygon and the second contour point of the polygon. The angle of this sector is determined (θn). If the absolute value of the angle of the sector is greater than 10°, it must be further subdivided by dividing the angle by the maximum angle allowed (10°). This value is rounded up to the next positive integer.

Then, the original angle is divided by this integer value. For example, if the original theta value is 18°, then 18° would be divided by 10° which would yield 1.8. Rounding this up to the next integer would yield 2.0. Then, taking 18° and dividing by 2 would yield 9.0°. Therefore, the original sector would be divided into two 9° subsectors.

The theta values can be either positive or negative depending on the direction from the previous point. The length of the line segments connecting the calculation point to the contour at each side of the subsector are determined and averaged together with the length of the line segment connecting the calculation point to the second contour point. This will be the average rho (Rn) for this sector and this value is always positive.

The penumbra value for each subsector of the field defining element (whether a block or a cutout) is:

exp 2ncp n* {1 - [-( * ]})B R(2* )

θπ

⎡ ⎤⎢ ⎥⎣ ⎦

where,

θn Is the angle of the subsector

Bcp Is the beta factor for the customized port, which can be either a cutout or block (see following page for calculation)

Rn Is the average rho value for the subsector

This expression is evaluated for each subsector and then the values for each subsector are summed together to produce one penumbra value for each polygon. (The individual sector values can be either positive or negative depending on the sign of θ for that sector.)


32

NOTE: In this way the sum of the subsector angles seen from a point outside a closed polygon is zero. The sum of the product of the angles and the scatter for that radius will yield the net scatter from within the polygon.

The general equations for the cutout and block penumbra are given below. There can be only one cutout and so there will be only one value for PT(;0,0,cutout;r,th,s). However, there can be multiple blocks and therefore the penumbra terms for each of these PT(;0,0,cust.block;r,th,s) must be calculated.

PT(;0,0,cutout;r,th,s) is calculated as:

2cp n

cutout cutout

sides sectorsm -( * )B R

m=1 n=1

PT(;0,0,cutout;r,th,s) = + (1 - )TRAN TRAN

* [1 - ]e2θπ

Δ∑ ∑

where,

Bcp

Is equal to cp

p

std2 * * AL sd * ( - std)s

TRANcutout

Is the transmission outside of the cutout region

ALcp

Is a user-defined source size scaling parameter for customized port (block edge AL)

std Is the source-to-blocking tray distance (where customized port is mounted)

Δθm Is the sector angle for the mth side of the cutout

Rn Is the radius (from calculation point) of the nth sector on the

mth side of the cutout

PT(;0,0,cust.blocks;r,th,s) is calculated as:

2cp n

cust.blocks

kk=1

sides sectorsm -( * )B R

m=1 n=1

PT(;0,0,cust.blocks;r,th,s) = (1 - )TRAN

* [1 - ]e2θπ

Δ

∑

∑ ∑

where,


XiO® 33

TRANk Is the transmission of the kth block defined in customized port

Δθm Is the sector angle for the mth side of the kth block defined in customized port

Rn Is the radius (from calculation point) of the nth sector on the mth side of the kth block defined in customized port.

Note that the penumbra calculation produces different sectorization for the different cutout types (customized port or multileaf collimator). For example, consider a customized port defined by the following coordinates (-3, 3), (-3, -3), (3, -3), and (3, 3).

(-3,3)

(-3, -3)

(3,3)

(3, -3)

These four coordinates would be stored for the customized port and used in the sectorization process to determine penumbra.

If this same cutout is shaped by an MLC with 1 cm leaves, the leaf endpoints would be stored and each shared endpoint would be stored twice, as shown in the following illustration.

(-3,3)

(-3, -3)

(3,3)

(3, -3)

{{{

}}}

1 cm

1 cm

1 cm

1 cm

1 cm

1 cm

SAR/SPR Integral The discussion in this subsection dealing with cutout and block scatter centers on customized ports. Again, if multileaf collimators are used, only a cutout will exist.


34

Phantom scatter is the scatter radiation from the opening defined by the field defining elements, collimators and cutout, minus the scatter reduced by blocks or portions of blocks lying within the field.

.

1

( ; , , ; , ) * ( ; , ,0; , )(1 ) * ( ;0; , ; , )

(1 )* ( ;0, , . ; , )

cutout

cutoutcut blocks

k kk

SAR d fd fco ba r th TRAN SAR d fd fco r thTRAN SAR d fco cutout r th

TRAN SAR d fco cust blocks r th=

=+ −

− −∑

where,

SAR(l;fd,fco,0;r,th) Is the SAR from open field (defined by collimator only). SPR(l;fd,fco,0;r,th) would be used in a TPR calculation.

TRANcutout Is the transmission of the area outside of the cutout

TRANk Is the transmission of the kth block defined in customized port; the value is assigned by the user

NOTE: The transmission outside collimators is ignored in the

SAR/SPR integral.

The scatter at a point from one irregular-shaped field is calculated by dividing the field into sectors and integrating the scatter reaching the calculation point from each of the sectors. See Figure 2. The mathematical formula (for scatter-air and scatter-phantom ratio) are

sec

1 1( ;0, , ; , ) ( ; ;0)

2

sides torsm

nm n

SAR d fco one polygon r th SAR d Rθπ= =

Δ= ∑ ∑

where,

Δθm is the sector angle at calculation point sustained by the mth side of the irregular field

Rn is the radius (from calculation point) of the nth sector on the mth side

SAR(l;Rn;0) is the central axis scatter-air ratio for a circular field of radius Rn at depth l. By definition,

n nSAR(d; ;0) = TAR(d; ;0) - TAR(d;0;0)eR 3

where TAR(l;en;0) is the TAR for equivalent square of side en. If the dimensions are expressed in centimeters,

n nn = (1.782 + 0.00184 )*e R R


XiO® 35

If SPRs are used:

SPR(l;Rn;0) is the central axis scatter-phantom ratio for a circular field of radius Rn at depth l. By definition,

nn n

PSCF( )eSPR(d; ;0) = TPR(d; ;0)* - TPR(d;0;0)eR PSCF(0)

where TPR(l;en;0) is the TPR for equivalent square of side en. If the dimensions are expressed in centimeters,

n nn = (1.782 + 0.00184 )*e R R

Figure 2.

xb

yb

P

bRbi

c

Rci

Rdi

d

Rn

Δθm

This figure shows an outline of an irregular-shaped field projected to the depth of calculation point p. The origin of the coordinate system, 0, is the point where thecentral ray passes.

O

F1_00256.DRW

n sectorth

m sideth

The scatter from a cutout ( a hollow field defining element through which the useful radiation passes) is calculated by dividing the cutout into sectors and summing as shown in the above formula. The scatter being excluded from the calculation point from each block (a solid element around which the useful, radiation passes) is (1 - block transmission) times the scatter calculated from the block by the above


36

equation. The radius of the sector will be clipped by the collimator if the polygon extends beyond the collimator boundary. See Figure 3.

Figure 3

Rn

Rn clipped by collimator

Collimator

Polygon representing acutout or a block

F1_00221.DRW

These assumptions are made in this scatter calculation.

• The Clarkson algorithm assumes a homogeneous patient for the purpose of scatter calculation.

• The primary component of the radiation beam is uniform cross the beam. Since the scatter contribution only depends on the distance between the dose point and the field edge, and not on the relative intensity of the beam at the source of scatter, the scatter contributions may be inaccurate for cases where the beam profile is not flat.

• None of the block contours overlap the cutout or other blocks. This, however, is the user's responsibility and is not verified by the calculation program. If overlapping regions do exist, the scatter in these regions will be considered twice.

• There is a homogeneous scattering medium over the region defined by the field. Therefore, if the dose point lies at the edge of a vertical boundary of the patient, the scatter will be overestimated at this point because of missing tissue.

• While the cutout and block scatter is evaluated by sector summation as described above, Day's rule is used to approximate the collimator scatter (See the section “Flattening Filter Correction” and Appendix C of this document for more details).


XiO® 37

Flattening Filter Correction

The flattening filter causes variations in the beam’s intensity. XiO provides an intensity function, INT(l;;r), to compensate for the variations. The intensity function, INT(l;;r), is the relative beam intensity along the beam fanlines with the effects of the adjustable collimators and phantom scatter removed because they are modeled separately. INT(l;;r) is assumed to be radially symmetric about the central axis. The intensity function on the central axis will always be equal to one because intensity values along the same depth are normalized to the central axis value at that depth.

The diagonal OCD data entered in Source File Maintenance is measured along a 45° diagonal for the user-defined maximum square field size. This OCD data is measured at the machine reference distance which can be different from the current beam's source to skin distance. The intensity correction used should be at a radial distance (r), measured from the central axis divergent to the same depth below the surface in OCD measurement geometry. The calculation point's radiological depth (l) is used in computation. See Figure 4.

Figure 4

dr

P(x,y)

Patient

P'r'

Central axis

l

the source-to-skindistance at which diagonalOCR is measured

ssdOCD

d = physical depthl = radiological depth

Flattening Filter Correction at P = Flattening Filter Correction at P'

Flattening Filter

S

F1-00234.DRW ' '

' '

( ; ;0) * ( ; ; ', 45, )( ;; )

( ;0;0) * (; ; ', 45, ) ( ; ;0)p p OCD

p OCD p

TAR l emx OCD l emx r ssdINT l r

TAR l PT emx r ssd SAR l emx=

+

or

pp p OCD

p OCD p

PSCF( )emxTPR(l; ;0)* OCD(l; ;r ,45, )*emx emx ssd

PSCF(0)INT(l;;r) = TPR(l;0;0)* PT(; ;r ;45, )+ SPR(l; ;0)emx ssd emx

′′ ′

′ ′

′

′

where,


38

r is 22 + yx , the radial distance from the central axis to the calculation point P

ssdOCD is the source-to-skin distance (typically machine reference distance) where OCD is measured

r' is OCD + lssdr * s

, the radial distance projected to depth l in OCD

measurement geometry

emxp' is the square field for which diagonal OCD is measured (typically the machine's maximum allowable square field) projected to depth l, in OCD measurement geometry

OCD(l;emxp';r',45,ssdOCD) is the OCD at depth l along the same fan passing through P. This is calculated by table look up. (Note that the OCD table selected for interpolation will have an SSD closest to the SSD of the central axis)

PT(;emxp';r',45,ssdOCD) is the primary correction at point P' in OCD measurement geometry (collimator set to maximum square field size, no beam modifier). This is calculated by the product of PT for maximum width collimator and PT for maximum length collimator. For details of the calculation, refer to the section “Primary Correction Due to Penumbra,” earlier in this document.

pp

PSCF( )emxTPR(l; ;0) * emx

PSCF(0)′

′ is a table look-up of the modified TPR

for the maximum square field size at the radiological depth, l.

TPR(l;0;0) is a table look-up of the TPR (which for the 0x0 field size is equivalent to the modified TPR) for the 0x0 field size at the radiological depth, l.

TAR(l;emxp';0) is a table look-up of the TAR for the maximum square field size at the radiological depth, l.

TAR(l;0;0) is a table look-up of the TAR for the 0x0 field size at the radiological depth, l.

See Appendix B of this document for the derivation of the intensity function.

SAR(l;emxp';0) and SPR(l;emxp';0) can be calculated by sector summation (sum over a square field of side emxp') as shown earlier. In the Clarkson model implemented on XiO, the scatter is estimated using Day's rule by adding one-fourth central axis SAR or SPR from four rectangular fields. See Figure 5. The sides of the four rectangular fields are


XiO® 39

E1×E1 where E1 = emxp' + 1.4142 * r'

E2×E2 where E2 = |emxp' - 1.4142 * r'|

E1×E2

E2×E1

1 1p

1 2

2 1

2 2

SAR(l; ;0) = 0.25 * SAR(l; x ;0)emx E E

+ g * 0.25 * SAR(l; x ;0)E E

+ g * 0.25 * SAR(l; x ;0)E E

+ 0.25 * SAR(l; x ;0)E E

′

where

g is 1 if emxp' > 1.4142 * r' is -1 if emxp' < 1.4142 * r'

SAR(l;E×E;0) is TAR(l; E;0) - TAR(l;0;0)

SPR(l;E×E;0) is PSCF(E)TPR(l; E;0)* * TPR(l;0;0)PSCF(0)

More information on Day's rule can be found in Appendix C of this document.


40

Figure 5.

P

r'

emxp'

emx p

'

E1-----2E2-----2

E1-----2

E2-----2

(A) (B)

emxp'em

xp'

r'

PE2-----2

E2-----2

E1-----2

E1-----2

(A) is an example of the rectangular fields formed when point P lies within the field.(B) is an example of the rectangular fields formed when point P lies outside of the field.

F1_00238.DRW


XiO® 41

XiO® 42

Software Implementation

Source Data Accessed for the Clarkson Calculation

The Clarkson algorithm accesses necessary data entered in Source File Maintenance. The general machine parameters data, the collimator data and the irregular field data are read for the appropriate machine ID.

OCRs/OCDs

Raw OCRs entry is optional because they are not used directly by the Clarkson calculation. If raw OCRs are entered, the program will use these (together with the source diameter defined in the irregular field data) to calculate the estimated collimator constants (collimator edge AL and collimator transmission). The use of these constants in the Clarkson calculation is explained in the section “Primary Correction due to Penumbra.”

Computational OCR depths and fanlines are mandatory and are used to form the “partial” computational OCR file that forms the basis of the fanline/depthline grid. This will be discussed later in this document. The computational OCDs for the SSD closest to the central axis of the beam will be accessed during dose calculation and used in the computation of the intensity factor.

TARs/TPRs

Computational TARs are used directly in dose calculation. If computational TARs do not exist, then the computational TPRs are used to derive modified computational TPRs that are used in dose calculation.

Modified computational TPRs is the computational TPR multiplie by the

ratio of the phantom scatter correction factors Sp(ed)Sp(0)

, where Sp(ed) is

for the field size and Sp(0) is for the 0×0 field size. (Note that within the Clarkson calculation documentation the computational TAR or modified computational TPR is sometimes referred to as a TXR). See Appendix A of this document for more detail.

Patient Data Sets

Each patient data set used for calculation requires either a set of CT scans (.CT) or a set of user-defined contours and electron densities. Each of these patient data sets is discussed below.

Software Implementation Dose Calculation — Clarkson

XiO® 43

Patient Data — CT Input If CT data is used in isodose calculation, the CT data must be converted to electron densities first. In some cases, the number of pixels is reduced by averaging the CT values into a coarser grid before converting to electron density. If the pixel size of the original CT image is less than or equal to 1mm the images are reduced using the guideline that each pixel is no larger than 2mm and that each pixel will be a whole number of CT pixels. For example, in Figure 6, assume that the original CT image is 1024×1024 pixels and each square pixel has a side which measures 0.4mm. (The CT values for each pixel are shown in the figure). The CT values for sixteen of these 0.4mm pixels can be averaged to yield one pixel (1.6mm×1.6mm) with a rounded average CT value of 4.0:

672 + 3 + 4 + 5 + 1+ 8 + 3 + 2 + 9 +7 + 6 + 3 + 5 + 4 + 3 + 2 = 16

= 4.1875 4.0→

Figure 6

256 pixels

256

pixe

ls

2mm

Reduced Image

4.02m

m2 3 4 5

1 8 3 2

9 7 6 3

5 4 3 2

1024 pixels

1024

pix

els

0.4mm1.6mm

Original Image

F1_00244.DRW

The resultant reduced image will contain 256×256 pixels. The reduced (.CT) file values are then converted to electron densities and stored in a (.EP) file. Note: although 25 of the 0.4mm pixels yields one 2mm×2mm pixel in this example, it is not the choice because 1024 can not be evenly divided by 5.

If the pixel size of the original CT image is greater than 1mm, the number of pixels will not be reduced and the original CT values will be converted directly to electron densities and stored in a (.EP) file. For example, in Figure 7, assume that the original CT image is 1024×1024 pixels and each square pixel has a side which measures 1.1mm. The


44

resultant electron density will contain 1024 x 1024 pixels, each 1.1mm, which are stored in a (.EP) file.

Figure 7

1024 pixels10

24 p

ixel

s

1.1mm

Original Image

1024 pixels

1024

pix

els

1.1mm

Reduced Image

1.1m

m

F1_00299.DRW

The data used to convert the CT values to relative electron densities are entered in Source File Maintenance. This conversion file is user-defined and can contain up to 72 CT numbers with corresponding relative electron densities. All electron densities are interpolated from this set of CT and ED numbers. CT numbers smaller than the smallest value in the conversion table will use the smallest table value. Values larger than the largest in the table will use the largest table value. Any pixel outside the patient is assigned a relative electron density of zero.

Patient Data - User-Defined Contours and Relative Electron Densities

If user-defined contours and electron densities are used for dose calculation, a 2mm pixel grid is overlaid on these contours with the corner of a pixel superimposed with the cross section origin. Each pixel will be assigned the electron density of the contour at the center of this pixel and this electron density information will be stored in an (.EC) file. Target contours have no density and will be ignored. If any user-defined contour intersects or is a subset of another contour, the following rules are used to determine which contour's electron density is assigned to the pixel:

• If the pixel is outside the patient skin outline, assign an electron density of zero to the pixel, otherwise,

• Of all the contours in which the pixel lies, assign the electron density of the last entered contour which does not entirely enclose another of the contours which contains the pixel. In this


XiO® 45

way, the patient outline contour is used only if no other contours contain the pixel.

Examples are given which discuss some of the possible clinical situations. In these examples, capital letters will be used to denote contour labels and the corresponding lower case letters will be used to denote their assigned electron density.

Figure 8. Nested Contours

a

A

b

B

F1_00307.DRW

In this case, Contour A will use it's assigned relative electron density (a) and Contour B will use its assigned relative electron density (b) excluding the region defined by Contour A.

Figure 9. Overlapping Contours

Assume contour A was entered first, contour C was entered second and contour D was entered third. The overlapping region shared by contours C and D is inside all three contours A, B and C. It would assume the relative electron density of contour D. It does not belong to contour A because both contour C and contour D are entirely enclosed by contour A. It does not use contour C because it was entered before contour D (see rule 2).


46

Figure 10. Contours overlapping the patient contour.

F

f

E

e

0

F1_00309.DRW

Assume Contour E was entered second and Contour F (the patient outline) was entered first. In this case, the order of entry is ignored. Contour E assumes its own density, (e), except for the region outside Contour F which is assigned an electron density of zero (see rule 1).

Figure 11. Contour completely outside of the patient contour.

F

f

G

0

F1_00321.DRW

Contour G will assume a relative electron density of zero (see rule 1).

Density File Generation The density file contains patient relative electron density values that have been scaled to the range of 0 to 255. Zero is reserved for the area outside of the patient, while 1 (Emin) to 255 (Emax) are used to represent patient density. To maximize this range, the largest relative electron density value will map onto the maximum value of 255. The scaled electron density is computed from user-defined contours and assigned densities using the equation:

min max minRE = + * ( - )E E ERm

⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

1

where,

E is the computed scaled electron density.


XiO® 47

Emin is the minimum value allowed to represent an electron density, a constant equal to 1.

R is the assigned relative electron density of the contour.

Rm is the largest relative electron density value assigned to any anatomical structure in the studyset.

Emax is the maximum value allowed to represent an electron density, a constant equal to 255, which is constrained by the number of bytes/pixel.

A similar scaling equation is used for CT image data. Due to the limited range (1 to 255) used to represent electron density, the accuracy of the electron density may be reduced. In normal clinical situations, this is not noticeable. A case where it becomes noticeable would be in a studyset which contains a high density material. For example, assume the maximum contour relative electron density is 14.52 (Rm), and the contour under consideration has a relative electron density of 14.32 (R). The scaled density value can be calculated as:

14.32E = 1 + * (255 - 1)14.52

= 251.5014

= 252

⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Similarly, the scaled electron density for water (R=1) can be calculated as:

1E = 1 + * (255 - 1)14.52

= 18.49

= 18

⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

In the dose calculation, these computed (.ED) values are converted into densities relative to 1.00. Therefore, the relative electron density for the contour under consideration is computed as:

(252 - 1) = 14.76(18 - 1)

Therefore, the relative electron density entered was 14.32 and the relative electron density used in calculation, after scaling, is 14.76. When computing the radiological depth at 10 cm deep, when the entire 10 cm


48

passes through the inhomogeneity of density 14.32, the true radiological depth would be 143.2 cm and the calculated radiological depth is 147.6 cm, a difference of 4.4 cm. Again, in a normal clinical situation, an error of this magnitude would not occur. The density in this example is higher than lead.

Extension of Patient Data The dose calculation (regardless of the algorithm used) uses patient electron density files. If a “pixel-by-pixel” dose calculation is requested, (.EP) files are used, otherwise, (.EC) files are used.

The studyset directory is read for the patient and the number of slices (.EP or .EC files) stored for the current orientation is determined. The reference distances for each slice are sorted in ascending order.

When using a 2-D studyset, the slices are put into an equally spaced format and additional slices are added at either end to cover the extent of the fanlines. The purpose of adding additional slices at each end of the 2-D studyset is so that when calculating a single slice, the patient is still represented by a volume and will not be infinitely thin. When using a 3-D non image-based studyset, a slice at a distance of 1mm is added onto each end of the existing patient slices. Therefore, the patient slices will extend from a distance of 1mm smaller than the smallest reference distance to 1mm larger than the largest reference distance. For a 3-D image-based studyset, a distance equal to the slice thickness is added onto each end. The added slices are made identical to the nearest neighbor slice. The equally spaced three-dimensional patient data consists of original electron density patient data and the added electron density slices are referred to as the extended patient data.

Use of Patient Data in Dose Calculation In order to determine the electron density at any point within the designated calculation volume, the calculation will use the electron density at this point within the extended patient data. In a 3-D studyset, the dataset is interpolated to about two millimeter slices. When a point does not fall directly on one of the interpolated patient slices, the electron density at the point will assume the electron density of the closest slice.

The program assumes the patient data in the transverse orientation adequately covers the entire treatment volume. Therefore, only the patient electron densities in the transverse orientation are used in dose calculation. If the dose calculation on a sagittal or coronal plane is requested, the fanline/depthline dose matrix calculated based on the patient's transverse electron densities will be interpolated to produce a 2-D dose matrix on the (sagittal or coronal) plane requested.

If a treatment plan is in two-dimensional planning mode, the program assumes the patient data in the transverse orientation does not necessarily


XiO® 49

cover the entire treatment volume. Therefore, the dose calculation will be based on the patient electron densities in the same orientation as the calculation plane. In other words, if the dose on a sagittal plane is requested, the patient's sagittal electron densities will be used in dose calculation.

Since the Clarkson algorithm calculates along diverging fanlines originating from the “source” of radiation, this patient density information must be converted into a diverging fanline/depthline matrix.

Calculation of the Beam Density Matrix

To simplify dose calculation, a density matrix is built that shares the same coordinate system as the beam. This diverging fanline/depthline matrix is called the beam density matrix. The following steps are involved in the calculation of this beam density matrix:

1. Determine the entry and exit points of each fanline (the point where the density changes from zero to non-zero and the point from non-zero to zero) in the extended patient data.

2. If the depthlines on the beam grid do not cover the deepest exit point, additional depth lines will be added to the beam grid.

3. Determine the density on the fanline and depthline intersections as follows:

• If the heterogeneity correction is requested, the density at the intersection will be the average density along the ray between that depth line and the next depth line nearer the source. In this way, the product of the density and the distance to the depth line nearer the source becomes the accurate radiological path length between the depths. See Figure 12..

• If the heterogeneity correction is not requested, the density at the intersection will be set to 0 or 1 depending on whether the relative electron density of the voxel containing the point is zero (outside the patient) or greater (inside the patient).


50

Physical and Radiological Depths

For each depth, for each fanline, the radiological depth (l) assigned to this point is the sum of the scaled depths from where fanline intersects the patient to this point. The scaled depth is the distance between two adjacent depthlines (measured along central axis) times the average electron density along the fan line between two adjacent depthlines. See Figure 12 below. Refer to Beam Density Matrix for electron density on beam calculation grid.

Central Axis

Surface

Fan Lines

Prdep7 = bdens*delzbdensdelz

Depth Lines

Dmax depth on CAX

Wt pt depth on CAX

Calculation of radiological depth, where

bdens is the electron density of the distal fan line/depth lineintersection point

delz is the perpendicular projection of the length of the fanline segment on the central axis

The radiological depth of this segment (rdep 7) is equal to bdens*delz.If point P were the calculation point, the radiological depth to point P wouldbe determined by calculating rdep=bdens*delz for each segment, and thensumming these products.

Last exit depth line

F1_00322.DRW

Figure 12.

The radiological depth (l) to a point is calculated and used in SXR integral and the zero field size TXR.


XiO® 51

Normalization Point

The location of the normalization point (weight point) is available from the beam setup information and is also displayed on the Source Index.

The physical depth (dw) and effective depth (lw) to the weight point as well as the {TXR} as the weight point are also listed on the Source Index. The TXR term is a composite term equal to:

{ ( ;; ) * [ ( ;0;0) ( ;0;0)* (; , , ; , , )

( ; , , ; , )( ; , , ; , )]}

w

w

w

INT lw rw TAR lw or TPR lwPT fdw fco ba rw th swSAR lw fdw dco ba rw th

or SPR lw fdw fco ba rw th+

where ba=0 for open normalization. The {TXR} value times 1/SW2 * WEGFAC(;;rw,th) or CMPFAC(;;rw,th} is the relative dose at the weight point.

Points on Fanline/Depthline Grid

The program loops through all fanlines and calculates the dose along each fanline, at each fanline/depthline intersection point. The dose at any arbitrary fanline/depthline intersection point becomes:

21 * WEGFAC(;;r;th) * {INT(l;;r)s

* [TAR(l;0;0) * PT(; fd, fco,ba;r,th,s)

+ SAR(d; fd, fco,ba,r,th)]}

⎛ ⎞⎜ ⎟⎝ ⎠

(or TPR/SPR can be substituted for TAR/SAR)

As illustrated, the relative dose at points on the fanline/depthline grid are calculated in the exact setup including beam symmetry and all beam modifiers. If a compensating filter is used, the CMPFAC(;;r,th) would also be in this equation.

Relative Dose

The relative dose, along each ray, at each depth point is calculated as:


52

D * Beam WeightDw

where,

D is the relative dose at anypoint in the fanline/depthline grid

DW is the relative dose at the normalization point (or weight point)

Patient Dose Matrix

If the calculation mode is either ’selected planes’ or ’volume’, the program transforms the relative doses stored in the fanline/depthline matrix into a rectilinear patient dose matrix specified by the user using an eight point linear interpolation. The rectangular grid may be either a volume, which is a matrix with NX columns, NY rows and NZ slabs or selected planes, each of whose size is specified by two of the appropriate values (NX,NY,NZ) as specified by the user. For points (on rectilinear grid) located outside the farthest fanlines of the fanline/depthline grid, the dose will be set to 0. Since depthlines are added to fanline/depthline grid to cover the deepest patient exit point, only interpolation (no extrapolation) occurs in depth direction when relative doses are transformed from the fanline/depthline grid to the rectilinear grid. The values on this patient dose matrix are used to display isodose surfaces or isodose curves. The precision of the display can be improved by defining a finer rectilinear grid.

Interest Point Calculations

Interest points can be entered with a cross section or can be defined in F1 Anatomy (located under global function SF3 EDITPLAN) in Teletherapy. The dose to these points is interpolated from the fanline/depthline relative dose matrix.

When F1 Isocurve (located under global function SF4 DISPDOSE) is activated in Teletherapy, a cursor representing a reference point can be moved around the calculated volume/area. The dose at the current reference point location is displayed in the right upper corner of the screen (along with the patient coordinates of the reference point). This dose, however, is interpolated from the rectilinear patient dose matrix, which was interpolated from the fanline/depthline grid. The value of the dose returned can be different from interest point dose. In this case, the interest point dose (which can be displayed or printed by activating F2 Weight) has higher precision. The accuracy of the reference point dose can be improved by defining a finer rectilinear grid or defining a grid with a grid point at the same location as the reference point.


XiO® 53

XiO® 54

Clarkson Manual Calculation

The following section will describe a manual calculation using the Clarkson algorithm. An asymmetric beam is used with a wedge and a customized port. The patient data includes a transverse and a sagittal cross section. This example is stored in the XiO Demo-only data clinic under the patient ID ‘tutpatient2’ and plan ID ‘manualcalc’. Computational data tables used in this calculation are included in Appendix E.

Treatment Plan Setup

The non image-based studyset ID manualcalc is used in this example. The studyset consists of two cross-sections: one transverse and one sagittal. The cross-sections are cuts from a 40x40x40cm cube phantom. The phantom contains a 10cm thick slab placed parallel to the top surface and across the entire phantom, the slab is located from 2cm to 12cm deep, the density inside the slab is 0.5 and all other areas have a density of 1.00. The origin of the patient is at the geometrical center of the phantom. The transverse cross-section is through reference 0cm, and the sagittal cross-section is at 9.9cm from the origin. One interest point is defined on sagittal cross-section at x=9.9cm, y = -3.3cm, z = 6cm. This interest point is our manual calculation point. See Figure 13.

Figure 13.

28

10

2

40

40

Y

Z

Interest point on a sagittal plane

Figure B

density = 1.0

density = 0.5

40

40

40

10

2

28

9.9

X

YZ

Figure A

F1_00325.DRW

The treatment plan setup, summarized below, is shown in Figure 14.

Source File Maintenance

Clarkson Manual Calculation Dose Calculation — Clarkson

XiO® 55

Machine geometry Machine reference distance: 100 cm Source-to-collimator distance (scd): 49.5 cm Source-to-blocking tray distance (std): 61.6 cm Irregular Field Source diameter (sd): 0.2 cm Collimator edge AL (Alc): 0.2 Collimator transmission (TRANc): 0.01 Block edge AL (ALcp): 0.15 Isodose Plan Beam setup Machine ID: photon1 Setup: SAD Collimator jaw: asymmetric Treatment distance: 100 cm Source-to-skin distance: 96.0 cm Field size defined at: Isocenter LW( cm): 0.;0 RW (cm): 12.0 UL (cm): 12.0 LL (cm): 12.0 Collimator angle: 180 deg Gantry angle: 180 deg Isocenter (cm): x:0.0 y:0.00 z:16.00 (4cm below skin) Weight defined at: isocenter depth field center Weight: 1000.0 Calculation parameters Algorithm: Clarkson Heterogeneity correction: yes Calculation mode: selected planes Treatment aids: Wedge Wedge ID: manualcalc wedge in (heel toward gantry) open normalization wedge type = fixed source-to-wedge dist (swd) = 50 cm max. width (cm) = 25 max. length (cm) = 25 heel position along: length linear attenuation coefficient ( μ 4) = .75 cm-1 wedge transmission factors: .45 for 4x4 .47 for 10x10 .48 for 15x15 .49 for 25x25 wedge heel coordinates in positive wedge coordinates (cm): (-12,0), (12,0), (12,-2), (0,-1), (-12,0)


56

Port ID

2 4

1 2

F 1 _ 0 0 3 2 6 . D R W

9 6

Figure 14.

Port ID: manualcalc (one cutout & one block defined at isocenter)

Cutout size (cm): 18x18 (5 HVL or TRANcutout = .03) centered over collimators (with 3 cm extending outside of the collimator width edges and symmetrically placed over central axis in length direction)

Block size (cm): 9x6 (1 HVL or TRANk = .5) with the left edge flush against left side of the cutout and the block symmetrically placed over central axis.

Figure 15 illustrates the dimensions and calculation point projected at 100/110cm respectively. 100cm is where the weight point is located and 110cm is where the interest point (or calculation point) is located. All dimensions given are in centimeters.


XiO® 57

Figure 15.

5 hvl

5 hvl

Y

X

12/13.2

18/1

9.8

1 hvl

The vertical rectangle (thick line) represents the collimator boundary.The small square represents the cutout.The horizontal rectangle inside of the cutout represents the block.

3/3.3

3/3.3

Collimator

Cutout

Block

Outside Cutout = 5 hvl

represents the field centerrepresents the calculation point

represents the central axis24

/26.

4

18/19.8

9/9.9

6/6.

6

F1_00327.DRW

Dose Calculation

In setup described, the following parameters can be defined:

Weight point:

sw = 100cm

coordinate with respect to the central axis (cm): (6,0)

rw = sqrt(62 + 02) = 6cm

th = tan -1 (0/6) = 0

dw = 4cm (the physical depth)

lw = (2 * 1 + 2 * 0.5) = 3 cm (the effective depth)

fco (cm): (6,0)

Calculation point:

s = 110cm


58

coord wrt central axis (cm): (9.9,-3.3)

r = sqrt(9.92 + 3.32) = 10.44cm

th = tan -1 (-3.3/9.9) = -18.43 deg

d = 14cm

l = (2 * 1 + 10 * 0.5 + 2 * 1) = 9cm

fco (cm): (6.6,0), where 6.6 = 6 * 110 /100

The machine ID ‘photon1’ has a TAR table, therefore, equation 1C will be used:

2

2

D(d,l; fd, fco,ba;r,th,s)TAR(dc;ec;0) 1MU * = * WEGFAC(;;r,th)

Dc(dc;ec;0,sc) sTRAY * scMU

* {INT(l;;r) * [TAR(l;0;0)

* PT(; fd, fco,ba;r,th,s)

+ SAR(d; fd, fco,ba;r,th)]}

⎛ ⎞⎜ ⎟⎝ ⎠

In all cases, the equivalent square used in the TAR, SAR table look-up for a rectangular field is calculated by:

2* W * LE = (W + L)

where

W and L are the width and length of the rectangle.

E is the side of the rectangle's equivalent square.

The “dose” at the weight point (Dw) and the “dose” at the calculation point (D) will be illustrated step by step below, followed by the relative dose (D/Dw) calculation.


XiO® 59

Weight Point Since the wedge is set to open normalization in the treatment plan, the wedge transmission calculation (WEGFAC(;;r,th)) will be ignored at the weight point. The XiO calculated relative dose is always normalized to the dose at the weight point without any blocking (i.e. ba = 0). Therefore, equation 1C can be rewritten as:

2

2

1Dose at Weight Point = * {INT(lw;;rw)sw

* [TAR(lw;0;0) * PT(; fdw, fco,0;rw,th,sw)

+ SAR(dw; fdw, fco,0;rw,th)}

1Dose at Weight Point = * {INT(3;;6)100

* [TAR(3;0;0) * PT

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

(;12x24,6,0;6,0,100)

+ SAR(4;12x24,6,0;6,0)]}

TAR for Zero Field Size

TAR(3;0;0) = 0.9783


PT(;12x24,6,0;6,0,100) = PT(;12x ,6,0;6,0,100) * PT(; x24,0,0;6,0,100)∞ ∞

ccscd = 2 * * AL sd * (s - scd)

49.5 = 2 * 0.2 * 0.2 * (100 - 49.5)

= 1.96

β


60

c c

c c

PT(;12x ,6,0;6,0,100) = +(1 - ) * 0.5TRAN TRAN

12 12 * {erf[ * (6 + - 6)] - erf[ * (6 - - 6)]}2 2

= 0.01+(1 - 0.01)* 0.5 * {erf[1.96 * 6] - erf[1.96 * (-6)]}

= 0.01+(1 - 0.01)* 0.5* [erf(11.76)+ erf(11.76)]

= 1.0

β β

∞

c c

c c

PT(; x24,0,0;6,0,100) = +(1 - ) * 0.5TRAN TRAN

24 24 * {erf[ * (0 + - 0)] - erf[ * (0 - - 0)]}2 2

= 0.01+(1 - 0.01)* 0.5* {erf[1.96 * 12] - erf[1.96 * (-12)]}

= 0.01+(1 - 0.01)* 0.5* [erf(23.52)+ erf(23.52)]

= 1.0

β β

∞

PT(;12x24,6,0;6,0,100) = 1.0 * 1.0 = 1.0

SAR Integral

The SAR at the weight point is calculated using integration. To avoid the complicated manual sector integration in this simple geometry, an approximation of the integral is given here by following Day's Rule:

SAR from collimators (no contribution from outside of collimators):

SAR(4;12x24,6,0;6,0) = SAR(4;12x24,0,0;0,0)

= SAR(4;16x16,0,0;0,0)

= 1.0104 - 0.9362

= 0.0742


INT(3;;6) is the flattening filter correction for a fan passing through any point on a 6cm radius circle around the central axis at 100cm. The plane containing this circle is overlaid by a 3cm thickness of tissue. This correction is assumed to be the same as the points on a 6.18cm radius circle around the central axis at 103 cm and overlaid by 3cm tissue. INT(3;;6) will be calculated using the geometry at 103cm (which is the ssd where OCD data are measured plus 3cm for tissue thickness). See Figures 16 and 17.

Figure 16.


XiO® 61

6

10097

Treatment Plan Geometry

6.18

103

100

6

OCD Measurement Geometry

F1_00328.DRW 2

TAR(3;41.2;0)* OCD(3;40;6.18,45,100)INT(3;;6) = TAR(3;0;0)* PT(;41.2;6.18,45,103)+ SAR(3;41.2;6.18,45)

OCD

Since the OCD data (measured at 100cm) has ratio step size 0.007071 for a field size 56.5685cm (diagonal distance for a 40×40 square) at 100cm, that means the fanlines are spaced 2mm apart at 100cm (or 2.06mm at 103cm). OCD data required to calculate INT(3;;6) would be 31st

( 6 + 10.2

or 6.18 + 10.206

) fan from the central axis at a depth of 3cm. From

table lookup, the OCD value at this point is 1.0293.

Figure 17. BEV at 103cm.

41.2

41.24.37

4.376.18

F1_00329.DRW

3 3

TAR terms


62

Since there is no extrapolation in TAR table lookup, TAR(3;41.2;0) = TAR(3;40;0) = 1.0559 and TAR(3;0;0) = 0.9783.

Penumbra (PT)term

ccscd = 2* *AL sd * (s - scd)

49.5 = 2* 0.2*

0.2* (103 - 49.5)

= 1.85

β

c c

c c

c c c

c

PT(;41.2;6.18,45,103) = { +(1 - )* 0.5TRAN TRAN

41.2 41.2 * {erf[ * 0 + - 4.37 ] - erf[ * 0 - - 4.37 ]}}2 2

41.2 * { +(1 - )* 0.5* {erf[ * (0 + - 4.37]TRAN TRAN 2

41.2 - erf[ * (0 - - 4.37)]}}

2

= {0.01+

β β

β

β

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(1 - 0.01)* 0.5* {erf[1.85* 16.23] - erf[1.85* (-24.97)]}} * {0.01+(1 - 0.01)* 0.5* {erf[1.85* 16.23] - erf[1.85* (-24.97)]}}

= 1.0* 1.0

= 1.0SAR

The SAR in the flattening filter correction calculation is for the (largest) open square field size defined by the collimators, and it is computed by Day's Rule:


XiO® 63

SAR(3;41.2;6.18,45) = 0.25 * [SAR(3;32.46x32.46;0) + SAR(3;49.94x49.94;0) + SAR(3;49.94x32.46;0) + SAR(3;32.46x49.94;0)]

= 0.25 * [SAR(3;32.46;0) + SAR(3;40;0) + SAR(3;39.35;0) + SAR(3;39.35;0)]

= 0.25 * [0.0791 + 0.0776 + 0.0777 + 0.0777]

= 0.0780

Use all calculated values to compute INT(3;;6).

Therefore, 1.0559* 1.0293INT(3;;6) = = 1.02890.9783* 1.0 + 0.0780

“Dose” at Weight Point

-4

{1.0289 * [0.9783* 1.0 + 0.0742]}Dw = 10000

= 1.0829x10

The value in braces { }, 1.068, is the TAR value for effective depth reported in treatment plan printout or in F5 SourceIndx.

NOTE: Unlike the computer calculation, the SAR sector integral is approximated by Day's rule in this hand calculation. We anticipate little difference between the computer generated and hand calculated TARs. See Appendix C of this chapter for details on scatter calculation using Day's rule.


64

Calculation Point

From equation 1C:

2

2

1DoseatCalculationPoint = * WEGFAC(;;r,th) * {INT(l;;r)s

* [TAR(l;0;0) * PT(; fd, fco,ba;r,th,s) + SAR(d; fd, fco,ba;r,th)}

1 = * WEGFAC(;;10.44,-18.43)110

* {INT(9;;10.44

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

) * [TAR(9;0;0) * PT(;13.2x26.4,6.6,0;10.44,-18.43,110) + SAR(14;13.2x26.4,6.6,0;10.44,-18.43)]}

Wedge Transmission

To calculate the segment of the fanline at the calculation point that passes through the wedge, refer to Figure 18. The origin is placed at the central axis at the source-to-wedge distance, the x and y axes are parallel to beam width and toward gantry respectively, and the z axis is pointing toward the source. The wedge is drawn based on its coordinates (the wedge thickness is 1cm at central axis and is 0 at y=-12cm and beyond). The coordinates of the projected calculation point (to source-to-wedge distance) are (4.5, -1.5, 0).

In triangle ABC (which contains a segment of the central axis), point D is at (0, -1.5, 0), segment DE is the section of the fanline inside the wedge. Triangle FGH is parallel to triangle ABC and passes through x=4.5 cm. Point D' is at (4.5, -1.5, 0), segment D'E' is parallel and equal to segment DE. If the fanline through point D' (by connecting source point S and point D') exits the bottom of the wedge at point E", the distance D'E" is the segment of the fanline that passes through the wedge for the calculation point.


XiO® 65

Figure 18.

X

Y

Z

S

5 0

A

CF

H

B

G

D 1 .5

E

D '

E 'E '' 1

4.5

F 1 _ 0 0 3 3 4 .D R W

Equation for segment DE (or line passing through (0,50) and

(-1.5,0)):

y - 0 z - 50 = -1.5 - 0 0 - 50

or

y = 0.03 * (z - 50)

Equation for line BC (or line passing through (0,-1) and (-12,0), see wedge coordinates):

y - 0 z - (-1) = -12 - 0 0 - (-1)

or

y = - 12 * (z + 1)

Solving these two equation for point E coordinates (y = -1.5262 and (z = -0.8728):


66

( ) ( )

( ) ( )

2 22

2 2

' '

0 1.5 1.5262 0.8728

0.8732

50 1.5

50.0225 50.0225 0.8732

50.8957

Distance D E DistanceDE

Distance SD

Distance SE

=

⎡ ⎤= + − + + −⎣ ⎦=

⎡ ⎤= + −⎣ ⎦== +=

From similar triangles SDD' and SEE":

2 2

22

D D SD = E E SE

50.8957Distance E E = 4.5 * 50.0225

= 4.5786

DistanceD E = [(Distance D E + (Distance E E ]) )

= [0. + (4.5786 - 4.5 ])8732

= 0.8767

′′′

′′

′ ′′ ′ ′ ′ ′′

The wedge factor for a wedged beam with a customized port would be looked up for the blocked equivalent square field size. This blocked equivalent square field size is calculated by a “reverse” Clarkson calculation, meaning that the scatter at the reference depth along the weight point fanline is calculated using sector integration. The blocked equivalent square field size is the field size that produces this calculated amount of scatter.

The calculation of the net SAR for a beam with a customized port consists of three parts:

• The SAR from the cutout (clipped by the collimator).


XiO® 67

• The SAR from the area inside of the collimator but outside of the cutout (the scatter from the transmitted radiation)

• The missing SAR due to the blocks inside of the collimator.

To calculate the SAR from a cutout, the cutout is divided into sectors by first determining the angles subtended by the sequential contour points. If the angle between two contour points is larger than 10°, the sector is divided into subsectors of equal angles.

In the particular example, the cutout is an 18cm x 18 cm square at the weight point (96 + 4cm) or a 17.55cm × 17.55cm square at the reference depth (96 + 1.5cm). The angle between the first and second contour points is 90°, so this is divided into nine 10° sectors. The radius of the sectors for calculating the cutout scatter equals the distance along the sectors from the calculation point to the boundary of the unblocked portion of the field. The scatter is accumulated for each sector at a depth of 1.5cm. The scatter from a given sector is a fraction (equal to the sector's angle divided by 360°) of the scatter from a circular field of that sector's radius. The resultant SAR for the cutout equals 0.0434.

To calculate the SAR between the area inside of the collimator and outside of the cutout, the cutout transmission factor will be multiplied by the difference of the SAR between the collimator and cutout. The collimator SAR is calculated at the reference depth using Day's Rule. In this example, the 12cm × 24cm collimator field size at the weight point (96 + 4cm) back projects to 11.7cm × 23.4cm at (96 + 1.5cm). An equivalent square of 15.6cm is used in the SAR table look-up, which at the depth of 1.5cm yields an SAR equal to 0.0464. The SAR from the area between the collimator and cutout is ([0.0464 - 0.0434) * 0.03].

To calculate the missing SAR from a block, the block is sectorized in a manner similar to the cutout. The block is a rectangle of dimensions 9cm × 6cm at the weight point (96 + 4cm). This back projects to 8.775cm × 5.85cm at the reference depth (96 + 1.5cm). The angle between the first and second contour points is 72°, which can be divided into eight sectors of 9° each. (In this example, some of the sector rays will be clipped at the collimator boundaries.) The scatter is accumulated for each sector at a depth of 1.5cm. The resultant SAR for the block equals 0.0139. The missing SAR due to the block is [(1 - 0.5) * 0.0139].


68

The net SAR at the reference depth along the weight point fanline is:

SAR(1.5;11.7x23.41,5.85,0;5.85,0) =

0.0434 + 0.03(0.0464 - 0.0434) - (1 - 0.5)(0.0139)

= 0.0434 + 0.00009 - 0.00695

= 0.0365

Table look-up at a depth of 1.5cm for SAR 0.0365 yields a field size of 11.25cm (at 97.5cm). The printout will return the blocked equivalent square field size at the “Field Defined At” depth, which is 100cm in this example. Therefore, 11.54cm would be returned on the Source Index for the blocked equivalent square field size.

To calculate the WEGFAC term, the central axis WFs are defined for the field size at the surface. Therefore, for field size 11.08cm (11.54cm projected to the skin surface or 96cm), the central axis wedge factor (WF) is 0.47216. Thus, the wedge factor through the calculation point is

( 0.75*0.8767)

( 0.75*1)

(;;10.44, 18.43) *

0.47216 *

0.5179

t

TeWEGFAC WFe

ee

μ

μ

−

−

−

−

− =

=

=

Note that if this example had not used a customized port, the wedge factor for the collimator equivalent square field size would be used to calculate WEGFAC.

TAR for Zero Field Size

TAR(9;0;0) = 0.7250


The primary correction due to penumbra consists of three terms: one from the collimators, one from the cutout and one from the block.

Penumbra from the Collimators PT(;13.2x26.4,6.6,0;10.44,-18.43,110) = PT(;13.2x ,6.6,0;10.44,-18.43,110)

* PT(; x26.4,0,0;10.44,-18.43,110)

∞

∞


XiO® 69


49.5 = 2 * 0.2 *

0.2 * (110 - 49.5)

= 1.64

β

c c

c

c

PT(;13.2x ,6.6,0;10.44,-18.43,110) = +(1 - )* 0.5TRAN TRAN

13.2 * {erf[ * (6.6 + - 9.9)]2

13.2 - erf[ * (6.6 - - 9.9)]}

2

= 0.01+(1 - 0.01)* 0.5* {erf[1.64* 3.3] - erf[1.64* (-9.9)]}

= 1.0

β

β

∞

( ) ( )

{ }

; 26.4, 0, 0; 10.44, 18.43, 110 1 0.5

26.4{ 0 332

26.40 33 }2

0.01 (1 0.01) 0.5[1.64 16.5] [1.64 ( 9.9)]

1.0

c c

c

c

PT TRAN TRAN

erf

erf

erf erf

β

β

∞ × − = + − ∗

⎡ ⎤⎛ ⎞∗ ∗ + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞− ∗ − +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

= + − ∗

∗ ∗ − ∗ −

=

PT(;13.2x26.4,6.6,0;10.44,-18.43,110) = 1.0 * 1.0 = 1.0

Penumbra from Cutout Since the cutout is a square, we can avoid the complexity of integral calculations and calculate it the same way as the collimators.

PT(; ,6.6,0,19.8x19.8;10.44,-18.43,110) = PT(; ,6.6,0,19.8x ;10.44,-18.43,110)

* PT(; ,0,0, x19.8;10.44,-18.43,110)

∞

∞


70

cpcpstd = 2 * * AL sd * (s - std)

61.6 = 2 * 0.15 *

0.2 * (110 - 61.6)

= 1.9091

β

cutout cutout

cp

cp

PT(; ,6.6,0,19.8x ;10.44,-18.43,110) = +(1 - )* 0.5TRAN TRAN

19.8 * {erf[ * (6.6 + - 9.9)]2

19.8 - erf[ * (6.6 - - 9.9)]}

2

= 0.03 +(1 - 0.03)* 0.5* {erf[1.9091* 6.6]

- erf[1.9091* (-13.2)]}

β

β

∞

= 1.0

cutout cutout

cp

cp

PT(; ,0,0, x19.8;10.44,-18.43,110) = +(1 - )* 0.5TRAN TRAN

19.8 * {erf[ * (0 + + 3.3)]2

19.8 - erf[ * (0 - + 3.3)]}

2

= 0.03 +(1 - 0.03)* 0.5* {erf[1.9091* 13.2]

- erf[1.9091* (-6.6)]}

=

β

β

∞

1.0

PT(; ,6.6,0,13.2x26.4;10.44,-18.43,110) = 1.0 * 1.0 = 1.0


XiO® 71

Penumbra from Block With respect to the field center, the center of the block is located at (-4.95,0) at 110 cm.

PT(; ,6.6,0,9.9x6.6;10.44,-18.43,110) = PT(; ,6.6,0,9.9x ;10.44,-18.43,110)

* PT(; ,0,0, x6.6;10.44,-18.43,110)

∞

∞

2 * ** ( )

61.62 * 0.15 *0.2 * (110 61.6)

1.9091

cp cpstdAL

sd s stdβ =

−

=−

=

k

cp

cp

PT(; ,6.6,0,9.9x ;10.44,-18.43,110) = (1 - )* 0.5TRAN

9.9 * {erf[ * (6.6 - 4.95 + - 9.9)]2

9.9 - erf[ * (6.6 - 4.95 - - 9.9)]}2

= (1 - 0.5)* 0.5* {erf[1.9091* (-3.3)]

- erf[1.9091* (-13.2)]}

= 0.25*

β

β

∞

{-erf[6.3] + erf[25.2]}

= 0


72

k

cp

cp

PT(; ,0,0, x6.6;10.44,-18.43,110) = (1 - )* 0.5TRAN

6.6 * {erf[ * (0 + + 3.3)]2

6.6 - erf[ * (0 - + 3.3)]}2

= (1 - 0.5)* 0.5* {erf[1.9091* 6.6]

- erf[1.9091* 0]}

= 0.25

β

β

∞

4

Therefore, the penumbra from the block =

PT(; ,6.6,0,9.9x6.6;10.44,-18.43,110) = 0* 0.25 = 0

The total penumbra correction = 1.0 * (1.0 - 0) = 1.0

SAR Integral

The SAR at the calculation point is calculated using a sector integration technique. An approximation of the integral is given here using Day's Rule:

SAR from collimators (no contribution from outside of collimators):

(9;13.2 26.4, 6.6,0;10.44, 18.43) 0.25 *[ (9;19.8 33.0;0)(9;19.8 19.8;0) (9;6.6 33;0)(9;6.6 19.8;0)]

0.25 *[ (9; 24.75;0) (9;19.8;0)(9;11;0) (9;9.9;0)

0.25 *[0.192 0.172 0.13 0.

SAR SARSAR SARSAR

SAR SARSAR SAR

× − = ×+ × + ×+ ×

= ++ +

= + + + 12]0.1535=

SAR from cutout (excluding area outside collimators):

where,


XiO® 73

(9; , 6.6,13.2 19.8;10.44, 18.43) 0.25 *[ (9;19.8 26.4;0)(9;19.8 13.2;0) (9;6.6 26.4;0)(9;6.6 13.2;0)]

0.25 *[ (9; 22.63;0) (9;15.84;0)(9;10.56;0) (9;8.8;0)]

0.25 *[0.18452 0.152

SAR x SAR xSAR x SAR xSAR x

SAR SARSAR SAR

− =+ ++

= ++ +

= + 04 0.12648 0.1090.14301

+ +=

SAR to be removed due to block (excluding area outside collimators):

(9;,6.6,6.6 6.6;10.44, 18.43) 0.25*[ (9;19.8 13.2;0)(9;6.6 13.2;0) (9;19.0 0;0)(9;6.6 0;0)]

0.25*[ (9;15.84;0) (9;8.8;0)(9;0;0) (9;0;0)]

0.25*[0.15204 0.109 0 0]0.01076

SAR x SAR xSAR x SAR xSAR x

SAR SARSAR SAR

− =+ ++

= −+ −

= − + −=

TTotal SAR:

(9;13.2 26.4,6.6, & ;10.44, 18.43)SAR from cutout (excluding area outside collimator)SAR from area within collimator but blocked by cutout

SAR from block inside collimator

SAR cutout block× − =

+−

(9; , 6.6,13.2 19.8;10.44, 18.43)[ * (9;13.2 26.4, 6.6, 0;10.44, 18.43)

(9; , 6.6,13.2 19.8;10.44, 18.43)][(1 ) * (9; , 6.6, 6.6 6.6;10.44, 18.43)]

0.14301 [0.03* 0.1535 0.14301] [(1 0

cutout

k

SARTRAN SARSAR

TRAN SAR

= × −+ × −− × −− − × −

= + − − − .5) * 0.01076]0.13794=

5


INT(9;;10.44) is the flattening filter correction for a fan passing through any point on a 10.44cm radius circle around the central axis at 110cm. The plane containing this circle is overlaid by a 9cm thickness of tissue. This correction is assumed to be the same as the points on a 10.35cm radius circle around the central axis at 109cm and overlaid by 9cm tissue. INT(9;;10.44) will be calculated with the geometry at 109cm (which is the ssd where OCD data are measured plus 9cm tissue thickness).


74

OCD

Since the OCD data (measured at 100cm) has a ratio step size of 0.007071 for a field size 56.5685cm (diagonal distance for a 40×40 square) at 100cm, that means the fanlines are spaced 2mm apart at 100cm (or 2.18mm at 109cm). The OCD data required to calculate

INT(9;;10.44) would be between the 48th and 49th data 10.35 + 10.218

6 fans

from central axis at a depth of 9cm. By table lookup and interpolation:

OCD(9;40;10.35,45,100) = OCD(9;40;10.25,45,100) + [OCD(9;40;10.46,45,100)

10.35 - 10.25 - OCD(9;40;10.25,45,100)] * 10.46 - 10.25

10.35 - 10.25 = 1.012 + (1.013 - 1.012)*10.46 - 10.25

= 1.0125

TAR terms

Since there is no extrapolation in TAR table lookup,

TAR(9;43.6;0) = TAR(9;40;0) = 0.9261, and TAR(9;0;0) = 0.7250.

Penumbra correction


49.5 = 2 * 0.2 *

0.2 * (109 - 49.5)

= 1.66

β


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43.6(; 43.6;10.35, 45,109) { (1 ) * 0.5 *{ [ * (0 7.32)]2

43.6[ * (0 7.32)]}}*{ (1 ) * 0.52

43.6 43.6* [ * (0 7.32)] [ * (0 7.32)]}}2 2

{0.01 (1 0.01) * 0.5 *{ [1.66 *14.48]

c c c

c c c

c c

PT TRAN TRAN erf

erf TRAN TRAN

erf erf

erfer

β

β

β β

= + − + −

− − − + −

+ − − − −

= + −− [1.66 * ( 29.12)]}}*{0.01 (1 0.01) * 0.5*{ [1.66 *14.48] [1.66 * ( 29.12)]}}

1.0 *1.01.0

ferf erf

− + −− −

==

SAR

SAR in flattening filter correction calculation is for (largest) square field size defined by collimators, it is computed by Day's Rule:

SAR(9;43.6;10.35,45) = 0.25 * [SAR(9;58.24x58.24;0)

+ SAR(9;28.96x28.96;0) + SAR(9;56.28x28.96;0)

+ SAR(9;28.96x58.4;0)]

= 0.25 * [SAR(9;40;0) + SAR(9;28.96;0)

+ SAR(9;38.68;0) + SAR(9;38.68;0)]

= 0.25 * [0.2011 + 0.1948 + 0.2004 + 0.2004]

= 0.1992 0.9261 * 1.0125INT(9;;10.44) = = 1.0146

0.7250 * 1.0 + 0.1992


76

“Dose” at Calculation Point

DoseWEGFAC(;;r,th)* {INT(l;;r)

s* [(TAR(l;0;0)* PT(; fd, fco,ba;r,th,s)) SAR(d; fd, fco,ba;r,th)]}

s

0.5179*{1.0146 * [(0.7250* 1.0) 0.13794]}

121000.3747 10

2

2

4

=

+

=+

= × −

Relative Dose In the isocurve display, Dw will be scaled to the user-defined beam weight, which is 1000 in this example. Therefore, the relative dose at the interest point on sagittal cross-section will be

D 0.3822 * 1000 = * 1000Dw 1.0829

= 352.94

6

This can be verified in F2 Weight (under SF4 DISPDOSE).

Unlike the computer calculation, the SAR sector integrals are approximated by Day's rule in this hand calculation. The dose at the interest point is directly calculated and not interpolated. Therefore, a small difference (within 1%) between the computer and hand calculated interest point doses should be expected.


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Appendix A: TPR Equation Derivation

The derivation of equation (1B) from equation (1A) is discussed in this Appendix.

When using TARs and SARs in dose calculation, the following equation (1A) is used:

D(d,l; fd, fco,ba;r,th,s)MU

=

Dc(dc;ec;0,sc)MU

TAR(dc;ec;0) *

scs

* INT(l;;r) * WEGFAC(;;r,th) * CMPFAC(;;r,th) * TRAY * [TAR(l;0;0) * PT(; fd, fco,ba;r,th,s) + SAR(l; fd, fco,ba;r,th)]

2⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎛⎝⎜

⎞⎠⎟

It is necessary to derive a similar equation when TPRs and SPRs are used in dose calculation. To do this, we need to establish the relationship between TAR and TPR and also the relationship between SAR and SPR.

When TARs are calculated from raw PDDs and backscatter factors, the following relationship is used: (See Khan, The Physics of Radiation Therapy, equation 9.23).

PDD = 100 * TAR(d;ed;0)TAR(dmx;edmx;0)

* (s+ dmx)(s+ d)

2⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

When TPRs are calculated from raw PDDs and Phantom Scatter Correction factors, the following relationship is used:

(See Khan et.al., Medical Physics 7(3), 1980, Eq A8.

PDD = 100 * TPR(d;ed;0)TPR(dmx;edmx;0)

* (s+ dmx)(s+ d)

* Sp(ed)Sp(edmx)

2⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

If TARs and TPRs are used to calculate PDD values, these PDDs should be identical regardless of the data used in their generation.

Therefore, equating the right sides of the equations above:

Appendix A Dose Calculation — Clarkson

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100 * TAR(d;ed;0)TAR(dmx;edmx;0)

* (s + dmx)(s+ d)

= 100 * TPR(d;ed;0)TPR(dmx;edmx;0)

* (s+ dmx)(s+ d)

* Sp(ed)Sp(edmx)

2

2

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

This equation can be reduced to:

TAR(d;ed;0)TAR(dmx;edmx;0)

= TPR(d;ed;0)TPR(dmx;edmx;0)

* Sp(ed)Sp(edmx)

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

If the equation is rearranged, this yields the relationship between TAR and TPR.

TAR(d;ed;0) * TPR(dmx;edmx;0) * Sp(edmx)TAR(dmx;edmx;0)

= TPR(d;ed;0) * Sp(ed)⎡

⎣⎢

⎤

⎦⎥ Set

k to

k = TPR(dmx;edmx;0) * Sp(edmx)TAR(dmx;edmx;0)

At this point we need to establish a relationship between TAR and TPR. The one place that the TAR value is always known is the 0×0 value at the Dmax depth. This value is always 1.000. So, we can use this fact to establish the relationship between TARs and TPRs by looking at the case where edmx=0:

From the equation above when (edmx)=0,

k = TPR(dmx;0,0) * Sp(0)TAR(dmx;0;0)

The value TAR(dmx;0;0) is always equal to 1.000; therefore,

k = TPR(dmx;0;0) * Sp(0)

Then, substituting k back into the equation yields the relationship between TAR and TPR:

TAR(d;ed;0) * [TPR(dmx;0;0) * Sp(0)] = TPR(d;ed;0) * Sp(ed)

TAR(d;ed;0) = TPR(d;ed;0) * Sp(ed)TPR(dmx;0;0) * Sp(0)

Note that the TPR(dmx;0;0) term remains in the equation at this point because it cannot be assumed that this value will equal 1.000 unless the reference depth is equal to Dmax.


80

By definition (see Khan, The Physics of Radiation Therapy):

SAR(d;ed;0) = TAR(d;ed;0) - TAR(d;0;0)

Using the relationship between TARs and TPRs previously established:

TPR(d;ed;0) * Sp(ed) TPR(d;0;0)* Sp(0)SAR(d;ed;0) = - TPR(dmx;0;0) * Sp(0) TPR(dmx;0;0)* Sp(0)

TPR(d;ed;0)* Sp(ed) TPR(d;0;0) = - TPR(dmx;0;0)* Sp(0) TPR(dmx;0;0)

Sp(ed)TPR(d;ed;0)* - TPSp(0)

=

⎡ ⎤⎢ ⎥⎣ ⎦

R(d;0;0)

TPR(dms;0;0)

By definition of SPR:

Sp(ed)SPR(d;ed;0) = TPR(d;ed;0) * - TPR(d;0;0)Sp(0)

⎡ ⎤⎢ ⎥⎣ ⎦

By substitution, the relationship between SAR and SPR is established:

SPR(d;ed;0)SAR(d;ed;0) = TPR(d;0;0)

Substituting TAR with its relation to TPR and substituting the SAR with its equivalent expressed in SPR into Equation 1A:

2

D(d,l; fd, fco,ba;r,th,s) Dc(dc;ec;0,sc) TPR(dmx;0;0) * Sp(0) = * MU MU TPR(dc;ec;0) * Sp(ec)

sc * * INT(l;;r) * WEGFAC(;;r,th)s

TPR(l;0;0) * CMPFAC(;;r,th) * TRAY * [

TPR(dmx;0;0)

* PT(; f

⎛ ⎞⎜ ⎟⎝ ⎠

SPR(d; fd, fco,ba;r,th)d, fco,ba;r,th,s) + ]TPR(dmx;0;0)


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Finding a common denominator for the final term in brackets[] and then canceling the term TPR(dmx;0;0) with the numerator of the first factor, and rearranging terms yields the following equation (which is the same as equation (1B) previously discussed):

2

Dc(dc;ec;0,sc)D(d,l; fd, fco,ba;r,th,s) (Sp(0)MU = *

MU TPR(dc;ec;0) Sp(ec)

sc * * INT(l;;r) * WEGFAC(;;r,th)s

* CMPFAC(;;r,th) * TRAY * [TPR(l;0;0)

* PT(; fd, fco,ba;r,

⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥

⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

⎛ ⎞⎜ ⎟⎝ ⎠

th,s) + SPR(d; fd, fco,ba;r,th)]

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Appendix B: Derivation of the Intensity Function

To calculate the measured dose rate in phantom for an open square field WEGFAC(;;r,th)* CMPFAC(;;r,th) = 1.0 the following equation is used:

2Dc(dc;ed;0,sc)

D(d;ed;r,th,s) scMU = * MU TAR(dc;ed;0) s

* TAR(d;ed;0) * OAR(d;ed;r,th) * TRAY

⎡ ⎤⎢ ⎥ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦

where OAR is the off-axis ratio or the ratio of the measurement at point (r,th) over the measurement at the central axis at the same depth.

From equation 1A for symmetric, open beams in phantom with WEGFAC(;;r,th)* CMPFAC(;;r,th)= 1.0 and ba=0, fco=0 and l=d:

D(d,l; fd, fco,ba;r,th,s)MU

=

Dc(dc;ec;0,sc)MU

TAR(dc;ec;0) *

scs

* INT(l;;r) * TRAY * [TAR(l;0;0) * PT(; fd, fco,ba;r,th,s) + SAR(l; fd, fco,ba;r,th)]

2⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎛⎝⎜

⎞⎠⎟

The dose rate should be the same regardless of the beam model used so the above equations can be set equal to one another. Reducing and rearranging terms:

INT(d;;r) = TAR(d;ed;0)* OAR(d;ed;r,th)TAR(d;0;0)* PT(; fd,0,0;r,th,s)+ SAR(d; fd,0,0,r,th)

If the measurement is for the maximum square field (fd=emx, where emx is the maximum field size at depth) and the points measured are along a 45° diagonal (th=45° and OAR=OCD), then:

Appendix B Dose Calculation — Clarkson

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INT(d;;r) = INT(d;;r,45)

TAR(d;emx;r) * OCD(d;emx;r,45) = TAR(d;0;0) * PT(;emx;r,45,s) + SAR(d;emx;r,45)

where r,45 indicates that the point lies on a transverse line at an angle 45° to the collimator axes.

If TPR data is used:


TPR(d;emx;0) Sp(emx) * * OCD(d;emx;r,45)TPR(dmx;0;0) Sp(0)

= TPR(d;0;0) SPR(d;emx;r,45) * PT(;emx;r,45,s) +

TPR(dmx;0;0) TPR(dmx;0;0)

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

Finding a common denominator (TPR(dmx;0;0)) and then canceling this with the denominator of the first term yields:


Sp(emx)TPR(d;emx;0) * * OCD(d;emx;r,45)Sp(0) =

TPR(d;0;0) * PT(;emx;r,45,s) + SPR(d;emx;r,45)

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Appendix C: Scatter Calculation Using Day's Rule

To calculate scatter for a point located away from the central axis in a rectangular field, it is simpler to calculate scatter by using Day's rule rather than sector integration.

Two situations are discussed below:

• The calculation point is located within the rectangular field

• The calculation point is located outside of the rectangular field.

Based on these two situations, a general equation for the calculation of scatter using Day's rule can be formalized.

Refer to Figure 19 in which the calculation point is located within the rectangular field. The calculation of SXR can be evaluated by the sum of the SXRs from four rectangles. Each component of the sum can be thought of as the rectangle formed by 2*the distance to a collimator edge in the width direction and 2*the distance to a collimator edge in the length direction with the calculation point (Q) as its center. Assume in this example that the field is represented by the rectangle EGOM in Figure 19. The following relationships are established:

SXR(Area EGOM) = SXR(AreaSGTQ) + SXT(Area QTOV)

+ SXR(Area ESQR) + SXR(ARea RQVM)

= [0.25* SXR(Area FGKJ)] + [0.25* SXR(Area BCON)]

+ [0.25* SXR(Area EHLI)] + [0.25* SXR(Area ADPM)]

Appendix C Dose Calculation — Clarkson

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Figure 19.

In the case where the calculation point is outside of the rectangular field, as shown in Clarkson-20, the same relationships are still observed:

SXR(Area EGOM) = SXR(AreaSGTQ) - SXR(Area QTOV)

- SXR(Area ESQR) + SXR(ARea RQVM)

= [0.25* SXR(Area FGKJ)] - [0.25* SXR(Area BCON)]

- [0.25* SXR(Area EHLI)] + [0.25* SXR(Area ADPM)]

In summary the scatter from all four rectangles is added if the calculation point is within the rectangular field. If the calculation point is outside the field, two of the rectangles are added and two of the rectangles are subtracted.

The SXR value for each of the fields in the above equation must be determined. The rectangular field sizes are converted into equivalent square field sizes using the relationship:


86

2* L* WL +W

Figure 20.


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Appendix D: Equivalent Radius

The equivalent radius of a square field is determined using the relationship:

diameter = 1.123 - 0.00067(sEq)sEq

where sEq is the side of the (equivalent) square in centimeters. Solving for the diameter:

diameter = 1.123(sEq) - 0.00067(sEq )

radius = 2.246(sEq) - 0.00134(sEq )

2

2


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Appendix E: Limitations of the Clarkson Algorithm for Small Field Dose Calculations

When using the Clarkson algorithm for small field dose calculations, the TAR/TPR values could be reduced when compared to measurements. The TAR/TPR term is derived from the sum of primary TAR/TPR and scatter components. The primary TAR/TPR is computed by multiplying the TAR/TPR for the zero open field by the penumbra intensity correction (Penumbra term: PT). The PT is determined as a function of the machine source diameter, source-to-collimator distance, collimator alpha, and collimator transmission values entered in Source File Maintenance. The PT value is assumed to be 1.0 on the beam central axis and decreases to the transmission value in the tail region outside of the field opening. Depending on the parameters listed above, the width of the penumbra will vary.

If the field size is reduced to be smaller than the penumbra width, the PT does not reach 1.0 on the beam central axis. For example, a machine is set with a source diameter of 0.2 cm, a source-to-collimator distance of 36 cm, a collimator alpha of 3.0, and a collimator transmission of 0.02. If this machine is used for a 2×2 cm field and the beam weight is defined at the beam central axis, the PT value at the beam central axis becomes 0.98. This causes the TAR/TPR at the weight point to be 2 percent less than expected. The PT value at the beam central axis decreases significantly for these particular parameters if the field size is reduced smaller than 2×2 cm. The PT is decreased to 0.77 for the 1×1 cm field opening.

Therefore prior to applying the Clarkson dose calculation for field sizes smaller than 4×4cm, CMS strongly suggests that the doses be validated, especially for the smaller fields, via the commissioning function in Source File Maintenance. In order to do this, small field PDDs and OCRs must be measured and fit to calculated data by utilizing the commissioning function.


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