Intensity-modulated radiotherapy by means of static tomotherapy: A ...

Intensity-modulated radiotherapy by means of static tomotherapy: 

A planning and verification study 

Mark Oldham and Steve Webb 

Joint Department of Physics, Institute of Cancer Research and the Royal Marsden NHS Trust, 

Downs Road, Sutton, Surrey, SM2 5PT, United Kingdom 

Received 25 July 1996; accepted for publication 15 March 1997 

There is currently much research interest in developing, evaluating, and verifying intensitymodulation 

techniques. Of particular interest is how well the delivery of intensity-modulated profiles 

can be simulated by planning algorithms, and how accurately these profiles can be delivered 

given the specification constraints of linear accelerators. In this paper we present a planning and 

verification study based on delivering radiation in ‘‘static-tomotherapy’’ mode via the NOMOS 

MIMiC Multileaf intensity-modulation collimator, which sheds some light on these issues. An 

inverse-planning algorithm was used to compute intensity-modulated profiles for a 9-coplanar-field 

plan for a body phantom. The algorithm makes several approximations about the form of the 

elementary fluence profile through bixels during delivery. Specifically, it is independent of the state 

of adjacent bixels i.e., open or closed and obeys the superposition principle. From the standpoint 

of comparing the predicted versus the delivered dose, these assumptions were made irrelevant by a 

final one-step forward dose calculation performed using the optimized intensity profiles. This forward 

dose calculation took into account the penumbral characteristics of the delivery system by 

decomposing the intensity profiles into the set of delivery components. Each component was assigned 

the appropriate penumbral functions thereby ensuring that the calculated dose distribution 

closely predicted the delivered dose distribution. The nine intensity modulated fields were delivered 

to a perspex phantom with the same geometry, containing a verification film. In general good 

agreement was found between the predicted and the measured delivered dose distributions. All the 

main features of the predicted dose distribution are seen in the delivered. The 90% isodoses were 

consistently in spatial agreement to within 3 mm. At the 50% isodose level consistent spatial 

agreement was again found to within 3 mm, the largest deviation being about 5 mm. The close 

correspondence between the predicted and measured dose distribution demonstrates the potential of 

the MIMiC delivery system. Our results indicate the level of dose conformation that is achievable 

in practice and the accuracy of the dose computation algorithm. However, this study only concerned 

delivery of radiation to a2cmthick slice, and the dose distribution was only verified in the central 

plane of the phantom where the film was placed. We therefore cannot comment as yet on what 

happens to the dose distribution away from the central film-plane. © 1997 American Association 

of Physicists in Medicine. S0094-24059700506-3 

Key words: intensity modulation, radiotherapy, conformal therapy, verification, tomotherapy, film 

dosimetry 

I. INTRODUCTION 

New techniques to modulate the intensity of radiation in 

external-beam radiotherapy enable much more precise tailoring 

of the spatial distribution of the delivered high-dose region 

to the shape of the tumor than can be obtained with 

conventional therapy. Many physical methods have been 

proposed that could achieve this modulation. The methods 

differ in their complexity and in the hardware necessary to 

deliver the radiation. All of these methods can be classified 

into one of three groups, each group having as the common 

denominator a type of hardware used to modulate the intensity 

of radiation. These three hardware groups are a 

multileaf-collimators, b tomotherapy-type slit collimators, 

and c moving-bar collimators. In group a the intensity of 

geometrically shaped radiation fields is modulated either by 

dynamically moving the leaves of the MLC during irradiation, 

or by multiple-static segmented irradiation. In group 

b, a slit of radiation is rotated transaxially about the patient 

with the long axis of the slit aligned transaxially. The radiation 

along the slit is modulated by short vanes that move into 

and out of the slit. This modulation can take place either with 

continuous gantry rotation or with the gantry at a series of 

fixed orientations. The latter we call static-tomotherapy. In 

group c a small bar is moved across the field with varying 

velocity during irradiation. All of these techniques aim to 

improve on the use of compensators, the traditional way to 

obtain intensity-modulated beams IMBs. 1 

There is currently much interest in developing, evaluating, 

and verifying all of the intensity-modulation techniques. 

Several questions are of particular significance. These include: 

1 How much of an improvement can intensity modulation 

make to radiotherapy dose distributions? 2 What 

827 Med. Phys. 24 (6), June 1997 0094-2405/97/24(6)/827/10/$10.00 © 1997 Am. Assoc. Phys. Med. 827

828 M. Oldham and S. Webb: Intensity-modulated radiotherapy 828 

FIG. 2. The planning problem studied in this paper dimensions given in the 

text. The PTV is the ‘‘bean-shaped’’ invaginated region resulting from the 

removal of parts of two circles the OARs from an ellipse. The problem 

represents a prostate PTV with bladder and rectum OARs. The ‘‘I values’’ 

are the importance-factors used in the inverse planning. 

FIG. 1. A schematic illustration of the slit beam of the MIMiC and its bixels 

in relation to the patient. Open bixels are shown as white spaces and closed 

bixels as black spaces. The vanes of the MIMiC are not shown, but five of 

the actuating mechanisms which can open or close the vanes are shown 

schematically. 

level of intensity modulation is optimal? 3 Where does the 

trade-off start to become unfavorable between increasingly 

complex and expensive treatments and improved dose distributions? 

And 4 which tumor types and disease sites will 

the new techniques be most useful for? Two of the most 

central questions are how well can the intensity-modulated 

dose delivery be simulated with current planning algorithms, 

and how accurately can an intensity-modulated plan be delivered 

in practice given the specification constraints of linear 

accelerators? In this paper we present a planning and 

verification study, in the static-tomotherapy framework, 

which sheds some light on the latter two issues. Investigating 

the dosimetry of the MIMiC in static tomotherapy mode is 

easier than for rotational tomotherapy due to the lack of 

smearing of the dose distribution and issues of the stability 

of MU delivery per degree. This study is therefore a necessary 

first step in investigating the dosimetry of the MIMiC. 

II. METHOD 

In this paper we present a planning and verification study 

based on delivering radiation in ‘‘static-tomotherapy’’ mode 

via the NOMOS 2591 Wexford Bayne Road, Suite 315, 

Sewickley, PA, 15143 MIMiC 2–7 Multileaf Intensity- 

Modulation Collimator Fig. 1. The MIMiC device attaches 

to the blocking tray of the linear accelerator and positional 

adjustment screws and bolts enable the collimation slit to be 

accurately aligned with the cross wires. The center of the 

cross wires coincides with the center of the slit, the edges of 

which are made parallel with the respective cross wire. The 

intensity along the slit beam of the MIMiC can be modulated 

by small absorbing vanes illustrated in Fig. 1 that move in 

and out of the slit under computer control. The vanes are 

arranged in two parallel banks, one superiorly and one inferiorly 

to the slit. The MIMiC can be used in either rotational 

mode, where the vanes move as the gantry rotates, or static 

mode, where the vanes move but the gantry is fixed at some 

orientation. The former is conventionally termed tomotherapy 

and the latter we refer to as static tomotherapy. 

The planning problem addressed is an idealized model of 

the treatment of an invaginated prostate patient illustrated in 

Fig. 2. The patient contor was assumed elliptical with major 

and minor axes of lengths 29 and 25 cm, respectively. The 

planning-target-volume PTV was part of an ellipse, major 

and minor axes of 9 and 7 cm, the center of which was 

offset vertically below the center of the contour by 2 cm. 

Two circular organs-at-risk OAR were located above and 

below the PTV with radii 4 and 1.5 cm, respectively. The 

centers of the OARs were positioned 5 cm above and 4 cm 

below the center of the PTV, the latter being defined as the 

isocenter. The PTV was the invaginated region created by 

the subtraction of parts of the circular OARs from the ellipse. 

A circular radius 7.25 cm region of sensitive tissue was 

defined enclosing the PTV and both OARs as shown. The 

axial thickness of the phantom was specified as extending for 

12 transaxial slices each of thickness 1 mm, with no change 

in the contours of any of the structures. 

Fitting the high-dose region to such an invaginated PTV, 

with directly abutting OARs constitutes a very difficult planning 

problem, and one that would not be achievable with 

conventional radiotherapy techniques. 8 

A. Inverse planning and forward dose calculation by 

components 

A relative dose prescription of 1.0 in the PTV, 0.1 in the 

two small OARs and 0.3 in the circumscribing sensitive tissue, 

was specified in the planning problem of Fig. 2 see 

Appendix B for a discussion of these units. Then a plan 

consisting of nine equi-spaced co-planar fixed slit fields was 

designed, with gantry angles of 0°, 40°, 80°, 120°, 160°, 

200°, 240°, 280°, and 320°. A set of intensity-modulated 

beams was computed by solving the inverse problem by 

least-squares iteration. 9 The planning software was written 

in-house to exactly simulate the geometry of the MIMiC, and 

Medical Physics, Vol. 24, No. 6, June 1997


the inverse algorithm assumed an elemental fluence profile 

for each bixel which incorporated the penumbral effects of 

the surrounding closed vanes. Bixel is the term used to refer 

to the cross-sectional area of a vane of the MIMiC through 

which radiation can pass if the vane is open. The planning 

algorithm is essentially similar to that used in the PEACOCK 

planning system. It was written to provide the potential to 

independently study effects such as bixel profile variation, 

inhomogeneity, and gantry sag. In this study the elemental 

fluence profile was found by fitting the measured dose data, 

delivered by a single open vane at 10 cm depth in water, to a 

mathematical function which obeys the superposition principle. 

This function is a convolution of a double exponential 

with a rect function equal to the bixel size. 10,11 The resulting 

profile, which we call the ‘‘stretched-fit’’ profile, has the 

property that when the fluence profiles of adjacent open bixels 

are combined corresponding to two bixels being open at 

the same time the resulting fluence profile is flat across the 

two bixels as it must be in reality. The terminology stretched 

fit signifies that the fitting procedure actually widens the elementary 

fluence profile since superposition of the measured 

profiles leads to regions of low fluence between bixels. This 

fitting procedure is discussed in detail in an earlier report. 11 

The dose computation grid was set at 1 mm resolution, with 

dimensionsof300 300 12. 

Once the intensity-modulated profiles were determined 

for the nine static ports, a forward dose calculation was performed 

to predict the delivered dose taking into account the 

characteristics of the delivery method. This is a crucial point 

we wish to elaborate on. The key point is that during planning 

the elemental fluence profile used for each bixel assumes 

that the neighboring bixels are closed. This is necessary 

as a priori information of the intensity-modulated 

profile is not available. When it comes to the delivery of the 

fluence through a bixel however, the neighboring bixels may 

well be either open, closed, or open for part of the irradiation. 

In essence, the penumbral effects assumed at the planning 

stage do not reflect what happens during the delivery. 

To circumvent this discrepancy, the first step of the forward 

dose calculation is the decomposition of the intensitymodulated 

profiles into their respective components. 11 Each 

component is a combination of open and closed vanes 

through which an amount of radiation is delivered. The delivery 

of any intensity-modulated profile must, in practice, 

adopt this decomposition concept. The alternative is to deliver 

each bixel independently, with all other bixels closed, 

which would be far too time consuming and give too high a 

leakage dose. As with the planning software, this forward 

dose calculation was written in-house to allow independent 

study of effects such as the penumbral characteristics during 

delivery. The NOMOS planning system PEACOCK does not 

use our component delivery method to model the penumbral 

effects during delivery. 

The forward dose calculation used in this study first computed 

the component bixel patterns for each gantry angle 

Appendix A. Then the appropriate fluence profiles were 

assigned to each component. Between adjacent open bixels 

the stretched-fit profile was assigned ensuring a flat profile. 

FIG. 3. The variation in relative-output-factor ROF between different bixels 

along the MIMiC for the PHILIPS SL 75/5. ROF’s are given in percentages, 

defined with respect to a 10 10 cm 2 open field measured at the dose 

maximum. 

Between an open and closed bixel, the appropriate measured 

penumbral profile was assumed. By accurately modeling 

how the MIMiC delivers an intensity-modulated profile via 

components, this forward dose calculation takes account of 

the penumbral characteristics of the delivery system. Furthermore, 

the variation in relative-output-factor ROF for 

bixels at different positions in the slit was incorporated by 

assigning each bixel a unique ROF obtained from fitting all 

bixels to the measured data for the open slit field Fig. 3. It 

is noted that only the centrally positioned bixels numbers 

6–15 have a significant amount of radiation passing through 

them in this study. An example of an intensity-modulated 

profile and its corresponding delivery components are given 

in Fig. 4 and Table I. Note that this is only one of many 

possible decomposition patterns. 12 However, as the MIMiC 

was operated by initially opening as many leaves as possible 

FIG. 4. The intensity-modulated profile found by inverse-planning for the 

field at gantry angle 0°. 



TABLE I. This table illustrates the decomposed components, and the number 

of monitor units delivered to each component, of the intensity-modulated 

profile of Fig. 4. 

Component Open bixels Monitor units 

1 6–8, 12–15 1 

2 6–8, 13–15 6 

3 6–7, 14–15 6 

4 6–7, 14 1 

for a profile and gradually shutting them as required, this is 

the actual decomposition pattern delivered by the MIMiC. 

The intensity profiles found by inverse planning for all nine 

gantry angles are shown in Fig. 5. The forward dose calculation 

was performed with the IMB profiles discretized to the 

nearest integer MU since fractions of a MU cannot be delivered 

in practice. The absolute values were chosen to give 

0.82 Gy to the maximum dose point Appendix B. 

B. Delivery and film measurement 

The intensity profiles of the nine fields Fig. 5 were delivered 

to a perspex phantom with the geometry given in Fig. 

2. The perspex phantom was constructed in two parts, each 

part having the dimensions of Fig. 2 with a thickness of 2 

cm. These two parts were then placed back-to-back on the 

treatment couch sandwiching a piece of Kodak X-OMAT V 

standard verification film between them. For each component 

of each field, the vane configuration was set manually via 

interaction with the touch-screen computer that is attached to 

the MIMiC. The same IMB was fed into each bank at any 

one orientation, and the MIMiC was attached to a PHILIPS 

SL 75/5 accelerator. The appropriate number of monitor 

units of dose was then delivered through that component’s 

vane configuration. This procedure necessitated that an operator 

entered the treatment room to set the MIMiC for each 

configuration. In all there were 62 component-vane configurations 

set making the total treatment time completely unfeasible 

for clinical routine about 2 h but acceptable for a 

one-off experimental demonstration of principle. The manual 

FIG. 5. The intensity profiles found by inverse-planning for the MIMiC fields at gantry angles 0°, 40°, 80°, 120°, 160°, 200°, 240°, 280°, and 320°. The 

profiles were discretized to the nearest MU. 



setting of vane configurations was necessary as the control 

software to automatically deliver components in this way is 

not yet available. The functionality to automatically deliver 

a sequence of component vane configurations, without the 

need for an operator to enter the treatment room, does exist if 

the intensity-modulated profile to be delivered has been produced 

by the PEACOCK planning software. 6,13 

At delivery the perspex phantom was positioned vertically 

on the couch so that the flat plane of the phantom was in the 

plane of gantry rotation. The couch was then adjusted so that 

the plane of the verification film sandwiched between the 

two halves of the phantom coincided with the AB cross wire. 

The film was thus positioned in the same plane as the source, 

and in the middle of the slit of radiation defined by the two 

banks of the MIMiC. During delivery, both banks of bixels 

were treated identically, so that, in effect, the nine intensitymodulated 

fields were delivered to a2cmtransaxial slice of 

the phantom. The verification film was located centrally in 

this 2 cm slice. 

The complex nature of the above nine field irradiation 

made an accurate comparison of the absolute predicted and 

measured dosimetry difficult. A second irradiation experiment 

was therefore performed to verify the absolute dosimetry 

of the forward dose calculation for the case of a single 

intensity modulated field: That at gantry angle zero Fig. 4 

and Table I. In this simpler problem the confusing effects of 

dose smearing from multiple-fields, gantry sag, flatness, and 

output variation with gantry angle etc., are reduced. The 

number of monitor units in Table I were increased by a factor 

of 3 for this irradiation to give an appropriate dose to the 

film in the absence of the other eight beams. A new high 

performance verification film was used which is reported 14 to 

be highly linear between 0–0.5 Gy, to have negligible energy 

response, and to have a 2% inter and intrafilm consistency. 

A further advantage of this film is that it is available in 

vacuum-sealed water-proof packaging removing problems 

associated with air spaces trapped in the film packaging and 

enabling depth-dose measurements in water. Our measurements 

confirm the superior performance of this film calibration 

curve is shown in Fig. 6a, and a comparison of the 

depth-dose curve in water measured by film and ion chamber 

in Fig. 6b, however we observed saturation at the much 

lower dose value of about 0.6 Gy compared to the reported 

0.9 Gy. 14 Both the CEA and Kodak film sensitivity were 

greater when the film was oriented perpendicular to the incoming 

radiation Fig. 6a. In both experiments the film 

was actually oriented parallel to the incoming radiation and 

so the parallel orientation calibration curves were used to 

convert from optical density to dose. It is clear from Fig. 6b 

that the sensitivity of both types of film remains independent 

of any spectral hardening with depth. It was not feasible to 

use the CEA film for the nine field irradiation experiment 

because its low-dose saturation property would necessitate 

delivering less MU for each field and much of the fine detail 

would have been lost. 

FIG. 6. a Calibration curves for the CEA dotted lines and Kodak 

X-OMAT V solid lines film. The two calibration curves for each film 

correspond to whether the film is perpendicular or parallel to incoming 

radiation. Radiation sensitivity is higher for perpendicularly oriented film in 

both cases. b Comparison of depth-dose measurements in water taken with 

CEA film triangles, Kodak X-OMAT V film stars and an ion chamber 

squares. The data were normalized at 5 cm depth. The close agreement 

indicates the negligible energy response of both the CEA and Kodak film. 

III. RESULTS 

The measured and predicted absolute isodose lines for the 

single field irradiation are shown in Fig. 7. Predicted isodoses 

were found by interpolation from dose points on a 2 

mm grid spacing. Note—As the distribution was only computed 

for the region inside the circular outer OAR of Fig. 2, 

the predicted isodoses do not extend to the full extent of the 

measured. Excellent agreement is observed between the film 

and the forward dose calculation prediction, although the effect 

of the increased spatial resolution of the film is visible in 

the regions of high dose gradient at the edges. The close 

agreement between measurement and prediction indicates 

that the component delivery CD forward dose calculation 

model is accurate to within a few percent. Use of the new 

high performance CEA film enabled a more accurate measurement 

than would have been possible with conventional 

verification film. 



FIG. 7. Comparison of predicted dotted lines with measured solid lines 

absolute isodose lines cGy for the single intensity modulated field irradiation. 

The predicted dose distribution corresponding to the nine 

intensity-modulated fields as found by the forward dose calculation 

step outlined in Sec. II A is shown in Fig. 8a. 

Isodoses, interpolated from dose points on a2mmgrid spacing, 

are shown after the distribution has been normalized to 

the maximum dose in the plane. As the distribution was only 

computed for the region inside the circular outer OAR of 

Fig. 2, the very low isodoses follow this OAR at the edges. 

In reality the isodoses will not assume this circularity at the 

edge. At delivery, Fig. 5 shows that a total of almost 200 

monitor units were delivered to the perspex phantom via the 

62 component vane configurations. This arrangement led to a 

maximum dose in the PTV as predicted by the planning software 

of 0.82 Gy Appendix B. On completion of the delivery, 

the film was developed and the delivered dose distribution 

was scanned Fig. 8b via an optical densitometer 

Wellhöffer WP102. The scan performed was a 2D net scan 

with each line of the scan containing 16 values per mm along 

its length, and with line spacing of 0.5 cm. The 2D netscan 

lines were performed in both the x and y directions. Conversion 

from optical density values to dose values was achieved 

using a calibration curve obtained from irradiations of other 

verification films in the same pack. 

The predicted and measured dose distributions Figs. 8a 

and 8b were compared by photocopying the predicted distribution 

onto an OHP transparency so that it could be overlaid 

on top of the measured distribution. The general agreement 

between the two dose distributions is very good. The 

90% isodose line of the delivered distribution clearly shows 

two concavities where the OARs are adjacent to the PTV. 

FIG. 8. Isodoses of the computer-calculated and film-measured dose distributions. 

Isodose lines on both plots are starting from outter lines and moving 

inwards 30%, 40%, 50%, 60%, 70%, 75%, 80%, 85%, 90%, 95%, 

100% of the maximum dose. A scale-drawing of the PTV and OAR’s has 

been superimposed onto the isodose lines. a The predicted dose distribution 

as found by the forward dose calculation which takes into account the 

penumbral characteristics of the delivery technique. b The delivered dose 

distribution as measured by verification film. 

Lower isodose lines also clearly show that the dose in the 

OARs has been kept relatively very low compared to the 

dose in the PTV. All the main features of the predicted dose 

distribution are seen in the delivered except for the artificial 

circularity of low-dose isodoses at the edges. The 90% isodoses 

are consistently in spatial agreement to within 3 mm 

except for in the lower right hand region where the predicted 

isodose makes a small kink into the body of the PTV. At the 

50% isodose level consistent spatial agreement is again 

found to within 3 mm, the largest deviation being about 5 

mm. Although the general shape of the distributions agree 

well, the delivered distribution is asymmetrically hotter on 



FIG. 9. Plot of the relative dose-per-MU arbitrary units against the number 

of MU delivered for Gantry settings of 0° and 90°. 

the left hand side. The reason for this asymmetry is probably 

due to any combination of a small air spaces in the film, b 

misalignment of the film in the narrow 2 cm treatment plane, 

c gantry sag, d the asymmetric interference of a metal 

tubular bar on the underside of the couch that will have 

slightly attenuated the two most posterior fields, and e nonlinearity 

of dose per MU for small MU see below. 

Due to the limitations of using film to measure absolute 

dosimetry, this study was primarily concerned with investigating 

the relative dosimetric agreement between the planned 

and delivered distributions. Errors in excess of 10% in the 

absolute measurement may be introduced by any of (a – e) 

above together with uncertainty in the film calibration. 

Against this background of uncertainty we measured a maximum 

dose on the film of (0.97 0.08) Gy compared to the 

predicted (0.85 0.04) Gy the uncertainty quoted here is 

that of the measurement of the ROF for a single bixel—see 

Eq. A9. One reason for this underestimate in the predicted 

dose can be traced to the nonlinearity of dose per MU for 

small MU deliveries Fig. 9. Figure 9 shows that in the MU 

region of the majority of the component deliveries in the 

plan out of 62 components, 49 were 6 MU and 38 were 

3MUthe dose per MU was 4%–7% higher than that used 

in the dose calculation algorithm. 

IV. DISCUSSION AND CONCLUSIONS 

The first part of this study was a software simulation of an 

intensity-modulated radiotherapy treatment. An inverse optimization 

algorithm was used to compute intensity-modulated 

profiles for the nine fields of the plan. This algorithm makes 

several unrealistic assumptions about the form of the elementary 

fluence profile through bixels during delivery. 

From the standpoint of comparing the predicted versus the 

delivered dose, however, these assumptions were made irrelevant 

by the final one-step forward dose calculation performed 

using the optimized intensity-modulated profiles. 

This forward dose calculation step took into account the penumbral 

characteristics of the MIMiC delivery system, attached 

to a PHILIPS SL75/5 accelerator, by decomposing 

the intensity profiles into the delivery components. Each 

component was assigned the appropriate penumbral functions 

thereby ensuring that the calculated dose distribution 

should predict as closely as possible the delivered distribution. 

The second part of the study concerned the delivery of 

the intensity-modulated plan, its measurement, and the comparison 

of the delivered dose with the predicted. Both the 

inverse planning and the forward dose calculation delivery 

model includes ‘‘out-of-line-of-sight’’ scatter and therefore 

our study does not suffer from the problems highlighted by 

Mohan et al. 15 

The major uncertainties in the simulation are: a How 

well the inverse-planning optimization algorithm performed 

i.e., how optimal are the intensity-modulated profiles predicted 

for the nine fields, and b how well the forward dose 

calculation has modeled the actual delivered dose. In this 

paper we are not concerned so much with a as we take as 

our starting assumption that the optimized intensitymodulated 

profiles found by the inverse-planning algorithm 

are near optimal. Note we use ‘‘optimal’’ here to mean that 

set of intensity profiles that produce a dose distribution that 

most closely matches the prescription. This assumption is 

not entirely valid because the stretched-fit elementary fluence 

profile assumed at the planning stage is a fit of the measured 

profile to a function obeying the superposition principle. This 

means that although the fluence profile across adjacent open 

bixels of the same intensity is accurately modeled, for adjacent 

bixels with nonidentical intensities the penumbral effects 

are only approximated. Intuitively one would expect 

this effect to be small, a conclusion which is supported by 

the fact that the planned dose distribution Fig. 8a appears 

so good in terms of homogeneity of dose to the PTV while 

sparing the OARs. 

Uncertainties and omissions in the forward dose calculation 

compared with the real delivery situation are i mechanical 

factors such as gantry and collimator sag, ii the 

uncertainty on the ROF parameter for a single 1 cm 2 bixel 

Eq. A9, iii the nonlinearity of dose per monitor unit for 

small monitor unit doses, and iv errors in positional setup 

of the phantom. At this stage we are not able to comment 

further on the implications or magnitudes of these effects. In 

future work we intend to expand the planning software so 

that it can investigate and predict their magnitude. 

This paper is more concerned with b above i.e., establishing 

how well the forward dose calculation has modeled 

the actual delivered dose. We are also interested in investigating 

the feasibility of the delivery method, and establishing 

the level of dose conformation achievable with the MIMiC in 

static tomotherapy. The close agreement between the predicted 

and measured absolute isodose lines in the single field 

experiment of Fig. 7 indicate the high accuracy of our CD 

mode forward dose calculation. 11 The close correspondence 

between the predicted and measured dose distribution, 

shown in Figs. 8a and 8b, demonstrate the potential of the 

MIMiC delivery system. These figures indicate the level of 

dose conformation that is achievable in practice and the ac- 



curacy of the dose-computation algorithm. It is pointed out 

however, that this study only concerned delivery of radiation 

toa2cmthick slice, and the dose distribution was only 

verified in the central plane where the film was placed. We 

therefore cannot comment as yet on what happens to the 

dose distribution away from the mid plane. A further important 

question-mark hangs over the significance of the 

matchline problem. The latter concerns the situation where it 

is necessary to index the couch in order to treat a PTV that 

extends outside of the 2 cm delivery slice. To investigate 

these areas a full 3D map of the delivered dose is required 

such as could be obtained by gel dosimetry. 

ACKNOWLEDGMENTS 

We are very grateful to the NOMOS corporation for the 

loan of a MIMiC and for the help given during installation 

and commissioning. In particular to Dr. Mark Carol, Dr. 

Alan Bleier, and Dr. Alexis Kania. Dr. James Bedford ICR 

gave timely help with the delivery. Thanks also to Dr. Philip 

Evans, Dr. Dave Convery, Professor W. Swindell, Dr. Alan 

Nahum, Mr. Jim Warrington, Mr. Glyn Shentall, Mr. Carl 

Rowbottom, and Ms. Vibeke Hansen, of ICR/RMNHST for 

discussions and proofreading. 

APPENDIX A: FORWARD DOSE CALCULATION BY 

COMPONENT DELIVERY 

The purpose of this Appendix is to explain in more detail 

the principle of dose computation by component delivery 

CD as introduced by Webb and Oldham 1996. 

At the inverse-planning stage of determining the set of 

IMBs, the delivery of radiation through any one bixel is considered 

independent of the state of opening or closing of 

adjoining bixels so-called ‘‘independent-vane IV model’’ 

since there is no other option. Each ith bixel delivers an 

elemental dose distribution j f p 

i to the jth dose point D j 

weighted by its weight w i and the resulting dose distribution 

is simply the superposition of all independent contributions. 

The superscript P distinguishes from the corresponding 

quantity at the delivery stage. That is, 

N 

D j C 

i1 

jf i P •w i , 

A1 

FIG. 10. A beam’s-eye-view illustration of the principle of component delivery 

CD with two banks of bixels. Consider the delivery of the radiation 

through the superior bixel whose intensity is B. This bixel is flanked inferiorly 

by a bixel whose intensity is B I and flanked in the same bank by a 

right bixel with intensity B R and a left bixel with intensity B L . When the 

four intensities are ranked as shown, the delivery requires the three separate 

components shown, each with different components of intensity arrowed 

and different wall conditions. Note ‘‘right’’ and ‘‘left’’ refer to the beam’seye-view 

where converts from dimensionless bixel weight to bixel 

MU, and C converts from MU to dose in Gy see Appendix 

B. 

However, at the treatment delivery stage, while the radiation 

through a particular bixel is being delivered, the state of 

opening and closing of adjacent bixels will change, since it is 

this change which characterizes the intensity modulation. 

Consequently the bixel will experience changes in the state 

of its collimation. For example, if all surrounding bixels are 

closed, the bixel has four lead ‘‘walls’’ surrounding it and a 

corresponding penumbra. However if one or more adjacent 

bixels opens, one or more walls disappears, changing the 

state of collimation. 

To accommodate this a forward dose calculation, subsequent 

to inverse planning, was performed in ‘‘componentdelivery 

CD mode’’. The delivery through any one bixel i 

was broken down into Q i components such that the component 

intensities w i,q sum to the total bixel intensity w i , but 

the collimation, fixed for each component, changes between 

components. Thus the delivered dose distribution becomes 

N 

D j C 

i1 

Q i 

q1 

jf D i,q •w i,q , 

A2 

where the quantity j f D 

i,q is the dimensionless fractional contribution 

from the qth component of the ith bixel to the jth 

dose point and the superscript D distinguishes this from the 

corresponding quantity j f p i at the planning stage. The component 

‘‘wall conditions’’ or penumbrae are embodied in 

jf D i,q . This principle is illustrated in Fig. 10. 

It should be noted that for any particular IMB, quantised 

into L discrete intensity levels, there are L! 2 entirely equivalent 

possible decompositions of the profile Yu et al. 1995. 

The one illustrated in Table I is just one convenient decomposition. 

It is not the same decomposition pattern as the forward 

dose calculation in CD mode described above, which 

was carried out bixel-by-bixel. Provided the accelerator output 

is linear with increasing MU, the two are equivalent. 

However nonlinearities in accelerator output remove this 



equivalence and are a possible source of comparative error. 

The code to make CD calculations is not set up to model this 

problem. 

APPENDIX B: CALIBRATION AND 

DISCRETIZATION OF IMB SETS 

Inverse planning creates the bixel intensities or weights 

in dimensionless units but with meaningful relative values 

such that, when the radiation through the bixels is delivered, 

the dimensionless dose distribution created matches the dimensionless 

prescription as closely as possible. 

The purpose of this Appendix is to explain how the bixel 

intensities in the IMB sets were determined in units of MU. 

Let there be N bixels, labeled by i with w i the weight of 

the ith bixel determined at the inverse-planning stage and 

M i the corresponding value in MU. Let the corresponding 

component weights be w i,q and M i,q see Appendix A 

where, with the ith bixel delivered in Q i components 

w i 

q1 

M i 

q1 

Q i 

w i,q , B1 

Q i 

M i,q . B2 

Let the dose in Gy at the jth dosepoint be D j and the maximum 

dose in the plan be (D j ) max . Let j f D 

i,q be the dimensionless 

fractional contribution from the qth component of the 

ith bixel to the jth dosepoint. The superscript D distinguishes 

this from the corresponding quantity j f P i at the planning 

stage—see Appendix A. Let convert from dimensionless 

bixel weight to bixel MU. Let C convert from MU 

in beam space to dose in Gy in dose space C was determined 

by definition and experiment—see later. 

Then for the ith bixel, the dose to the jth dose point is 

Q i 

D j C jf D i,q •M i,q . 

B3 

q1 

The total dose to the jth dose point from all N bixels is 

N 

D j C 

i1 

By definition 

Q i 

q1 

M i,q w i,q . 

jf D i,q •M i,q . 

So, combining Eqs. B4 and B5 

N 

D j C 

i1 

Q i 

q1 

jf D i,q •w i,q . 

The maximum dose in the plan is 

D j max C 

N 

i1 

 

Q i 

q1 

B4 

B5 

B6 

jf D i,q •w i,q . B7 

max 

Inverting Eq. B7 gives the conversion factor 

N 

Q i 

D j max /C 

i1 

q1 jf D i,q •w i,q . B8 

max 

Radiotherapy machines which deliver simple open fields are 

calibrated and setup so that, by definition, 100 MUs gives a 

dose of 1 Gy to the maximum of the depth-dose curve for a 

field of size (10 10) cm 2 where, C 0.01 Gy MU 1 . For 

the MIMiC, however, we measured the relative output factor 

ROF, determining that, for a single open bixel, 100 MU 

delivered a dose of 0.822 Gy, whereby, from Eq. B3 with 

Q i 1 and j f D 

i,q 1, C 8.22 10 3 Gy MU 1 . In inverse 

planning the dose distribution was normalized so 

N 

 

i1 

 

Q i 

q1 

jf D i,q •w i,q 1.0. 

max 

It was decided that, in order to arrange that the dose distribution 

fell on the linear part of the film response curve 

(D j ) max was set to 0.822 Gy in the central plane of one leaf 

bank, whereby from Eq. B8, 100. 

Thus the weights w i and their components w i,q determined 

by inverse planning were multiplied by 100 Eq. B5 

to arrive at the bixel intensities M i and their components 

M i,q in MU. The IMB sets were then discretized so each 

bixel intensity was at the nearest integer number of MUs 

since the accelerator can only deliver integer numbers of 

MUs. In passing we may note that, from Eqs. B6 and B7 

isodoses in the plan are defined such that the pth isodose, 

expressed as a fraction, is 

D j 

p 

D j max 

N 

 

i1 

Q i 

q1 

N 

jf D i,q •w i,q 

i1 

 

Q i 

q1 

jf D i,q •w i,q , B9 

max 

and an alternative, but equivalent, definition of normalizing 

to the dose (D j ) max at the isocenter, is 

N 

D j iso /C 

i1 

Q i 

q1 

jf D i,q •w i,q . B10 

iso 

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