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British Journal of Radiology (2003) 76, 678-689
© 2003 British Institute of Radiology
doi: 10.1259/bjr/65676879

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Review article

The physical basis of IMRT and inverse planning

S Webb

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


    Abstract
 Top
 Abstract
 Background
 What is IMRT?
 History of IMRT
 Physical basis of IMRT:...
 Inverse planning for IMRT
 The future of planning...
 References
 
Intensity-modulated radiation therapy (IMRT) can sculpt the high-dose volume around the site of disease with hitherto unachievable precision. Conformal avoidance of normal tissues goes hand in hand with this. Inhomogeneous dose painting is possible. The technique has become a clinical reality and is likely to be the dominant approach this decade for improving the clinical practice of photon therapy. This Series will explore all aspects of the "IMRT chain". Only 15 years ago just a handful of physicists were working on this subject. IMRT has developed so rapidly that its recent past is also its ancient history. This article will review the history of IMRT with just a glance at precursors. The physical basis of IMRT is then described including an attempt to introduce the concepts of convex and concave dose distributions, ill-conditioning, inverse-problem degeneracy, cost functions and complex solutions all with a minimum of technical jargon or mathematics. The many techniques for inverse planning are described and the review concludes with a look forward to the future of image-guided IMRT (IG-IMRT).


    Background
 Top
 Abstract
 Background
 What is IMRT?
 History of IMRT
 Physical basis of IMRT:...
 Inverse planning for IMRT
 The future of planning...
 References
 
The X-ray was discovered on 8 November 1895 and radiotherapy began the year afterwards. Possibly there was more hope than skill at those times but by the year 1903 the first textbook of radiotherapy had been printed by Freund, arguably the "father of radiotherapy" [1]. The first half of the 20th century saw a systematic development of the practice of radiotherapy assisted by physicists, technologists, radiographers and electrotechnical companies [27].

By the mid-20th century the treatment energy had been raised from keV to MeV, the linac was beginning to be a standard delivery tool, imaging was harnessed to radiotherapy through clever classical tomography, treatment planning by hand was possible even for rotational and non-coplanar fields, blocks, wedges and compensators were in use and radiation dosimetry was well developed. Systems of technical education were well established. Most importantly, radiophysicists and doctors understood the need to "concentrate" the radiation in the tumour whilst sparing normal structures (see for example Kieffer's "radiation concentrating device" [8]). Today we call this conformal radiotherapy (CFRT) and many imagine it to be a new concept. However, the concept was understood 50 and probably 100 years ago. Sadly, in those days, it was unachievable and it is today's achievability of CFRT, through synergistic advances in mathematics, therapy planning techniques, radiation technology, computing and three-dimensional (3D) imaging, that gives it its somewhat unwarranted modern flavour.

Intensity-modulated radiation therapy (IMRT) is a special form of CFRT and this paper will review its history, the reason it is needed and just one aspect of its radiophysics, the determination of modulated fluence through planning techniques. Other articles in the series will cover delivery, quality assurance, practicalities and clinical application. Conveniently bridging these topics, this paper will hint too at future developments.


    What is IMRT?
 Top
 Abstract
 Background
 What is IMRT?
 History of IMRT
 Physical basis of IMRT:...
 Inverse planning for IMRT
 The future of planning...
 References
 
IMRT is the delivery of radiation to the patient via fields that have non-uniform radiation fluence. Arguably the terminology has become incorrectly established because strictly it is fluence not intensity that is modulated [9]. The spatial modulation can be performed either: (i) by creating a spatial variation by continuously interrupting an otherwise uniform flow of X-rays via collimation and/or compensation or (ii) by creating a spatial variation by temporally modulating the fluence and varying the temporal modulation in space. In this paper CFRT means geometrically conformal radiotherapy in which the outlined shape of each beam can be controlled but the fluence is uniform across the field. IMRT adds the modulation possibility to CFRT, i.e. combines geometrical and fluence shaping (Figure 1Go). The author's three linked texts [1012] on CFRT and IMRT provide a wealth of detail and references whilst other shorter reviews cater for different needs (physics account for clinicians [13], lay-physics account [14], pure history account [15], review and possibly totally inaccurate speculation on the future [16]).



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Figure 1. Illustrating the key differences between (a) conventional radiotherapy, (b) conformal radiotherapy (CFRT) without intensity-modulation and (c) CFRT with intensity modulation (IMRT). For almost a century radiotherapy could only be delivered using rectangularly-shaped fields with additional blocks and wedges (conventional radiotherapy). With the advent of the multileaf collimator (MLC) more convenient geometric field shaping could be engineered (CFRT). The most advanced form of CFRT is now IMRT whereby not only is the field geometrically shaped but the intensity is varied bixel-by-bixel within the shaped field. This is especially useful when the target volume has a concavity in its surface and/or closely juxtaposes organs-at-risk, e.g. as shown here in the head-and-neck, where tumours may be adjacent to spine, orbits, optic nerves and parotid glands.

 

    History of IMRT
 Top
 Abstract
 Background
 What is IMRT?
 History of IMRT
 Physical basis of IMRT:...
 Inverse planning for IMRT
 The future of planning...
 References
 
Definition of history and the meteoric rise of IMRT
When one thinks of "history" one generally thinks of times long past, of ancient Egypt, of Kings and Queens of England and parodies such as the book "1066 and all that". It is a testament to the sheer speed of development of IMRT and its huge anticipated clinical worth that a subject that was virtually unheard of 15 years ago can now merit a historical review. In 1990 there was just a handful of physicists working on IMRT. In the mid-1990s the main techniques had been established but all deliveries were "one-offs", to phantoms not patients, and in university hospitals. The subject had not become interdisciplinary. By the late 1990s all the major manufacturing companies were offering embryonic IMRT planning and delivery systems and now one might observe "everyone wants IMRT". The rise of interest in IMRT has been astronomical. It has become truly interdisciplinary requiring the merged skills of doctors, radiographers, engineers, physicists and technologists. There are now thousands of papers on the subject. Indeed maybe 40% of all papers at radiotherapy and radiotherapy physics conferences have IMRT somewhere in their title. It promises a major change of approach to clinical practice. The Royal College of Radiologists Working Party on CFRT [17] highlighted that education, training, practical experience and implementation of IMRT should become non-negotiable features of the future of radiotherapy. In the USA a Collaborative Working Group has recently reported at length with similar recommendations [18].

The language of IMRT
In the early days we spoke and wrote about "beam modulation", "variable fluences", and a variety of other phrases meaning the same thing. No official body standardized the language and the acronym "IMRT" has entered our language (about 1996) without anyone being able to recall its first use. Certainly no-one has come forward to claim ownership. Ned Sternick's book [19] was the first to have the words IMRT in its title.

Pre-history of IMRT
With the glorious wisdom of hindsight some very early developments – we may call them pre-history – might be considered part of the development of IMRT. The mathematician George Birkhoff showed in 1940 that any drawing could be made up of lines of varying pencil thickness so long as negative pencils were allowed [20]. If we read "X-rays" for "pencils" and "dose distribution" for "picture" the analogy with IMRT is clear. Sadly there are no negative X-rays or uncomplicated tumour control would be 100% guaranteed. In the 1950s Shinji Takahashi was using beams shaped by a primitive multileaf collimator (MLC) with rotation to achieve CFRT [21]. The Royal Free Hospital developed the tracking cobalt unit [22] and MGH Boston tracked the jaws of a linac to shape fields [23]. Proimos [24] developed "gravity blocking", a form of binary on-off IMRT in which the target was always in the beam's-eye-view (unless part of this included a view of organs at risk). Synchronous shielding of organs at risk (ORs) was a feature.

The beginnings of modern IMRT
There is very little dissention from the view that IMRT began in 1982 with the seminal publication from Anders Brahme and colleagues in the Karolinska Institute in Stockholm [25]. This paper showed that for a very special 2D inverse problem with complete circular symmetry it was possible to generate an annulus of uniform dose around a completely blocked central circle by rotating a modulated beam profile.

This first paper on IMRT showed how to begin with a desired dose distribution and arrive at a specification of the required fluence modulation to create it. Today we refer to this process as inverse-planning. The paper showed how to obtain a uniform annulus of dose D with outer radius R surrounding a central circle of radius r0 of zero primary dose. It was shown that this would be achieved by a rotation technique in which the fluence was zero from the origin out to the inner radius r0 of the annulus and which then rose suddenly (Figure 2Go) to infinity (or in practice "large") followed by a falloff with distance x>r0 according to:Go


where µ is the X-ray linear attenuation coefficient. Chapter 2 of reference 10 develops the algebra in detail and shows further extensions from this concept. It is possible from this to generate the algebraic form of the point rotation kernel and this in turn leads on to the ideas of creating dose distributions via convolution and superposition techniques.



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Figure 2. A diagram adapted from that in paper [25] showing how a uniform annular dose distribution surrounding a circle of zero dose could be obtained by rotating an intensity-modulated beam (see text for more detail).

 
The next milestone in inverse planning in 1988 was also by Brahme [26] in which was explained the notion of creating fluence profiles from dose distributions by first inspecting the dose distribution in dosespace, then deconvolving the point-spread dose kernel from this to create the density of fluences required. The next step was to backproject the density of fluence back into fluence space to create the fluence profiles. Then, from the profiles so generated, the deliverable dose distributions could be formed by a process of projecting the fluence back into dosespace. The projection and backprojection processes both involved exponential (±) depth operations. However, projection and backprojection are not inverse processes and so the technique was only approximate. Also, the process could generate negative fluences and so required some form of truncation to zero or the addition of constant terms in fluence space. The work was a landmark because it established the idea of this kind of planning viewed as a form of inverse computed tomography, a term which actually was used for a while but later abandoned.

It is important to make a few observations on the "position" in 1988. (i) There was no equipment capable of delivering a modulated beam (other than a compensator). Computer controlled multileaf collimators were only just beginning to be available and it would be 6 more years before the "translation equations" linking modulated fluence to field patterns would be known. Even then it would be another 3 years before commercial equipment would be available. (ii) There was just this one inverse-planning technique. (iii) There were no commercial inverse-planning algorithms or equipment. (iv) The subject was generally regarded as the research interest of physicists. Doctors, radiographers, commercial technologists played no part.

The next steps in inverse planning in the late 1980s/early 1990s
At the end of the 1980s there were just a handful of physicists working on inverse planning for IMRT. At the Royal Marsden Hospital in London, Webb had developed IMRT inverse planning by simulated annealing by 1989 [27]. At Memorial Sloan Kettering Cancer Center in New York, Mageras and Mohan has made similar independent developments [28]. At DKFZ, Heidelberg Bortfeld, working with Schlegel, was developing an analytic gradient-descent technique that mimicked the inverse of computed tomography. His first paper appeared in 1990 [29]. These techniques generated 2D dose distributions from 1D modulated fields. In the early 1990s techniques were extended to generate 3D dose distributions from 2D modulated fields.

The invention of IMRT delivery techniques
The story of the development of techniques for IMRT delivery will be covered by other writers in this Series. For completion, and just listing techniques without detail, the developments were: (i) the NOMOS MIMiC (1992) [30]; (ii) the multiple-static-field MLC technique (1994) [31]; (iii) the dynamic MLC (dMLC) technique (1994) [3234]; (iv) the scanning attenuating bar technique (1995) [35]; (v) intensity-modulated arc therapy (IMAT) (1995) [36]; (vi) spiral tomotherapy (1993 (concept) –2003 (actuality)) [37]; (vii) robotic IMRT (2001) [38]. The first deliveries were "one-offs" in specialist university hospitals. Then came commercial interest leading to wider participation in development and now we have tools available so that, in principle, almost any centre with an MLC can perform IMRT. The story of the first practical implementations and how these led to commercial developments has been recounted elsewhere [1012, 15].

In the early days of IMRT (pre-circa 1997) the two processes of the generation of modulated profiles and the practical realization of modulated beams were entirely separated. The latter process serially followed the former. Then it was realized that, because of: (i) the non-point-like X-ray source; (ii) head scatter; (iii) phantom/patient scatter, leakage through and between collimator elements etc., the interpretation of inverse-planned fluence maps (through consideration of the primary radiation only) did not exactly match those planned. Two classes of developments then ensued: (i) some workers intelligently "patched up" the interpretation so the summed fluences (primary, scatter and leakage) matched the required planned total fluences [39]; (ii) others built the delivery process into the inverse planning itself. The former will be described in later papers in this series as this is a delivery concept. The latter planning concept will be described in the section on "Incorporating equipment constraints in inverse planning" of this paper.

Clinical IMRT milestones
The first clinical IMRT in the world by any technique (excluding historical attempts and those via the use of wedges, compensators etc.) was performed with the NOMOS MIMiC delivery technique at Baylor College of Medicine, Houston Texas in March 1994 [40]. The first dMLC treatments were those at Memorial Sloan Kettering Cancer Institute and Hospital starting in April 1996. The very first IMRT patient there only had the boost phase of a treatment for prostate cancer treated with IMRT but the second and subsequent patients had the full course of treatment planned and delivered as IMRT. The first five patients were treated between April and June 1996 [41]. The first IMRT of bladder cancer in the UK was at the Christie Hospital NHS Trust, Manchester in April 1999 [42]. The first patient treated for prostate cancer with IMRT in the UK was in September 2000 at the Royal Marsden NHS Trust [43].


    Physical basis of IMRT: concepts of cost, ill-conditioning and pseudodegeneracy
 Top
 Abstract
 Background
 What is IMRT?
 History of IMRT
 Physical basis of IMRT:...
 Inverse planning for IMRT
 The future of planning...
 References
 
Convex and concave dose distributions
Much of the literature of IMRT involves quite complicated mathematics. The papers from the late 1980s and early to mid-1990s were relatively easy to follow. Some recent papers seem to have become excessively mathematical in my opinion. In this section the concepts of IMRT will be presented as simply as possible. Firstly, let us try to establish the physical basis using words only.

If a number of beams are brought together around an isocentre and each beam, possibly shaped geometrically, is of uniform fluence, then the volume of intersection of such beams will be "convex", i.e. it will not contain any concavities—no dimples, no dips, no invaginations. For example, if a large number of circular fields were brought together from all directions then the volume of intersection would be a (convex) sphere. If say six square fields were directed from the six main compass point directions then the volume of intersection would be a (convex) cube. This describes conventional non-IMRT. If, conversely, the fluence were modulated across some or all of the fields the high-dose volume so created by superposition of beams can have invaginations (Figure 3Go). This is the physical basis of IMRT. Plans are designed so the high-dose volume "shrinkwraps" the planning target volume (PTV). By arranging that the ORs lie in the concavities of the high-dose volume they will receive a lower dose. Ideally they should receive no dose but this is not possible owing to the finite range of X-ray-generated electrons and the physics of the scattering interactions.



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Figure 3. Illustrating the principles of IMRT. A single (CT) slice of a patient is shown with the contours of the prostate (PTV, planning target volume), rectum (OR, organ at risk) and bladder (OR) outlined. The prostate has a concave outline and it is desired to achieve a high dose in the PTV sparing the ORs. Three intensity-modulated beams (IMBs) are shown. For simplicity they are shown as parallel beams and at arbitrary gantry orientations. It is not intended to suggest these three orientations are the most appropriate for the treatment. Each beam has a modulated fluence.

 
Concave dose distributions need modulated beams
Why does this concave high-dose volume arise? Brahme's 1988 paper gave the first clue. He argued that the fluence should be greatest where the length of intersection between a ray and the PTV was greatest. There should be an approximate proportion. Conversely, where the ray intersects significant OR en route to or after intersecting the PTV then it should have a lower fluence. This is not strictly mathematically correct but it is a good first approximation. In fact several recent papers (e.g. reference 44) have used this concept to prioritise "good" lines of attack into the PTV and to address the question of optimum beam orientation. Figure 3Go provides a simple 2D description. Several 1D beams are modulated and where they come together the contour of the 2D high dose is concave. Some readers may be happy just to accept that "this happens". For others we may note that the least-squares inverse-planning technique, at its most fundamental, can be regarded as a pure "cooking recipe" in which elements of fluence weight are offered to beam elements (bixels) and accepted if they improve the dose distribution. Without any formal mathematical attempt to modulate the beams anyone who codes this will find the beams "settle" to the appropriate modulated fluence pattern of the type shown. Of course there are lots of unaddressed details here, such as the effect of number and orientation of beams, fluence stratification and the "cost function" which drives the optimization, but to understand the fundamentals we can ignore these for now.

Simple explanations of ill-conditioning and degeneracy
Now to introduce the barest minimum of mathematics. We may write that: Go


where D is the 3D dose distribution, b is the vector of individual beamweights and A is the matrix linking each dose-space element to each beam-space element. The elemental entries into A depend on the physics of photon–tissue interaction and can be pre-calculated using either simple equations, or convolution and superposition algorithms or at best full Monte Carlo calculations. The "inverse problem" is that we can specify the required dose prescription D, we know A and we require to calculate b. At first sight it might seem logical to perform a simple inversion: Go


However, this is generally not done because: (i) it would generate some negative beamweights; (ii) it would require the construction, storage and inversion of a too large matrix A; (iii) it would take too long and; (iv) it would provide no method to control the behaviour of the beamweight vector b.

The matrix A is "ill-conditioned". This means that there are a very large number of different vectors b that when operated on by matrix A give the 3D dose distribution D. This is an inherent property of the radiation physics inverse problem. The property has permitted workers to develop numerous mathematical solutions to the problem, all of which have different interesting features. For example it is this ill-conditioning that has been exploited to control the behaviour of the fluence vector b.

Let us try to explain this with firstly a simple 2D planning problem. Suppose the PTV is a circle of radius rin about the isocentre 0, the origin of coordinates. Suppose an organ at risk is an annulus around this of outer radius rout and inner radius rin. Suppose the patient contour is also a circle of radius Rcont>rout, all three circles being concentric. Suppose the planner is allowed to use two beams from any orientation. Then it is self evident that the circular symmetry of the problem is such that any pair of directly opposing beams of width 2rin will assume the same beam profile and will generate exactly the same dose to the PTV and the OR, albeit that the spatial position of the PTV and OR dose will change with beam orientation. If the "cost" of the treatment plan is assessed in terms of the overall dose to PTV and OR it will be the same whatever the orientations of the beam. Hence, if the inverse-planning problem is not constrained by any other specifications it cannot decide between any of the infinity of plans with opposing beams. This is an example of degeneracy.

The concept of cost
The above problem is of course highly artificial. However, the same behaviour arises with real clinical problems in both 2D and 3D. Let us introduce the concept of "cost". Suppose the required prescribed dose distribution is Dp(x,y,z). This begs the question of how the prescription is arrived at, but for now we will assume that a medical doctor can specify this based on clinical experience. At first sight it might be thought that Dp should be simply a high uniform dose in the PTV and zero in the OR(s). However, experience has shown that demanding too low a dose in the ORs can compromise the goal of obtaining the required PTV dose [45]. Hence it is often considered acceptable to have some finite dose to ORs (lower of course than in PTV). It is at this stage that "dose-volume" constraints are sometimes imposed of the type "let no more than q% of the volume of OR receive more than dose ". However, to keep the discussion simple here, let us revert to a simple specification of prescription dose to each point. Now imagine that the plan is constructed iteratively. At some stage of iteration the actual delivered dose becomes D(x,y,z). Then a simple quadratic cost function would be: Go


where I(x,y,z) is the importance of each voxel specified by (x,y,z), allowing the planner to bias the plan selectively towards either the required dose in the PTV or that in the OR or something in between. For example, a plan with a much higher importance attached to PTV voxels than to OR voxels is called a tumour-conformal plan. Conversely, if the voxels in the OR are given a higher importance than those in the PTV then this is called conformal avoidance.

The iterative planning process might start off with empty beams. Given that D(x,y,z) will then be zero, the starting cost={Sigma}I(x,y,z)[Dp(x,y,z)]2 will be at its highest. Now consider offering grains of beamweight to each beam element in some random order. After each addition the cost is recalculated. Changes to beam elements are accepted if they lead to a lower cost and vice versa. As D(x,y,z) gradually builds up the cost will decrease. Simultaneously the modulation of the beams will develop. Generally such schemes offer very small grains (small compared with the finally accepted maximum bixel value) and many thousands of iterations in which bixels are randomly selected from all of the constituent beams. The goal of the optimization is to reach the global minimum of the cost function.

Cost-function space can be plotted on a graph. For the cost function above this would be a simple monotonic curve with no local minima. The ill-conditioning is represented by the very wide plateau that the curve would reach at low cost. Whilst there is some single global minimum this may be only a very small depression in an otherwise almost flat-bottomed plateau. The beam configurations corresponding to each value of the cost in this plateau will all be different. Some may be much more physically desirable than others. It is this ill-conditioning that is sometimes exploited to generate maximally smooth beams [46]. We will return to this point. The inverse problem is not strictly "degenerate" because there is only one true solution with global minimum cost but the ill-conditioning makes the problem pseudodegenerate (Figure 4Go).



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Figure 4. (a) The left graph shows a plot of a cost function for a problem with a well defined global minimum cost as well as several local minima. Cost is plotted on the vertical axis and position on the horizontal axis labels the particular stage of some iterative planning cycle. For example, the global minimum corresponds to having achieved the beams which deliver dose best matching the prescribed constraints. The very left hand position might represent the start of iteration when beams have not yet been properly formed. (b) The right graph conversely shows a cost function more typical of radiotherapy inverse planning problems. There is a wide plateau (basin) of beam arrangements all of which correspond to dose distributions that are much the same and best satisfy the planning constraints. There may be a small dip (global minimum) for the absolute best but continuing the iteration to find this might be futile when any position in the plateau would be acceptable.

 
The form of the curve will depend on the starting conditions and on the random number sequence used to address the bixels. If we arrange all possible cost-function curves in space they construct a "basin" with a very flat bottom but with a single small depression of cost minimum. Now imagine such a basin in reality and roll a "sticky marble" from its rim towards the bottom. Where it stops will depend on where it starts. Gravity will take it to the bottom but its viscosity will stop it rolling to the global minimum. Given that the cost function is almost flat and nearly minimum over so wide a bottom this will not matter. But it does explain degeneracy rather nicely. It explains how controlling "where the solution will end up" is dependent on the starting conditions and the trajectory. The trajectory of the inverse-planning problem can be controlled to physical advantage.

An analogy for ill-conditioning
The concept of ill-conditioning and degeneracy are well known to theoretical physicists and mathematicians but maybe less so to radiotherapists, radiologists and radiographers. Hopefully the above wordy discussion has demystified the process. As a final analogy let me tell this little story. Suppose there is a sell-out football match at Old Trafford. Each seat has a ticket allocated to it with seat row, number etc. Ahead of time these tickets are scattered throughout the country, maybe even abroad with their purchasers. On the day each purchaser travels from their home to the appointed seat and the purchasers all sit down in a pre-determined location. The stadium is full. Now think of the purchasers as bixels. Consider their journeys as beam orientations. Consider the full stadium as a dose distribution. If we know where each person lives (the bixels) and we know their ticket number specifying the journey from home to seat (irradiation), then the pattern of people in the stadium (dose distribution) can be precisely determined. This is forward-planning. But suppose we do not know the home places nor the journey. Suppose all we know is the pattern in the stadium. Suppose we want to "invert" this pattern to find out exactly where everyone has come from (assuming no-one can speak and tell us!). Then this is an ill-conditioned problem. We know this because if the ticket distribution had been geographically quite different, the tickets and their purchasers the night before would have been in totally different places. But on match day the tickets and their purchasers would all be in their appointed places corresponding to the seats. So there is a huge number of configurations of ticket sales (bixels and beam directions) that correspond to the same final location of tickets (dose distribution). The analogy goes further. As far as the players and clubowners are concerned the bixel values and trajectories are irrelevant. There is a big crowd and that is all that matters. Every seat is taken (perfect dose distribution). The cost function is minimized by the lack of empty seats. But maybe some journey patterns are simpler than others. If all the ticket holders live in Manchester then getting to the ground is easy. If they live all over the UK it is less easy. The radiotherapy analogy is that we are interested not just in the final dose distribution (the crowd) but in how this came to be achieved (the beam trajectories) using the appropriate mix of beams and collimation.


    Inverse planning for IMRT
 Top
 Abstract
 Background
 What is IMRT?
 History of IMRT
 Physical basis of IMRT:...
 Inverse planning for IMRT
 The future of planning...
 References
 
Planning constraints, physics constraints, human constraints
Many of the elements of inverse planning have already been introduced in this paper. The fundamental concept is that, instead of the planner trying a variety of configurations of beams, wedges and beamweights until a suitable match is found to the dose prescription (called "forward planning"), the reverse is attempted. The planner decides on the required dose prescription with clinical input. Most modern inverse-planning computer systems allow a specification of dose-volume constraints and/or dose limits (Figure 5Go). For example, an upper and lower bound to the dose to PTV can be specified. For ORs the specification is of the kind mentioned earlier that "no more than q% of the OR may receive dose or more". The specification "drives" the optimization and the computer code will "do the best it can" to satisfy the constraints. In this sense there is considerable automation of the planning process. However, the burden is returned to the doctor and planner to use their experience to decide appropriate constraints. Clearly the physics of photon-tissue interactions limits what can be achieved. Sadly no amount of specification of zero dose to ORs and high uniform dose to PTV will actually achieve this. The solution is to arrive at sensible compromises between what can be expected to be achieved and what the doctor wants. In this sense inverse planning has most certainly not made the mind of the planner redundant. Quite the contrary, there is now probably more work, not less, for the planner and doctor to perform. The experience at this centre is that, in introducing clinical IMRT, much experimentation has to be made and the most appropriate inverse plan chosen. As the number of patients with a particular tumour type accrues some "settling" on "class solutions" (similar beam arrangements and constraints for similar problems) is beginning to take place. If the 1980s quest for planning automation ever transcribed to the goal of somehow "doing away with the planner and doctor", nothing now could be further from the truth. In fact the chronic UK and international shortage of oncologists, physicists and radiographers is a major impediment to widespread rolling out of clinical IMRT (but this is a person power planning and political issue and no more will be said here [17]).



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Figure 5. Illustrating the concept of a dose–volume constraint in inverse planning. The figure shows an integral (sometimes called cumulative) dose–volume histogram for an organ at risk (OR). The clinician specified the starred points as dose–volume constraints and the algorithm "did its best" to meet them (not entirely succeeding as not all the stars are to the right of the curve). For example, 0.85 of the OR volume receives 20 Gy or more; 0.3 of the volume receives 27 Gy or more and so on.

 
What do we really mean by optimization?
Something that is optimum cannot be bettered by definition. Therefore the optimum plan is the best that could ever be obtained for treating a particular patient with a particular external shape, location of disease and arrangement of internal organs. I propose the view that this optimum plan is unachievable and that in practice this does not matter. To arrive at the optimum plan one would have to investigate the use of: (i) all types of irradiation (protons, carbon ions, photons...); (ii) of all energies (continuous not just those we have available); (iii) all possible numbers of beams from 1 to infinity; (iv) all possible ranges of fluence levels; (v) all possible beam geometry shapes; (vi) all possible fractionation schemes...and so on. It is totally apparent that as "optimizers" we cannot and do not do this. We are constrained by: (i) the beams available on our machines; (ii) the need to keep the number of beams deliverable within some specified delivery timeslot; (iii) the delivery mechanics available to us (which links in to the fluence level issue); (iv) the collimation available from the machine; (v) the need to treat in the daytime not at night; (vi) the time available for planning...and so on. In practice the planner and doctor have already considerably reduced the optimization search space by making a priori choices, e.g. the choice to work with photons at 6 MV, with say 5 beams from certain specified directions, with a preset number of fluence levels, with the MLC of a particular manufacturer, with a specified fractionation scheme. Then all that is left to optimize is the beam profile subject to the constraints set. When we write of "optimization" we mean most certainly "constrained optimization" [47]. Most studies of "optimization" have concentrated on finding the best beam directions and best modulation.

Simulated annealing as an example of iterative inverse planning; biological cost
"The concept of cost" section introduced the notion of a quadratic cost function based on prescribed and achievable dose distributions. This is an example of a physical cost function. Furthermore, known to have just a single global minimum, this cost function is minimized by the inverse-planning technique described whereby grains of beamweight offered to bixels at random are accepted if they lead to a lower cost and vice versa. Of course both positive and negative grains must be added in order to tune the modulation. The modulation must be constrained positive and this is easily achieved by such an iterative technique.

Dose is, however, just a surrogate for biological response. From any given 3D dose distribution D the tumour control probability (TCP) and normal tissue complication probability (NTCP) can be calculated via mechanistic models fed with biological data (see e.g. reviews in [10, 11]; note however that many do not accept the validity of the models and the data on which they stand are unreliable). Hence cost could alternatively be specified in terms of achieved TCP and NTCP. The inverse planning problem could be specified as one of: (i) maximize TCP subject to specified maximum NTCP; (ii) minimize NTCP subject to specified minimum TCP; (iii) maximize uncomplicated tumour control P+=TCP(1-NTCP). Hybrid specifications could be made in terms of cost depending on both physical and biological parameters. There is considerable debate about whether such cost functions possess local minima, and whether they are sufficiently close to the global minimum that finding the global minimum becomes inconsequential [48]. However, techniques exist to cope with local minima and, with modern fast computers, they can be invoked even in the absence of conclusive proof of the existence of local minima. The simulated annealing technique guarantees achieving the local minimum [27]. This invokes a small but important modification of the iterative least squares method previously described. Instead of rejecting all bixel changes which lead to an increase {Delta}V in the cost function, these are conversely accepted with a probability exp(-{Delta}V/kT) where k is the Boltzmann constant and T is temperature. At the onset of iteration T is set high so that a large number of "wrong way" or "uphill" changes are accepted and then as the iteration progresses the temperature T is gradually reduced until at the end of the iteration only downhill changes are being accepted. The temperature must be lowered slower than the reciprocal of the logarithm of the iteration number [49]. There are other faster simulated annealing methods in which grains are selected from a Cauchy distribution whose width depends on temperature and by allowing large grain changes "tunnelling" through cost-function barriers can replace uphill moves [50]. This technique was the basis of the first commercial inverse-planning system known as PEACOCKPLAN, now known as CORVUS from the NOMOS Corporation [19]. Figure 6Go shows an analogy I have published before in which the downhill search is analogous to a skier attempting to reach the foot (global minimum) of a slope (cost function) but impeded by snowbumps (local minima) further up the piste.



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Figure 6. An analogy for simulated annealing. The ski slope is the cost function. It has local minima (snowbumps) and a global minimum to be reached (finish flag). The skier has to occasionally ski uphill (or "tunnel through the snowbumps!") in order to avoid becoming trapped in a local minimum. The technique gets its name from the slow cooling of a solution in which the crystalline state is the global minimum and local minima are amorphous states. The grains referred to are the elemental changes in beamweight.

 
Gradient descent techniques
If the cost function is guaranteed to have no local minima then gradient descent techniques may be used instead. These require that the cost function can be written mathematically and that its derivatives exist. Then techniques similar to those learnt at school to find minima by Newton's method are used. The advantage of these techniques are that they can be very fast. However, they can generate negative beamweights and so require to have some a posteriori adjustment such as setting negative beamweights to zero or the addition of constant fluences, both of which effectively disturb the optimization.

Genetic algorithms
Genetic algorithms, as their name suggests, are based on the Darwinian theory of evolution. Categories of solution are proposed and via an iterative process, those solutions which are "fittest" survive through to the end of the planning process. This requires a definition of fitness (which is a kind of cost function) and a mechanism for evolution. Bixel intensity values can be considered like chromosomes and these are exchanged and/or mutated during the evolution. It has been shown that these techniques can yield conformal plans [51, 52]. They have not been used in clinical commercial planning systems.

Incorporating equipment constraints in inverse planning
As mentioned at the end of the "History of IMRT" section, the traditional approach to IMRT has been to first construct the modulated fluence patterns through some form of inverse planning and then to "interpret" these into the required patterns of collimation. Unfortunately, because the metal collimators deliver unwanted leakage, scatter and transmission, a component of radiation is added to the truly wanted modulation. In general this is a perturbation, i.e. is small, but it needs to be understood and accommodated. One way is to adjust the delivery patterns so that the sum of all the radiation components becomes the required planned modulation [39]. A better way is to build the delivery technique into the inverse-planning process itself. Of course this requires the a priori specification of which IMRT delivery technique will be used and needs data on the leakage, transmission and scatter, all of which can be measured. For iterative processes these contributions are then factored into the planning process as the iterations proceed, there then being no need to do any a posteriori "patching up" [5357]. This approach also has the simultaneous advantage that any equipment delivery limitations can also be incorporated into the planning process itself, e.g. some MLCs do not allow interdigitation or have minimum leaf-gap constraints. This technique is particularly appropriate when it is also required to limit the number of contributing components to delivering a modulated field. Essentially the modulation that is created via such constrained inverse planning will be the best possible with that limited number.

Segmented field optimization
Logically related to the above techniques are those which specify the field segments at the outset of the planning process and then optimize their weights. The advantage lies in being able to constrain the number, shape, size and disposition of these field components and to ensure a priori that they are deliverable. IMRT via segmented fields has the additional advantage that such fields can be verified by conventional techniques since the methodology is not too far removed from that of delivering conventional fields. The only difference is that several field shapes are delivered from each gantry orientation.

Two applications of note are to: (i) IMRT of breast cancer and; (ii) IMRT of head-and-neck cancer. The former has been pioneered at the William Beaumont Hospital, Royal Oak, MI. First, a non-modulated plan is created for tangential fields. Then field (sub) shapes are constructed to match the projections of specific isodose lines. Then the weights of these subfields are optimized [58]. Segmented breast-tangential fields have been designed by a quite different technique at the Royal Marsden Hospital through the use of electronic portal imaging [59]. The technique has formed the basis of one of the first randomized phase 3 trials of IMRT [60]. The segmented-field technique pioneered in Ghent, Belgium for treating head and neck cancer, creates apertures to match projections of structures (anatomy based segmentation), then weights and reorders them to practical delivery (SOWAT, Segment Outline and Weight Adapting Tool) [61].

Forward planning for IMRT
Whilst inverse planning has made a major contribution to the development of IMRT, it should not be regarded as synonymous with IMRT. The general argument is that, for full-bixel-modulated IMRT, inverse planning is essential because there are just too many unknowns to determine by any other means. Inverse planning grew up through the need to start with the required dose prescription and to algorithmically arrive at the beam modulations that best fitted it. However, forward planning is possible for some forms of IMRT, specifically those in which the number of field subcomponents is relatively limited (see section on "Segmented field optimization"). A number of groups are working on this basis; some call their techniques Direct Aperture Optimization [62].

Clearly forward-planned IMRT is only appropriate for those techniques employing a few beams each with just a few segments. Potential advantages include: (i) the segments can become the starting building blocks so removing the task of a posteriori segmenting a modulated beam; (ii) the full delivery physics (headscatter, beam penumbra, leaf transmission etc.) can be built into the planning process because of (i); (iii) in principle the fieldshapes can be adjusted during the planning; (iv) the technique is "more like" conventional radiotherapy planning; (v) the beams become fairly smooth by definition; (vi) the beams have little resulting noise; (vii) planning may be quicker.

Smooth beams
Beam modulation by definition means the creation and use of beams which are not smooth as uniform beams are. Hence the subtitle "smooth beams" may seem to be a diversion in the wrong direction for the efficacy of IMRT. However, it is well known that the more modulated the beam becomes, the more complex becomes the delivery. Also for the dMLC delivery technique the number of monitor units required directly relates to the sum of the rising fluence changes as the leaves move across the aperture. Hence it is desirable to achieve modulation that is no more than is required for beam conformality and with as little unwanted noise as possible. This goal has occupied many groups and there are several ways to achieve it. The most promising construct a cost function that combines a representation of the desired conformality in dose-space with a representation of the desired smoothness in beam-space. By combining these planning constraints with two controlling importance factors, the relative importance of the two can be adjusted. Clearly if the beam-space is constrained too smooth, conformality will worsen. Conversely, if the beam-space is not constrained at all then the conformality will be the best possible subject to all the other factors pertaining. There is a rich variety of approaches [6368]. A further advantage of smoother intensity modulated beams is that the IMRT treatment should be less susceptible to patient movement and ongoing research is investigating this.

Beam orientation optimization for IMRT
The problem of determining the optimum bixel intensities and the optimum beam orientations are coupled. In general it is not possible to generate and inspect all the plans for all coupled options and so hybrid techniques have been developed by which simulated annealing techniques are used to select beam orientations whereas faster gradient-descent methods are used to determine the bixel values at each orientation. In some algorithms a measure of "goodness of beam direction" is introduced a priori to select a subset of directions over which to inspect the bixel-optimization performance. The use of such beam's-eye-view volumetrics is becoming more common [69].

Inverse planning with commercial planning systems
Just a very few years ago there was only one commercial treatment-planning system (TPS) available with an inverse-planning code for IMRT, originally known as PEACOCKPLAN and now CORVUS from the NOMOS Corporation. It uses a simulated-annealing algorithm and was originally designed to plan for delivery with the MIMiC technique. Later it was extended to plan for MLC-based IMRT deliveries. Now virtually all the vendors of TPSs offer software to plan for IMRT and to do justice to these would require a separate review. The KONRAD system uses a gradient-descent technique. The HELAX system uses a technique of gradient descent directly optimizing the weights of field segments. By commercial necessity the details of exactly what each planning system does are sometimes far less transparent than techniques developed in-house and published in peer-reviewed literature. Hence the papers based on the use of commercial TPSs tend to be of the type "we did this and this happened". Experience here (with several different systems) has been that when unwanted planning outcomes arise it is sometimes hard to know what to do to overcome them. Even when a TPS is nominally based on some published algorithm (several are based on well known PhD theses) they have often mutated considerably by the time they reach the customer. They are also subject to continual upgrades so performance must be benchmarked to version status. Some of these systems allow Monte Carlo planning of IMRT to properly account for tissue inhomogeneities. This is a big research subject in itself [70].


    The future of planning for IMRT—image-guided IMRT (IG-IM-RT)
 Top
 Abstract
 Background
 What is IMRT?
 History of IMRT
 Physical basis of IMRT:...
 Inverse planning for IMRT
 The future of planning...
 References
 
Predicting the future is a dangerous game [16]. Some words from two invited speakers at UKRO2 (Bath: April 2003) stick in my mind: "They only ask you to predict the future when you are truly old and possibly won't be around to see if you are right". I have heard some young people ask if there is any serious IMRT science left to do. I have had this question too from my hospital and university chiefs and I have implicitly heard it from some grant review bodies. All my training as a physicist teaches me that just as one reaches such a period of questioning the next avenue opens up. I personally believe that development in IMRT is anything but concluded. We are good at shaping a high-dose distribution to a stationary target ("dead patient"/phantom) but we are still embryonic on solving the problem of irradiating the moving target [71]. Image guided-IM-radiotherapy (IG-IM-RT) will help us. Also some of us are developing simpler non-MLC IMRT delivery techniques [7274] and IMRT with 60Co machines [75]. There is the whole issue of how to determine target volumes from more than anatomical tomographic data. Multimodality imaging is hardly applied yet to IMRT [76]. Dose-painting for inhomogeneous target irradiation is in its infancy [77]. Perhaps when molecular genetics moves from the lab to the clinic there will be an enhanced response due to combination with radiation, which will be required to be selectively delivered to precise geometric sites.

Controversially, I would suggest that maybe we do not need more algorithms to generate modulated beams in traditional settings. In my opinion there is much re-inventing of the wheel going on and re-publication. What we do need is solutions of the problem of determining the gross tumour volume (GTV), clinical target volume (CTV) and PTV, accommodating errors in positioning, the moving target, intra- and inter-fraction variations and a greater integration with medical imaging, especially functional imaging [78]. If we want to permit IMRT with cobalt machines to maybe assist less fortunate countries then some form of non-MLC IMRT will be needed. In the UK, as clinical IMRT becomes more widely established in oncology centres, we should not fall into the trap of regarding the development of the physics and engineering of IMRT as somehow completed. IMRT is not some static product to be purchased. There are still many unsolved problems including those above. Ultimately its clinical value must be tested through phase 3 randomized trials.


    Acknowledgments
 
I am grateful to Dr Chris Nutting for comments on this manuscript. I am grateful to all my physics, radiographer and radiotherapist colleagues, too numerous to list, for discussion on the present status and future of our subject. Where views expressed are more opinion than science I take sole responsibility. Our subject generates controversy and there is a spectrum of opinion.


    Footnotes
 
The work of the Joint Department of Physics is supported by Cancer Research UK, Elekta, Nucletron/Nordion, ADAC, Varian, EPSRC, MRC, NOMOS and the British Council. Back

Received for publication May 30, 2003. Accepted for publication June 30, 2003.


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 Top
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 Background
 What is IMRT?
 History of IMRT
 Physical basis of IMRT:...
 Inverse planning for IMRT
 The future of planning...
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