Emeritus Professor of Human Oncology and Medical Physics, Medical School of University of Wisconsin, Madison, Wiscon- sin, USA; Former Director of the Gray Laboratory, Northwood, London, UK

1 Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics

Jack F. Fowler

J. F. Fowler , DSc, PhD

Emeritus Professor of Human Oncology and Medical Physics, Medical School of University of Wisconsin, Madison, Wiscon- sin, USA; Former Director of the Gray Laboratory, Northwood, London, UK

Present address:

150 Lambeth Road, London, SE1 7DF, UK

Glossary

D, alpha Intrinsic radiosensitivity. Log

of the number of cells sterilized non-repairably per gray of dose of ionizing radiation.

E, beta Repair capacity. Log

of the number of cells sterilized in a repairable way per gray squared.

DE, alpha/beta ratio the ratio of “intrinsic radiosensitivity” to “repair capability” of a specified tissue.

This ratio is large (>8 Gy) for rapidly proliferating tissues and most tumors. It is small (<6 Gy) for slowly proliferating tissues, including late normal-tissue complica- tions. This difference is vital for the success of radiotherapy. When beta (E) is large, both mis-repair and good-repair are high. It is the mis-repair that causes the cell survival curve to bend downward.

CONTENTS Glossary 3 1.1 Introduction 6

1.2 The Simplest Modeling 7

1.2.1 The Seven Steps to LQ Heaven – Brief Summary 7 1.3 The Seven Steps to LQ Heaven – The Details 8 1.3.1 Development of the Simple LQ Formula E = nd (1+d/D/E) 9

1.3.2 Biologically Effective Dose 10 1.3.3 Relative Effectiveness 11 1.3.4 Overall Treatment Time 12 1.3.5 Acute Mucosal Tolerance 13

1.3.6 To Convert from BED to NTD or EQD

1.3.7 One Example 14

1.3.8 What is the Standard of Precision of These Estimates of BED or NTD? Gamma Slopes 15 1.4 Rejoining Point for Those Who Skipped: How to

Evaluate a New Schedule – Brief Summary 15 1.5 Now Let Us Study Some of the Best-Known Schedules

for Head-And-Neck Tumor Radiotherapy 16 1.5.1 Standard Fractionation 16

1.5.2 Hyperfractionation 17

1.5.3 Radiation Therapy Oncology Group Four-Arm Fractionation Trial (RTOG 90-03) 17 1.5.4 Head-and-Neck Schedules That Were Initially

“Too Hot” in Table 1.2 18

1.5.5 Shortening the Wang 2-Fraction-a-Day Schedule Using BED to Adjust Individual Doses 19 1.5.6 General Considerations of

Head-and-Neck Radiotherapy 19

1.5.7 A Theoretical Calculation of “Close to Optimum”

Head & Neck Schedules: 3 Weeks at Five Fractions

per Week 20

1.5.8 Conclusions Re Head-and-Neck Schedules 21 1.5.9 Concurrent Chemotherapy 22

1.6 Hypofractionation for Prostate Tumors 22 1.7 Summary 23

1.8 Appendix: Is This a Mistaken Dose Prescription? 24

1.8.1 For Equal “Late Complications” 25 1.8.2 For Equal Tumor Effect 25 1.8.3 Acute Mucosal Effects 26

1.9 Line-by-Line Worked Examples:

Details of Calculations of the Schedules Discussed in This Appendix 27 References 29

This chapter is written mainly for those who say “I don’t understand this DE business – I can’t be both- ered with Linear Quadratic and that sort of stuff.”

Well, it might seem boring – depending on your per- sonality – but it is easy, and it makes so many things in radiation therapy wonderfully and delightfully clear.

Experienced readers can turn straight to Section 1.4.

Accelerated fractionated schedules with shorter overall times than the conventional 7 (or 6) fractionation weeks.

BED Biologically effective dose, proportional to log cell kill and therefore more conceptu- ally useful as a measure of biological damage than physical dose, the effects of which vary with fraction size and dose rate. Formally, “the radiation dose equivalent to an infinite number of infinitely small fractions or a very low dose-rate”. Corresponds to the intrinsic radiosensitivity (D) of the target cells when all repairable radia- tion damage (E) has been given time to be repaired. In linear quadratic modeling, BED=total dose×relative effectiveness (RE), where RE=(1+dDE), with d=dose per fraction, D=intrinsic radiosensitivity, and E=repair capacity of target cells.

bNED Biochemically no evidence of disease. No progressive increase of prostate specific antigen (PSA) level in patients treated for prostate cancer.

CI Confidence interval (usually ±95%).

CTV Clinical tumor volume. The volume into which malignant cells are estimated to have spread at the time of treatment, larger than the gross tumor volume (GTV) by at least several millimeters, depending on site, stage, and location. See also GTV and plan- ning treatment volume (PTV).

't Time interval between fractions, recommended to be not less than 6 h.

EBR External beam radiation.

EGFR Epithelial growth factor receptor, one of the main intracellular biochemical path- ways controlling rate of cell proliferation.

EQD Biologically equivalent total dose, usually in 2-Gy dose fractions. The total dose of a schedule using, for example, 2 Gy per fraction that gives the same log cell kill as the schedule in question. If so, should be designated by the subscript EQD2 Gy.

EUD Equivalent uniform dose. A construct from the DVH of a non-uniformly irradiated volume of tissue or tumor that estimates the surviving proportion of cells for each volume element (voxel), sums them, and calculates that dose which, if given as a uniform dose to the same volume, would give the same total cell survival as the given non-uniform dose. Local fraction size is taken into account by assuming an DE ratio for the tissue concerned.

Gamma, J-50, J-37 Slope of a graph of probability, usually tumor control probability (TCP), versus total fractionated dose (NTD or EQD), as percentage absolute increase of probability per 1%

increase in dose. The steepest part of the curve is at 50% for logistic-type curves and at 37% for Poisson-type curves. Tumor TCP is usually between a gamma-50 (or -37) of 1.0 and 2.5. The difference between J-50 and J-37 is rarely clinically significant.

Gy, gray The international unit of radiation dose: one joule per kilogram of matter. Com- monly used radiotherapy doses are approximately 2 Gy on each of 5 days a week.

Gy10, Gy3, Gy1.5 Biologically effective dose (BED), with the subscript representing the value of that tissue’s DE ratio=10 Gy for early radiation effects, 3 Gy for late radiation effects and 1.5 Gy for prostate tumors. The subscript confirms that this is a BED, proportional to log cell kill, and not a real physical dose.

GTV Gross tumor volume. The best estimate of tumor volume visualized by radiologi- cal, computed tomography (CT) scan, magnetic resonance, ultrasound imaging, or positron emission tomography.

HDR High dose rate. When the dose fraction is delivered in less than five or ten minutes; that is, much shorter than any half-time of repair of radiation damage.

Hyperfractionation More (and smaller) dose fractions than 1.8 Gy or 2 Gy.

Hypofractionation Fewer (and larger) dose fractions than 1.8 Gy or 2 Gy.

Isoeffect Equal effect.

LC Local control (of tumors).

LDR Low dose rate. Officially (ICRU), less than 2 Gy/h; but this is deceptive because any dose rate greater than 0.5 Gy/h will give an increased biological effect compared with the traditional 0.42 Gy/h (1000 cGy per day). For example at 2 Gy/h, the biological effects will be similar to daily fractions of 3.3 Gy and 2.8 Gy on late complications and on tumors respectively.

Linear effect Directly proportional to dose.

Ln log

Natural logarithm, to base e. One log

is equal to 2.303 log

.

Log

Common logarithm, to base 10. “Ten logs of cell kill” are 23.03 log

of cell kill.

LQ Linear quadratic formula: log

cells killed=Dudose+Eudose-squared.

Logistic curve A symmetrical sigmoid or S-shaped graph relating the statistically probable incidence of “events”, including complications, or tumors controlled, at a specified time after treatment, to total dose (NTD). This curve is steepest at the probability of 50%.

LRC Loco-regional tumor control. LC would be local control.

NTCP Normal tissue complication probability.

NTD Normalized total dose of any schedule. The total dose of a schedule using 2 Gy per fraction that gives the same log cell kill as the schedule in question. The NTD will be very different for late effects (with DE=3 Gy and no overall treatment time factor) than for tumor effect (with DE=10 Gy and an appropriate time factor).

Poisson curve A near-sigmoid graph of probability of occurrence of “events”, such as tumor control at X years, versus total dose or NTD. Based on random chance of successes among a population of tumors or patients, the probability of curve P=exp (–n), where an average of n cells survive per tumor after the schedule, but 0 cells must survive to achieve 100% cure. If an average of 1 cell survives per tumor, P=37%. If 2 cells sur- vive, P=14%. If 0.1 cells survive on average, P=90%. This curve is steepest at the probability of 37%.

PTV Planning treatment volume – larger than CTV to allow for set-up and treatment- planning errors.

PSA Prostate-specific antigen: can be measured in a blood specimen as a measure of activ- ity of the prostate gland. Often taken as a measure of activity of prostate cancer.

Quadratic Effect proportional to dose squared, for example from two particle tracks passing through a target.

QED Quod Erat Demonstrandum – Latin for “That’s what we wanted to show!”

RE Relative effectiveness. We multiply total dose by RE to obtain BED. RE=(1+d[DE]) where d is the dose per fraction.

RTOG Radiation Therapy Oncology Group, USA.

SF Surviving fraction after irradiation, usually of cells.

Tpot Potential doubling time of cells in a population; before allowing for the cell loss factor. Tpot is the reciprocal of cell birth rate. It can only be measured in a tissue before any treatment is given to disturb its turnover time.

Tp Cell doubling time in a tissue during radiotherapy; probably somewhat faster than Tpot.

Determined from gross tumor (or other tissue) results when overall time is altered.

Tk Kick-off or onset time: the apparent starting time of rapid compensatory repopula- tion in tumor or tissue after the start of treatment, when it is assumed that there are just two rates of cell proliferation during radiotherapy: zero from start to Tk, then constant doubling each Tp days until end of treatment at T days. Accelerating repopulation is discussed in Section 1.5.6.

TCP Tumor control probability.

In the early years of the development of the LQ formulation, there was no overall treatment-time factor (Douglas and Fowler 1976; Barendsen 1982; Withers et al. 1983). This was added later (Travis and Tucker 1987; van de Geijn 1989;

Fowler 1989), based on LQ-aided analyses of animal and clinical data (Denekamp 1973; Turesson and Notter 1984a,b; Thames and Hendry 1987). Since then, the strong effect of repopulation of tumor cells during radiotherapy has been well substanti- ated so that a repopulation term has been added for tumors (Fowler 1978, 1989; Withers et al. 1988;

Fowler and Lindstrom 1992; Hendry et al. 1996).

More recently, a different set of parameters has been described to predict acute mucosal reactions in human patients (Fowler et al. 2003c).

Although the accuracy and even the nature of the LQ factors has been queried a few times, for example whether the parameters are unique or distributed (King and Mayo 2000; Brenner and Hall 1999, 2000; King and Fowler 2002; Dasu et al. 2003;

Moiseenko 2004) , the LQ formulation has remained solidly useful and has aided in the design of clinical trials that have changed the practice of radiotherapy (Thames et al. 1983). Examples include the design of hyperfractionated (more and smaller fractions) and accelerated fractionation (shorter) trials, the avoid- ance of gaps in radiation treatment, the development of high dose-rate brachytherapy, and a better under- standing of when to use or avoid hypofractionation (fewer and larger fractions). The recent growth of stereotactic body radiotherapy is a subset of the latter category (Fowler et al. 2004).

One of the most interesting series of modeling investigations concerns oral and laryngeal cancers in which the overall times were deliberately short- ened until the acute reactions became too severe, in several well known schedules in different coun- tries. Each schedule was then moderated in some way until it became tolerable. The modeling then showed that not only the acute mucosal reactions fell into a narrow band of BED, but the modeled tumor responses were then all close to 11 log

of predicted tumor cell kill for a variety of different time–dose schedules. This story will be told in this chapter.

Although the numerical results of modeling depend to some extent on the values assumed for the parameters, ratios of parameters such as DEratios and time–dose trade-offs (grays per day) are often known sufficiently well for reasonable variations to lead to no clinically significant differences in pre- dicted total dose or BED or NTD. In the modeling 1.1

Introduction

It is well known that the simplest description of radi- ation dose, the total dose, is not adequate because its effect varies with size of dose per session (the dose fraction) and with dose rate. If we double the dose per fraction from 2 Gy to 4 Gy (keeping total dose constant), the effect is 20% greater for tumors but 100% greater for late complications. Further, if a given physical dose is spread evenly over 24 h instead of 2 min, its effect is reduced by 20% for most tumors, but to about half for late complications. We need a way of expressing radiation “dose” in some quantitative way that is more proportional to the observed biological effect. This is the object of cal- culating a biologically effective dose (BED), and an equivalent dose in 2 Gy fractions (EQD or NTD), so that a 20% increase or decrease of BED or NTD or EQD

will lead to a reasonable approximation of a 20% increase or decrease of the expected biological effect. The interesting point is that the same change in physical dose is likely to alter the incidence of late complications to double or half of its effect on tumors. So how can we deal with that?

The basic truth in radiotherapy is that any change in the schedule of dose delivery has a different effect on tumors from its effect on late complications, unless both dose per fraction and dose rate are kept constant. These differences provide some of the remarkable advantages of radiation therapy, and also some puzzles until they are explained. BED can take these differences into account, and preferably explain them.

In the 25 years since the linear quadratic (LQ) formula has been used for the evaluation of radio- therapy schedules, it has proved remarkably reliable.

It is now the main and generally accepted method of

rationalizing the improved time–dose-fractionation

schedules that have been developed to replace, in

some body sites, the standard “2 Gy given five times

a week for 6 or 7 weeks” schedules. It was first of all

useful in identifying the important difference in the

effect of dose-fraction size between rapidly prolifer-

ating tissues (most tumors) and slowly proliferating

tissues (most late complications). This explained the

blindly used, but not always wrong, predominance

of multi-small-fraction schedules, such as 1.8 or

2 Gy five times a week for 6 or 7 weeks. As explained

below, the theoretically ideal overall time would be

close to the time at which rapid repopulation in the

tumor kicks off, designated Tk days after starting

treatment.

described below, we take care to limit the assumed values of parameters to a small library of values selected from experience, avoiding “elegant varia- tion”. Then, results that are useful and self-consis- tent are obtained.

Unlike some other attempts at modeling, the number of initially viable cells per mm

is not essential in LQ modeling for time–dose evaluations, because it is largely cancelled out against radiosensi- tivity D in the standard BED formulation. This is also true of equivalent uniform dose (EUD; Niemierko 1997). Both BED and EUD enjoy similar stability for this reason. The most essential biological factor in LQ formulation is the DE ratio of the tissue concerned, which appears in the BED with weighting equal to dose per fraction. The other significant factors are the “kick-off time” of rapid tumor repopulation Tk and the doubling time of repopulating tumor cells Tp, which together with the D value all appear in the repopulation term which is usually, but not always, a small proportion of BED.

A glossary is attached at the beginning of this chapter of terms that are best for readers to know when reading about these topics.

1.2

The Simplest Modeling

Years ago, mathematical models were regarded with suspicion, or with derision as playthings for chil- dren – but not any more. Modeling has become an important scientific tool in the design and evalua- tion of topics from global warming to engineering design of aircraft and the pharmacological develop- ment of drugs, replacing expensive experimentation in many cases (Prof. Gordon Steel 1990, personal communication). With the aid of computers, math- ematical modeling using optimization and new imaging are continuing to revolutionize treatment planning in radiotherapy, as other chapters in this book will show.

“There are good reasons for believing that the primary effects of radiation on biological tissues are cell damage and cell depopulation in renewing populations” (Thames and Hendry 1987). This is still true whether the concern is damage to normal tissues or the elimination of every malignant cell in tumors. However, certain indirect biological end points, such as radiation sickness or extent of late fibrosis, do not appear to depend only on numbers of cells sterilized, although it does not mean that

they are not a strong function of cells sterilized. The phantom of immunological response keeps rear- ing its head, with little practical effect. The addi- tive effect of chemotherapy seems to apply most effectively when it is used concomitantly, and up to now amounts to some 10% of the total cell kill com- pared with radiotherapy. The strength of radiation as a treatment strategy is that it can and does reach wherever the physical plan puts it, with increasing efficiency of positional accuracy.

The object of the present modeling is to find simi- lar biological effects to the radiotherapy treatments with which we are familiar, from different schedul- ing of time and dose. That is the “isoeffect modeling”

that has much history, passing through the “cube root law” of the 1930s, the “Strandqvist slope” of the 1940s, and the “Nominal Standard Dose” (NSD) or

“Time-dose Factor” (TDF) of Dr Frank Ellis in the 1960s and 1970s – and then to evaluate methods of doing better. We shall start by writing down some ideas for explaining the non-linear action of ioniz- ing radiation in damaging biological cells.

Radiation sterilizes cells, meaning that they do not die immediately, but at the time of the next cell division, or a few divisions later. An important factor is the repair occurring in the cells between irradia- tion and their next cell division. This repair in cells, i.e., recovery in tissues, depends on the turnover rate of renewing tissues – a day or so for rapidly prolifer- ating tissues and most tumors, but many months for the organs that normally proliferate slowly. That is the important difference in the tissues that LQ mod- eling helped to bring out, with the help of the famous or infamous ratio D over E.

1.2.1

The Seven Steps to LQ Heaven – Brief Summary First, before explaining them in detail, we list here the “Seven (algebraic) Steps to LQ Heaven”, so that readers who wish to do so can skip several pages and turn to Section 1.4. That’s where the story gets exciting.

Alpha is the intrinsic radiosensitivity of the

cells, defined as how many logs (to the exponential

base “e”) are killed (sterilized) per gray, in a “non-

repairable” way. Beta is the repairable portion of the

radiation damage, requiring 6 h or more for com-

plete repair. It can be regarded as the result of two

charged-particle tracks passing through a sensitive

target in the cell nucleus in less than 6 h, so this term

has to be multiplied by d squared. E is the log

sum of

the non-repairable D term and the partly repairable E term. So for n fractions of d Gy dose each:

[1.1]

[1.2]

[1.3]

Dividing through by D to express DE as a ratio.

[1.4]

With no repopulation considered; as for most types of late complication.

This is the limited definition of BED. It applies to late effects. BED=E/D=total doseuRE where RE=(1+d/[DE@), and it is very useful.

Next, we subtract the log cell kill due to repopu- lation of any cells during radiotherapy, after the

“kick-off” or onset time Tk, where T is overall time and Tp is the average cell-number doubling time (in days) between Tk and T.

E = nd (D+Ed)–(T – Tk)u rate of repopulation per day [1.5]

E = nd (D+Ed)–log

2(T–Tk)/Tp [1.6]

(log

2 = 0.693)

To transform the total log cell kill E into the total BED requires the same division throughout by D that we carried out in step 3 (Eq. 1.3) above:

[1.7]

Final step = LQ heaven!

BED can be expressed as Gy

(or Gy

) for late complications, or as Gy

(or Gy

) for tumor or early normal-tissue reactions, the subscript referring to the DE value used in its calculation. Gy

and Gy

values must not be mixed, as USA and Canadian or Hong Kong dollars cannot. However, several segments of a schedule can have their Gy

values added together and, separately, their Gy

values added, for a compar- ison of total BEDs amounting to a “therapeutic ratio”

of Gy

Gy

– representing tumor cell damage divided by late normal-tissue damage. This notation should always be used, or confusion quickly sets in. It both reminds us that this is a BED, not the real physical dose; and confirms which DE ratio was used, thus

for which tissue each BED or EQD

or NTD is cal- culated. We should state each time “late” or “tumor”

or “early” BED or NTD (Fowler 1989).

A further measure of radiation damage from these formulae is:

Log

cell kill = E = BEDuD, so that in the

“common log to base 10 scale, log

cell kill = log

cell kill/2.303.

To convert from BED to EQD 2 Gy or NTD (total equivalent dose in 2 Gy fractions):

For late complications, divide Gy

by 1.667. For tumor or early effects, divide Gy

by 1.2.

The explanation comes from the identity BED1=BED2, where “1” is for “d Gy per fraction”

and “2” is for “2 Gy per fraction”, so for identical BEDs, we have:

Total dose 1uRE1=total dose 2u(RE for 2 Gy and the same DE).

NTDu(1+2/DE)=BED. Then solve for NTD!

1.3

The Seven Steps to LQ Heaven – The Details This is the section that experienced modelers might wish to skip by several pages, and go to Sections 1.4 and 1.5, where comparisons of actual schedules are tabulated and discussed.

The mathematics of the LQ formula are very simple and are taught in courses before the end of the high school syllabus at age 15 or 16 years. Solving an alge- braic quadratic equation and manipulating the con- version of an exponential to a logarithmic form are the only steps requiring an effort of memory and are only necessary if you get into more calculations than you will normally need. If you don’t deal with this sort of arithmetic every week, you normally need no more than the four simplest keys (+, –, u,) on your hand calculator, or even just the back of an enve- lope. The “Seven Steps to LQ Heaven” are intended to ensure that you, the reader, are never puzzled by such simple calculations again. Having followed, and understood, the seven steps that follow, you should be able to use them with confidence for any comparisons of radiotherapy schedules you wish to make in terms of BED (Fowler 1989, 1992).

BED is proportional to log cell kill for cells of the specified DE ratio, and so is a strong function of biological effect. BED is itself a ratio (E/D), the two E = nd ( α β + d )

E / α = nd ( 1+ d β α / )

E

α = nd + d =

E nd d

( 1 / ) ( 1 )

α β +

E = n ( α d + β d

)

BED nd d

T Tk Ε

α 0, 693

= = ( + − α −

/ ) ( )

1 α β Tp

parameters of which are not individually important.

Mathematically, E/D is simply a linkage term show- ing equivalence between two schedules – their “iso- effect” in the time-dose scale – as well as providing a useful number, the BED. Its numerical value is a convenient number, representing a dose without the repairable component, that is avoiding dose per fraction or dose rate, until relative effectiveness (RE) is built in. We could talk about D=0.35 log

per Gy and E=10 log

of cell kill, or equally D=0.035 log

per Gy and E=one log

of cell kill, and the BED would be the same because it is ED

1.3.1

Development of the Simple LQ Formula E = nd (1+d/D/E)

Cell depopulation is the main effect of radiation, both for eliminating tumors and for damaging normal tissues. In addition, some genes are activated, which can be relevant in those cells that survive irradia- tion, and some apoptosis (cell death independent of mitosis) may be caused. Although apoptosis does not appear to be the major effect of radiation, when it happens it adds to the effect. The promising strat- egy of damaging tumors by depleting their blood supply, with pharmacological or enzyme-pathway aid (Fuks et al. 1994), can be regarded as consider- ing the neovasculature (that is, rapidly proliferating endothelial cells (Hobson and Denekamp 1984) as legitimate oncogenic targets, as well as the directly malignant tumor clonogens (Hahnfeldt et al.

1999). Radiation can reach them all.

If a dose of radiation D sterilizes a proportion of cells in a given tumor or normal organ so that the number of viable cells is reduced from the initial No to Ns: the surviving proportion is Ns/No, which is designated S.

This process is represented as S=e

= exp(–D/Do), in its simplest form, where Do is the average dose that would sterilize one cell. This shows that the surviving proportion of cells is reduced exponentially with radiation dose. That is, each successive equal increment of dose reduces the surviving cells by the same proportion, not by the same number. The proportion of surviving cells would then decrease from 1 to 0.5, to 0.25, to 0.125, etc.; if higher doses per fraction were used, from 0.1 to 0.01 to 0.001.

Another way of writing this is: log

S= D/Do, which plots out as a graph of log

S vertically versus dose horizontally to give a straight line of slope

minus D/Do. This would be called a “single-hit”

curve. The dose Do would reduce survival by one log

which is to 37% survival. You can see that Do=1/D in the LQ formula, that is 1/(the dose to reduce cell survival by one log

).

Plotting graphically the proportion of surviving mammalian cells against single dose of radiation gives a curve, not a straight line. This curve starts from zero dose with a definite non-zero slope, gradu- ally bending downward into a “shoulder” at less than 1 gray of dose, and continuing to bend downward until at least 10 or 20 grays. At higher doses, the cell sur- vival curve appears to become nearly straight again but of course steeper than at the origin (Gilbert et al. 1980). This is the well-known “cell survival curve”

where the logarithm of the surviving proportion of cells is plotted downward, against radiation dose on the horizontal axis (Fig 1.1). It is the same curve as the negative logarithm of the fraction of cells “killed”

– actually “sterilized” so that they die later, after the next cell division or a few divisions.

This is the plot of log

proportion of cells surviv- ing: S=–Dd–Ed

Therefore also: log

proportion of cells sterilized (killed)=+Dd+Ed

.

It is this logarithm of the number of cells steril- ized that can be divided into one part proportional to dose Dd; and another part proportional to dose-

Fig. 1.1. The simple cell survival curve for linear quadratic cell kill versus radiation dose, for a single dose of radiation deliv- ered within a few minutes. The alpha component increases as shown linearly with dose. The beta component is added to this in a curving pattern, increasing with the square of the dose.

This example is numerically correct for the DE ratio of 3 Gy

squared Ed

(that is “quadratically”, where two sub- lesions combine, each produced in number propor- tional to dose). The logarithm (proportion) of lethal events caused by a dose d is then:

E=Dd+Ed

The linear component is found to be not repair- able beyond a few milliseconds after the irradia- tion, but this does not mean it cannot be altered, by oxygen if present at the time of irradiation as one major example. However, the dose-squared damage gradually fades over a few hours. It is repaired by several processes within the biological cells, mostly within the DNA, so that cell survival recovers toward the straight initial D slope. Cells and tissues are said to “recover” and biochemical lesions in DNA are said to be “repaired”.

Let us call this cell-number damage E, the loga- rithm of the number of cells sterilized by a dose d in grays (Gy). Then, we just write the first equation as:

E=Dd+Ed

where Dand E are the coefficients of the linear com- ponent and the dose-squared component, respec- tively. This is the first step on the ladder to explain- ing the LQ formulation, commonly taught as the

“Seven Steps to Heaven”. If you remember this little starting formula you will have no trouble at all with the next three steps. “LQ Heaven” is reached when

you understand the LQ steps well enough not to forget them within the next few days.

The next step is the simple one of adding several fractions of daily doses, each of d grays, to obtain the total dose if n fractions (daily doses) are given.

Figure 1.2 illustrates the sequence of equal fractions, giving a total curve that is made up of a sequence of small shoulders, in toto an exactly linear locus, of slope depending on the value of Eq. 1.1 at each dose d:

[1.1]

The dose per fraction d can be taken outside the parentheses, nd being of course the total dose:

[1.2]

We usually know the ratio of the two coefficients, DE, for given cells and tissues much better than we know their individual values, so the next step is to express E in terms of this ratio. It is most use- fully done by dividing both sides of Eq. 1.3 by D (Barendsen 1982). If instead we divided by E, then the resulting BED would be in terms of dose squared, which would be awkward.

E/D=nd(1+dED) [1.3]

which is also of course identical to:

[1.4]

Because D is defined as a number (log number of cells sterilized) “per gray”, the term E/D has the dimensions of dose. It is in fact the BED that we are seeking to calculate, so Eq. 1.4 gives it to us. We are halfway up the “Steps to Heaven” and this equation enables us to do many useful things in predicting biological damage (Fowler 1989; also Chapters 12

& 13 in Steel 2002).

1.3.2

Biologically Effective Dose

The basic concept of BED was defined by Barendsen (1982), who at first called it extrapolated tolerance dose (ETD), meaning that dose which, if given as an infinite number of infinitely small fractions (along the initial slope of the cell survival curve), or at a very low dose rate (so that all the quadratic damage has been repaired), would cause the same log cell kill

Fig. 1.2. The survival curve for four equal radiation doses given sequentially, with suffi cient time – at least 6 h – between them to allow complete repair of the beta component of radiation damage. Since the shape of each is then a repetition of the previous dose, the track of the result after each dose fraction is a straight line when plotted as log cell kill against dose as shown

E = n ( α d + β d

)

E = nd ( α β + d )

E

α = nd + d =

E nd d

( 1 / ) ( 1 )

α β +

as the schedule under consideration, thinking of the maximum dose that a normal tissue would tolerate.

Since it was obvious that this conceptual extrapola- tion to very small dose per fraction could be applied to any level of damage, not just to the maximum tolerated level or only to normal tissues, it was soon renamed extrapolated response dose (ERD) and later to the more general BED (Fowler 1989). It is illustrated graphically in Figure 1.3. Because BED is defined in relation to the initial slope, i.e., the linear component of damage, it is E/D Because D has the dimensions of 1/dose, E/D has the dimen- sions of dose, as we require. We are talking here about an averaged value of D during the weeks of radiotherapy. Different values of D could be applied to different segments of a schedule but there is not yet evidence to justify this.

Since the definition of BED is the ratio E/D, the individual values of E and D are irrelevant for esti- mating relative total doses. The ratio E/D is math- ematically a link function, signifying biological equivalence between two schedules having equal effect. The individual values of E or D have no nec- essary biological significance in LQ formulation.

Values of D are particularly vulnerable to variation of tumor size, stage of tumor, and accuracy of dose, but provided that the ratio between E and D (and E does not vary – between one prospective popu-

lation in a clinical trial and another – there are no effects on ratios of doses between schedules, which are what we want to compare. This is the important reason why BED, like EUD (Niemierko 1997), is robust. They are relatively stable if parameters are varied by modest amounts.

These biological ratios between log cell kill and cellular radiosensitivity have in fact been deter- mined in certain biological experiments concerning skin and intestinal clones in mice (Withers 1967, 1971), so these tissues can be put on to a more exact basis concerning number of cells per millimeter of mucosal surface. However, the use of a number to designate log cell kill in tumors depends on a specific assumption of how many viable malignant cells were present per mm

of tumor volume, which is usually unknown. The number has wildly varied between ten thousand million (10

) and one-tenth per tumor among various authors for various tumors (the latter being curable with 90% probability). Because of the real variability between tumors and patients, a value for D [derived for example from a gamma- 50 slope of dose versus tumor control probability (TCP)] cannot be regarded as a reliable guide to cell numbers, as some writers have assumed and argued about. Heterogeneity decreases the extracted value of D from populations. What is interesting is that, as more detailed and precise descriptions for stage of tumors are developed, as they are by various inter- national agreements, the recorded gamma-50 slopes from clinical data become steeper (or D values higher), illustrating less heterogeneity (Hanks et al. 2000; Regnan et al. 2004). The log

cell kill values quoted in the tables below are all assuming D=0.35 log

per Gy.

1.3.3

Relative Effectiveness

The BED thus comprises the total dose “nd” multi- plied by a parenthesis (1 + d  D  E). This parenthetic term is called the RE, and is one of the most useful concepts of the LQ formulation. RE is what deter- mines the strength of any schedule when multiplied by total physical dose. BED is simply total dose mul- tiplied by RE:

BED = nd (1 + d  D  E) = nd u RE [1.4]

or

BED = total dose × RE

Fig. 1.3. Graphical illustration of the concept of biologically

effective dose. The dashed line represents the log cell kill of

20 (or more) equal fractions of radiation. BED is the dose that

would cause the same log cell kill E as the schedule with nud

fractions, if it could be delivered in an infi nite number of infi -

nitely small fractions, or at very low dose rate. All the beta com-

ponent would then have been repaired, and the straight line

representing the log cell kill versus total dose would follow just

the initial slope alpha as shown, its slope determined by the

value of alpha only. The ratio of BED to the real physical dose

is of course the RE, relative effectiveness=BEDtotal dose

When dose rate varies, the term containing E includes, instead of just d, the dose rate and the recovery rate of tissues as a ratio, as discussed in detail elsewhere (Barendsen 1982; Thames and Hendry 1987; Fowler 1989).

The concern of RE is obviously dose-per-fraction size, in relation to the D  E ratio for different tissues.

RE is larger for larger dose per fraction and for smaller ratios of D  E. Dose per fraction is therefore one of the three major factors in LQ formulation, the co-equal second being the D  E ratio. The third is overall time, which we deal with in the remaining three “steps to LQ heaven” below. The major biological importance of RE depends on the fact that D  E is large for rapidly proliferating tissues such as most tumors and small for slowly proliferating tissues such as late complica- tions. This is attributed to the fact that slowly prolif- erating tissues, with long cell cycle times, have more time to carry out repair of radiation damage than do short cell cycles. The exact molecular causes of these differences in ratio of radiosensitivity to repair are not yet known but inhibition of epidermal growth factor receptors (EGRFs) is likely to be involved Fowler 2001). It is these differences that enable conventional radiotherapy to succeed, often using a large number of small fractions such as 30 or 35F u 2 Gy. We shall discuss the differences in D  E in the next section.

RE is of course high when d is high and D  E is low, for example as in hypofractionation (fewer and larger fraction sizes than normal), which could therefore cause excessive damage in slowly respond- ing, late-reacting tissues, i.e., in late complications.

In contrast, RE is naturally low for hyperfraction- ation (more and smaller fractions) and for low dose- rates (lower than 100 cGy/h; ICRU can be criticized for defining low dose rates as “up to 200 cGy/h”, because their BEDs differ significantly from those for 50 or 100 cGy/h). It is the difference in doses per fraction above or below 2 Gy that has biased radio- therapy in the direction of many small fractions.

Total doses at 2 Gy per fraction should be above 60 Gy for cure of all except a few radiosensitive types of tumor; and preferably above 70 to 90 Gy NTD

for many tumors of stage II or III sizes. Equations 1.4 and 1.4a enable a great many aspects of radiotherapy to be compared simply and quickly.

A question often asked is “what new total dose should be used if dose per fraction is changed from d

to d

?” This is easily answered by taking the inverse ratio of the two values of RE. If the two total doses are D

and D

, of which D

is known and D

is the value sought, both of the BEDs have to be equal, by definition:

BED=total dose × RE = D

× RE

= D

× RE

;

So that [1.4b]

This is an example of a comparison of schedules where the only biological factor is the ratio D  E, treated as if it were a single parameter.

1.3.4

Overall Treatment Time

The major effects of radiation are diminished by any proliferation occurring in the cell populations during the weeks of radiotherapy, obviously by replacement of some of the cells that were steril- ized by irradiation. In slowly proliferating tissues, naturally those in whom the reactions appear late, this replacement by repopulation is negligible (as in nerve tissue) (Stewart 1986). In rapidly proliferat- ing tissues, however, including most tumors, it can counteract up to one-third of the cell-killing effect of the radiotherapy, although there is a delay of some days before it begins in tumors; it is more rapid in normal mucosa, at 7 days in oral mucosa.

This delay in tumor repopulation is because the tumor needs time to shrink due to the treatment, so as to bring more of the surviving cells into range of the generally poor blood supplies in the tumor. In solid tumors above a few millimeters in diameter, the daily production of cells is much faster than the volume growth rate would suggest. In most carcinomas, there is a cell loss factor of 70–95% due to nutritional failure and, to a lesser extent, due to apoptosis. This causes the volume doubling times of tumors to be as slow as months, although the number of cells born would cause the population to double in only a few days if no such cell loss occurred. Therefore, the clinically observed doubling time of a tumor is no guide to the cell repop- ulation rate inside the tumor. This is one of the curi- ous facts in cancer that has only been apparent from kinetic population studies in the 1970s, using originally radioisotope labels and recently immunological fluo- rescent labels with flow cytometry (Begg and Steel 2002). Most human carcinomas have volume doubling times varying from 1 month to 3 months, but their cell population birth rates yield cell-number doubling times of 2–10 days, with 3–5 days commonly found.

The cell doubling times are called Tpot (potential dou- bling times, meaning cell birth rate “in the absence of nutritional or apoptotic cell loss”). Prostate tumors are exceptionally slowly proliferating with median cell doubling times of 42 days (range 15 days to >70 days).

D D

RE RE

d d

2 1

2 1

1 2

= (1 + / α/β) (1 + /α/β)

=

So, if a number S of cells survives a certain phase of treatment, this number will increase at the rate of Se

per day and Se

in t days. This means the number of cells increases by a constant proportion per day. This rate is conventionally described as the time in days required to double the number of cells, Tp, so the population increases by log

2/Tp per day.

Therefore, the log

number of cells killed decreases by log

2  Tp per day, meaning its change is –0.693/Tp per day and –0.693 t/Tp in t days.

It is very important that we do not forget the delay in start of repopulation mentioned above, after a “kick-off” or “onset” time designated Tk days.

Allowing for this, the time available for repopula- tion is of course t = T – Tk days, where T is the overall time in days. It is important to note that the treat- ment begins at day 0, not day 1. The log

amount of repopulation is then 0.693(T – Tk) / Tp, and it has to be subtracted from the log

cell kill E that we calcu- lated above in Eq. 1.3. Thus, the total amount of log cell kill, allowing for repopulation, is of course:

E = nd (D + Ed) – (T – Tk) u rate of repopulation per day [1.5]

E = nd (D + Ed) – log

2(T – Tk) / Tp [1.6]

where T = overall treatment time (in days, the first day being 0 not 1), Tp = cell population doubling time during treatment, Tk = starting time of repop- ulation, and Tp = average cell-number birth rate during the irradiation. Note that Tp may be different from Tpot, which can only be measured before any treatment is given; Tp is possibly somewhat faster.

This simple modeling, assuming zero repopulation up to Tk days and a constant doubling time Tp after Tk, is certainly an approximation to a smooth curve of accelerating Tp, but several more complex models to add different times at which faster Tp is assumed have not succeeded in giving better correlations with clinical outcome (van Dyk et al. 1989; Denham and Kron 2001), although Denham and Kron (2001) made a persuasive case for some type of accelerating curve in tumors. Fenwick has recently proposed an ingenious Delayed Feedback (from the level of cell depletion reached) algorithm which deals elegantly with this concept (Fenwick 2006).

To transform E into the total BED requires the same division throughout by D that we carried out in step three (Eq. 1.3), and this is the final equation, the seventh step:

[1.7]

Putting it into words:

[1.7a]

It will be noticed that this LQ formula now involves three more parameters, so we might think that LQ has

“lost its innocence” when repopulation is included. A value must be chosen for radiosensitivity D and the start of rapid repopulation in tumors Tk, and for the doubling time Tp of viable cells thereafter. There are, however, some clinical data that enable ratios of these parameters to be estimated. (There is a practical snag in that it is easy to forget the D in the last term! But, if you remember that this repopulation term is usually about 0.5–0.8 Gy ₁₀ per day for rapidly proliferating tissues including most tumors, you can easily make a rough check on the subtraction from the simpler BED of step four (Eq. 1.4). This term is somewhat greater if doses per fraction are larger than 2 Gy.)

In radiotherapy practice, late-responding normal tissues generally have a normal turnover time of many months, and these are reflected first of all in their kick-off time of acceleration, Tk, after starting radiotherapy (Stewart and van der Kogel 2002), and also obviously in the value of Tp after compen- satory repopulation has started (Hendry et al. 1996;

Fowler and Chappell 2000), which may be a little shorter than Tpot measured, as it can only be, before any treatment. It is this repopulation aspect of radio- biological modeling that has brought most radio- therapy departments to pro-actively avoid gaps in treatment, and in many to choose shorter schedules (Hendry et al. 1996). To a radiobiology modeler, it is surprising that this took so long to realize (Fowler 1978; Withers et al. 1988; Fowler and Lindstrom 1982; Hendry et al. 1996). The huge difference between gross tumor volume doubling time and the cell birth doubling time, Tpot, is due to the high cell loss factor (70–90%) in carcinomas. The kick-off time for tumor repopulation, Tk, could be between 21 days and 32 days in human head and neck tumors (Brenner 1993; Roberts and Hendry 1999).

1.3.5

Acute Mucosal Tolerance

A particular application of Eq. 1.7 above has been a long-term development of a collection of data about

BED E

nd (1+ d = TP

α / β ) − 0, 693

= α −

α ( T Tk )

BED E

= = ×

α Total Dose Relative Effectiveness minus

(log 2)

×

×

(Days availible for repop)

alpha Cell Doubling Time

minus

head and neck radiotherapy schedules that are judged to be clinically just tolerable from the acute reaction point of view. A comprehensive review of published head and neck schedules with consequential recom- mendations for a “tolerance zone of early BED” was recently published by the present author and col- leagues (Fowler et al. 2003c). The best matching with these not particularly small treatment fields, for “nearly intolerable” acute mucosal reactions in many types of fractionation regime, was found to be at BED = 59–63 Gy

if the following specific param- eters in Eq. 1.7 were used: DE= 10 Gy; D= 0.35 Gy per log

; Tk (starting time of repopulation) = 7 days for oral mucosa (Dörr et al. 2002); and Tp = 2.5 days (average mucosal cell doubling time during irra- diation). There is provisional information that the range of BED may be somewhat higher in rectal mucosa, for example 64–69 Gy

together with a mucosal area limitation down to a few centimeters squared for doses of 78–80 Gy NTD (Huang et al.

2002; Vargas et al. 2005), but further confirmation is required. A practical conclusion from our acute mucosal review (Fowler et al. 2003c) was that if a new schedule causes too strong acute reactions, these can be avoided easily by extending the over- all time by a few days. Acute BED for mucosa is normally reduced at a rate of about 0.8 Gy

per day (with 2 Gy fractions), after the time Tk = 7 days from starting daily radiotherapy (Dörr et al. 2002).

1.3.6

To Convert from BED to NTD or EQD

We have talked in terms of BED in Gy

and Gy

mainly, and familiarity with these terms and num- bers is recommended. However, those who do not use them regularly are often more comfortable with their conversion into the biological equivalent of total dose in 2 Gy fractions. This conversion of BEDs into EQD

or NTD (normalized total dose/nor- malized to 2-Gy fractions; Maciejewski et al. 1986;

Joiner et al. 2002) is a further mathematical step, which experienced modelers may not wish to do, but it is very simple, as described below. It does, and this is important, give different EQD and NTD values for late complications from the EQD and NTD for tumors in the same schedule, so that subscripts for BEDs in Gy

or Gy

,

etc. must be kept rigorously through every step of every calculation. Further, the translation from “tumor BED” or “late complica- tions BED” into “tumor NTD” or “late complications NTD” must be made when the results are summa-

rized. Such designation of Gy

or Gy

, and “Late”

or “Tumor” or “Acute” is necessary every time, or readers (and you the writer!) will quickly become confused.

We can see, from the obvious equality of: BED of any schedule 1, called BED1, required to be equal in radiobiological effect to BED2, shows that: total dose D1 u “the RE for its dose per fraction” must be equal to NTD

u RE for 2 Gy fractions, with the appro- priate D  E ratio so we obtain the formula:

NTD

u RE2 = BED1; therefore NTD

= BED1 / RE2

= BED1  (RE for 2 Gy fractions). QED Therefore, an NTD

or EQD

can be derived from any BED by simply dividing that BED by the RE for 2 Gy fraction size and the same D  E ratio as used to calculate the BED (Table 1.1).

1.3.7

One Example

First let us look at the simple example of a 60 Gy dose given with different fraction sizes, starting with 2 Gy u 30F in 6 weeks. The “late complications BED”

is usually calculated first, assuming D  E = 3 Gy, and from Eq. 1.4 it is:

BED = nd (1 + d / (D  E))

60 (1 + 2 / 3) = 60 u 1.667 = 100 Gy3

The “tumor response BED” is calculated assum- ing D  E = 10 Gy and is therefore:

60 (1 + 2 / 10) = 60 u 1.2 = 72 Gy10

Both the late BED of 100 Gy

and the tumor BED of 72 Gy

correspond to an NTD to 2 Gy fractions or EQD

of 60 Gy, of course.

Table 1.1. Conversion of BED to NTD or EQD

Type of tissue DE for BED Divide the BED by:

Late complications 3 Gy (1+2/3)=1.667 Early and tumor effects 10 Gy (1+2/10)=1.2 Exceptions would be:

Late CNS with DE=2 Gy 2 Gy (1+2/2)=2.0 Prostate tumors

DE=1.5 Gy

1.5 Gy (1+2/1.5)=2.333

For a stronger schedule of 70 Gy = 2 Gyu35F the corresponding “late” (complications) BED and the early (tumor) BED would be 70u1.667 = 117 Gy3 and 70u1.2 = 84 Gy10, respectively. Both the late BED of 117 Gy3 and the tumor BED of 84 Gy10 correspond of course to the NTD or EQD

of 70 Gy. Those four values of BED form useful reference points when con- sidering other schedules. In particular, the late com- plications BED of 117 Gy3 should not be exceeded except in special circumstances, such as research or IMRT-related “new” schedules. The NTD can be obtained by dividing the BED by 1.667 for “late com- plications” or by 1.2 for the “early or tumor effects”, respectively, as described in Table 1.1 above.

Other examples have been worked out in detail in the Appendix (Section 1.8), starting with 60 Gy in 4 Gy fractions.

1.3.8

What is the Standard of Precision of These Estimates of BED or NTD? Gamma Slopes

How much margin of tolerance can we allow before being concerned about an overdose to normal tis- sues, above the arbitrary but currently practical limit of 117 Gy

? And how much should we be con- cerned about tumor BEDs below the 70 Gy NTD or EQD

, that is below a BED of 84 Gy

? These ques- tions require whole papers to answer them properly, involving the slopes of dose–response curves. These slopes are often called gamma-50 or gamma-37, a term derived from photographic film density, mean- ing a change in absolute percentage response for a 1%

change in total dose or NTD (and BED if fraction size is not changed). However, a common sense answer is that about a 7% change in BED or total dose at 2 Gy fraction size would probably be large enough to be clinically noticeable, and a 3% change would probably not, until very many hundreds of patients had been treated in the same way. The degree of anxiety about an over- or under-dose would also depend on the actual circumstance of what risk is most important for the individual patient.

Gamma-50 or gamma-37 is used to define the steepest part of a dose–response curve, which is at 50% probability for logistic modeling, but at 37%

for Poisson probability modeling. The difference is rarely important in comparison with the precision of clinical results. For many types of tumor gamma- 50 or -37 slopes found are about 1.5 to 2 times the percentage change in total dose (and BED if dose per fraction is unchanged). This is in the middle range

of the sigmoid response curve between about 20%

and 70% for local control (LC). So tumor response often corresponds to a gamma slope of 1.5 or 2. For doses above or below that middle range, the slopes will of course be smaller. For certain types of tumor that have been well classified in detail so that narrow risk categories are available, the change of response with BED may be greater than twice the percentage (Hanks et al. 2000; Regnan et al. 2004). In general a 10% change in tumor or NTD

or EQD means a change in LC of approximately 20%. Wadsley and Bentzen (2004) showed that there was a highly statistical relationship (P=0.00017) between loco- regional control (LRC) and later overall survival in head-and-neck cancer, 5-year overall survival aver- aging two-thirds of the 2-year LRC.

For normal tissues, the dose–response curves are often steeper than for tumors, because normal tis- sues are less heterogeneous in physiological proper- ties than tumors are. Gamma-50 slopes of three or four in the steepest range of late complications are found, although slopes are shallower at the lower incidences of a few percent that are common for serious complications; the “toe” of the curve (and at the top, should we see any result close to 90% or 100%).

1.4

Rejoining Point for Those Who Skipped:

How to Evaluate a New Schedule – Brief Summary

In case some of these details might have slipped your mind, we now summarize the three main weapons in our armamentarium for assessing any proposed schedule, before irradiating any patient.

For example, we may wish to add an introductory phase-I dose-escalation study of effects in normal tissues before deciding on a particular dose level in a long clinical trial. A calculation of BEDs or NTDs would enable any arm to be evaluated theoretically in relation to known schedules. Such theoretical comparisons do not replace phase-I clinical trials, but they can put them into critical perspective and save time-wasting steps with dose intervals that may be too wide or too narrow.

x Calculate late complications BED, DE= 3 Gy, with no time factor [Eq.1. 4], keeping below 117 Gy

. x Estimate tumor BED including repopulation,

as Gy10 [Eq. 1.7]. For head-and-neck or lung

tumors assume:

Tk = 21d, Tp = 3d, D= 0.35 log

Gy, and DE= 10 Gy, until further data allow the parameters to be re- estimated; or as log

cell kill = E = DuBED/2.303;

and compare with known schedules.

x Calculate acute mucosal BED Gy

; assume Tk = 7d, Tp = 2.5d, DE= 10 Gy, a = 0.35 [Eq. 1.7]; keeping below 59–63 Gy

, for oral and pharyngeal mucosa, until further evidence is obtained (Fowler et al.

2003c). Possibly higher than 63 Gy

for late rectal mucosa, but with strict volume limits.

1.5

Now Let Us Study Some of the Best-Known Schedules for Head-And-Neck Tumor Radiotherapy

Take a deep breath and read each line as slowly as you like! Make sure a glass or cup of your favorite drink is at your elbow. Stop skipping for once!

Baumann et al. (2002) have written an excellent general review of “Altered Fractionation”, which includes both hyperfractionation (smaller doses per fraction and higher total doses) and accelerated fractionation (shorter overall times and lower total doses). Bourhis et al. (2004) reviewed the results of 15 randomized trials between 1970 and 1998. They con-

cluded that an absolute benefit on LRC of 7%, from 46% to 53%, at 5 years was obtained, depending on type of schedule. Many of those trials and some more recent ones are included in the present analysis.

Table 1.2 lists some of the strongest published radiotherapy schedules that have been used to treat patients with oral and pharyngeal cancer. The first five were successful schedules, with one possible exception, but the bottom three in the table gave too many complications, so that they were later revised to become clinically acceptable. We can learn from these three pairs of later-modified schedules.

1.5.1

Standard Fractionation

The first line in Table 1.2 shows a standard treatment of 2 Gyu35F = 70 Gy in 7 weeks (SF). This schedule also gives the standard top level of “late BED” at 116.7 Gy

. The acute mucosal BED of 53.1 Gy

for this schedule is comfortably below our recently derived “tolerance zone” of 59–63 Gy

for oral and pharyngeal irradiations (Fowler et al. 2003c) and so acute mucosal reactions appear tolerable for the tissue volumes usual in head and neck radiotherapy, including some coning down, in accord with known clinical experience for 7-week schedules.

Table 1.2. Modeling landmark head-and-neck schedules. Parameters for tumor log

cell kill: DE= 10 Gy; D= 0.35 lnGy;

Tk = 21 days; Tp = 3 days; of which Dis inversely proportional both to log cell kill and to BED (= E/D) in this formulation; so its actual value is not a sensitive factor. For acute mucosal BED: DE= 10 Gy; D= 0.35 lnGy; Tk = 7 days; Tp = 2.5 days

Schedule (reference) Dose/fraction × number of fractions

Total dose (Gy)

Overall time (days)

Tumor log

cell kill

Late effect BED (Gy

)

Acute muco- sal BED (Gy

)

1. Standard 7 weeks 2 Gy × 35F 70 46 10.26 116.7 53.1

2. EORTC HFX 7 weeks (Horiot et al. 1992)

1.15 × 70F 81.5 46 11.13 113 58.9

3. RTOG HFX 7 weeks (Fu et al. 2000)

1.2 Gy × 68F 81.6 45 11.48 114 61.3

4. Concomitant boost 6 weeks (Knee et al. 1985)

1.8 Gy × 30F + 1.5 Gy ×12F 72 39 11.03 113 59.1 5. Wang split course 6 weeks

(Wang 1988)

1.6 Gy × 42F 67.2 39 10.04 103 52.6

6. 4–5 weeks

(Leborgne et al. 2000)

1.6 Gy × 42F–

y×– 44F

64–

–70.4 28–

–33

11.15 11.21

103 108

61.3 61.1 7. GORTEC 1 3 weeks

(Bourhis et al. 2000)

2 Gy × 32F 64 21 11.47 107 64.9 ??

8. CAIR 1 5 weeks (Maciejewski et al. 1996)

2 Gy × 35F 70 34 11.46 117+? 62.6 ?

9. HARDE 1 5 weeks 1.4×20F+1.6×10F+

(McGinn et al. 1993)

1.2 Gy × 20F + 2 Gy × 4F 76 33 12.01 112 66.2 ???

? “Probably too high to be clinically tolerable”

1.5.2

Hyperfractionation

The second row of Table 1.2 shows the EORTC hyper- fractionation clinical trial of 1.15 Gy twice a day for 70 F = 81.5 Gy in 7 weeks, which was the first random- ized clinical trial to show an advantage for a non- standard fractionation schedule, with 325 patients in two arms (Horiot et al. 1992). The control arm was 70 Gy in 7 weeks (Row 1). The increase of 5y LC at 3 years was from 40% to 59% with good significance at P=0.02. The difference in log cell kill with our parameters was 0.8 log

, so if the gamma-50 slope was 2.0 for these tumors, a difference in LC of about 16% might have been expected. The observed differ- ence of 19% was not statistically significantly dif- ferent from this expected gain. The late complica- tions were not significantly different in the two arms.

Although they were slightly higher in the hyperfrac- tionated arm, which was not expected, the low inci- dence of late effects did not allow the difference to be significant. The interval 't between two fractions per day is emphasized as being critical in any schedule.

The acute mucosal reactions in Gy

were strong at 58.9 Gy10, just below the lower end of our arbitrary

“tolerance zone” (Fowler et al. 2003c), but tolerable.

There is less repairable damage to worry about if DE is 10 Gy than if it is 3 Gy, of course, in the RE bracket.

It is also less for smaller doses per fraction.

1.5.3

Radiation Therapy Oncology Group Four-Arm Fractionation Trial (RTOG 90-03)

The Radiation Therapy Oncology Group (RTOG) was then encouraged to set up its four-arm randomized clinical trial 90-03, which ran from 1990 to 2000 and accumulated close to 280 patients in each arm (Fu et al. 2000). Table 1.2 shows that it consisted of:

Row 1: the standard 2 Gyu35F in 7 weeks, fi ve treat- ments a week (designated SF).

Row 3: a hyperfractionation arm [of 68Fu1.2 Gy twice a day (b.i.d.) = 81.6 Gy in 7 weeks (HFX)].

Row 4: the concomitant boost at 72 Gy in 6 weeks, with 1.8 Gy given daily for 6 weeks plus 1.5 Gy at two fractions a day in the fi nal 12 treatment days, designated AF-C.

Row 5: the CC Wang split-course accelerated sched- ule of 1.6 Gy 2 fractions a dayu42F = 67.2 Gy in 6 weeks, designated AF-S. This had a 2- week planned gap in it.

All these schedules are summarized in the first five lines of Table 1.2, including their “late BED”

in Gy

, tumor cell kill in log10, and “acute muco- sal BED” in Gy

. Two of these schedules in RTOG 90-03 yielded 54% 2y LC, and two yielded 46%, the difference being significant (Fu et al. 2000). Both of the 54% arms gave more than an estimated 11 log

cell kill, using our stated parameters. Both of the 46% arms gave barely over 10 log

of cell kill, as shown in Table 1.2. These results give a guide to the calibration of our currently used radiobiologi- cal model parameters for tumor BED. A difference of one log

represents a difference of about 10% in log cell kill, from 10 log

to 11 log

and, therefore, up to about 20% in predicted change of percentage LC for tumors whose gamma-50 is about two. Non- uniformity between tumor sizes and stages could flatten this dose–response gamma-50 from 2 to about 1, as appears to have happened in this trial.

About 30% of the patients were T4 or N2A, whereas those in the EORTC trial previously mentioned were T2–T3 and N

, so the difference in gamma-50 is easy to explain.

How do the parameters for complications in normal tissues hold up for consistency? Only one of the “late” BEDs in the table and none of these first five “acute mucosal” BEDs were above the limits we have described above (117 Gy

for “late”

and 59–63 Gy

for “acute mucosal”). However the acute Gy

for concomitant boost is in the modeled

“maximum mucosal tolerance zone” at its lower edge, which does not contradict its acute reactions reported as the highest of the four arms from the RTOG trial. Concomitant boost is an ingenious method of shortening a previous 8-week schedule so that the overall time was reduced to 6 weeks (Knee et al. 1985). The coned-down boost dose was inserted as a second dose of 1.5 Gy on the last 12 treatment days of the basic 6 weeks of 1.8 Gy frac- tions (with 't=6 h), so that two fractions a day were required in only the last 2.5 weeks.

The finally listed arm of RTOG 90-03 was the C.C. Wang schedule using 1.6 Gy fractions twice a day, with a planned break of 2 weeks after 32 Gy in 2 weeks (row 5, Wang 1988). This offered the lowest acute mucosal BED, the lowest “late effect” BED of 103 Gy

and, not surprisingly, the lowest predicted tumor log cell kill, equal to that of the standard 70 Gy in the 7-week schedule. The originator, Dr. C.C.

Wang, often added one or two fractions if reactions

in individual patients justified doing so, but this

was not done in RTOG 90-03. However, Drs. J. and

F. Leborgne in Montevideo have explored that elec-