Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics 3 1 Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics Jack F. Fowler CONTENTS 1.5.8 Conclusions Re Head-and-Neck Schedules 21 1.5.9 Concurrent Chemotherapy 22 Glossary 3 1.6 Hypofractionation for Prostate Tumors 22 1.1 Introduction 6 1.7 Summary 23 1.2 The Simplest Modeling 7 1.8 Appendix: Is This a Mistaken Dose 1.2.1 The Seven Steps to LQ Heaven – Brief Summary 7 Prescription? 24 1.3 The Seven Steps to LQ Heaven – The Details 8 1.8.1 For Equal “Late Complications” 25 1.3.1 Development of the Simple LQ Formula E = nd 1.8.2 For Equal Tumor Effect 25 (1+d/D/E) 9 1.8.3 Acute Mucosal Effects 26 1.3.2 Biologically Effective Dose 10 1.9 Line-by-Line Worked Examples: 1.3.3 Relative Effectiveness 11 Details of Calculations of the Schedules 1.3.4 Overall Treatment Time 12 Discussed in This Appendix 27 1.3.5 Acute Mucosal Tolerance 13 References 29 1.3.6 To Convert from BED to NTD or EQD2 Gy 14 1.3.7 One Example 14 1.3.8 What is the Standard of Precision of These This chapter is written mainly for those who say “I Estimates of BED or NTD? Gamma Slopes 15 don’t understand this DE business – I can’t be both- 1.4 Rejoining Point for Those Who Skipped: How to ered with Linear Quadratic and that sort of stuff.” Evaluate a New Schedule – Brief Summary 15 Well, it might seem boring – depending on your per- 1.5 Now Let Us Study Some of the Best-Known Schedules for Head-And-Neck Tumor Radiotherapy 16 sonality – but it is easy, and it makes so many things in 1.5.1 Standard Fractionation 16 radiation therapy wonderfully and delightfully clear. 1.5.2 Hyperfractionation 17 Experienced readers can turn straight to Section 1.4. 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 J. F. Fowler, DSc, PhD Using BED to Adjust Individual Doses 19 Emeritus Professor of Human Oncology and Medical Physics, 1.5.6 General Considerations of Medical School of University of Wisconsin, Madison, Wiscon- Head-and-Neck Radiotherapy 19 sin, USA; Former Director of the Gray Laboratory, Northwood, 1.5.7 A Theoretical Calculation of “Close to Optimum” London, UK Head & Neck Schedules: 3 Weeks at Five Fractions Present address: per Week 20 150 Lambeth Road, London, SE1 7DF, UK Glossary D, alpha Intrinsic radiosensitivity. Loge of the number of cells sterilized non-repairably per gray of dose of ionizing radiation. E, beta Repair capacity. Loge 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. 4 J. F. Fowler 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. Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics 5 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 loge Natural logarithm, to base e. One log10 is equal to 2.303 loge. Log10 Common logarithm, to base 10. “Ten logs of cell kill” are 23.03 loge of cell kill. LQ Linear quadratic formula: loge 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.