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

  • Jack F. Fowler
Part of the Medical Radiology book series (MEDRAD)


This 9-section chapter begins with an elementary explanation of the Linear Quadratic model of Radiation Response, to make sure readers haven’t missed out on understanding this robust and reliable way of comparing different schedules in Radiation Oncology. A detailed account of its many applications has recently been published as “21 Years of BED (Biologically Effective Dose)” (Fowler Br J Radiol 83:554–568, 2010). The essential feature of this modeling is that a given dose has very different biological effects on neighbouring but different tissues because of their biological alpha/beta ratios and their “kick-off” or “onset” times of repopulation during continuing irradiation. In Sections 4 and 5 comparisons of actual clinical trials are presented that have shaped the current and emerging schedules of treatments of Head & Neck tumors, going on to SBRT (Stereotactic Body Radiation Therapy), IMRT (intensity Modulated Radiation Therapy) and IGRT (Image Guided Radiation Therapy). Some insights into how the biological strategies of fractionated radiotherapy actually deliver therapeutic advantages are introduced. Section 6 explains how Optimum Overall Times can now be predicted, using basially least two constraints, one for Late Complications and the other for Acute Tolerance Doses. This is a fairly new break-through (2008). Section 7 goes into detail on why Overall Times might be too short or too long, with examples from modern schedules still being clinically trialed. Section 8 goes further into non-standard schedules, with updated emphasis on the Recovery Times of various tissues and tumors, intervals between fractions and extended fraction times. The chapter ends with an explanatory table of best and next-to-best schedules for Head and Neck radiation oncology. The continuing need to obtain data on individual tumor T-\( {\raise0.5ex\hbox{$\scriptstyle {1}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {2}$}} \) and repopulation is emphasized.


Late Complication Stereotactic Body Radiotherapy Relative Effectiveness Biologically Effective Dose Linear Quadratic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


α, alpha

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

β, beta

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

α/β, 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 complications. This difference is vital for the success of radiotherapy. When beta (β) is large, both mis-repair and good-repair are high. It is the mis-repair that causes the cell survival curve to bend downward

Accelerated fractionation

Fractionated schedules with shorter overall times than the conventional 7 (or 6) weeks


Biologically effective dose, proportional to log cell kill and therefore more 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 (α) of the target cells when all repairable radiation damage (β) has been given time to be repaired. In linear quadratic modeling, BED = total dose × relative effectiveness (RE), where RE = (1 + d/[α/β]), with d = dose per fraction, α = intrinsic radiosensitivity, and β = repair capacity of target cells


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


Confidence interval (usually ±95%)


Continuous low dose rate


Constructive lengthening: when adding a day (or two) followed by a not-too-small fraction (or two) adds to the accumulated radiation damage in the tumor, rather than allowing it to fall by tumor repopulation, or minimizes any loss


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 planning treatment volume (PTV)


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


External beam radiation


Epithelial growth factor receptor, one of the main intracellular biochemical pathways controlling rate of cell proliferation


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 added digit “2” EQD2 Gy


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 α/β ratio for the tissue concerned


18F Fluorothymidine, a radioactive label for freshly synthesized DNA to indicate actively dividing cells. The radioactive label 18F emits positrons

Gamma, γ-50, γ-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 γ-50 and γ-37 is rarely clinically significant


Dose rate factor. A number less than 1 that describes the decrease of biological effect if the duration of irradiation is longer than a few minutes

Gy, gray

The international unit of radiation dose: one joule per kilogram of matter. Commonly 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 α/β 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


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


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


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


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


Image Guided Radiotherapy. Using superimposed images from CT-scans or Magnetic Resonance Imaging or PET-scans


Intensity Modulated Radiotherapy: instead of a constant dose rate from all angles, the dose rate is made to vary with the angle from which it is delivered, by computerized dose planning; leading to deliberately non-uniform dose-planning and ‘dose-painting’ in tumors or ‘dose-avoidance’ of critical organs

IR, Incomplete Recovery

Residual radiation damage that may add to the effect of the next fraction if a too short interval occurred (Thames and Hendry 1987). Repair usually refers to intracellular repair. Recovery refers to other processes too and is a more general term


Equal effect


Local control (of tumors)


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 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


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


Linear quadratic formula: loge cells killed = α × dose + β × dose-squared


A linear cell survival curve suggested by some authors to replace the higher dose downward curvature of a standard LQ curve, which some authors fear, probably wrongly, but the issue is not yet resolved

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%


Loco-regional tumor control. LC would be local control


Magnetic Resonance Imaging. Scans of body tissues which can show the chemical state of molecules, instead of only their density as CT scanning does


Normal tissue complication probability


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 α/β = 3 Gy and no overall treatment time factor) than for tumor effect (with α/β = 10 Gy and an appropriate time factor)


Non-small-cell lung cancer


The use of a few large fractions, say 5–20 Gy or higher, and only a few of them, say ten or less (Ling et al. 2010)


Positron Emitting Tracer. A radioactive nuclide that emits positrons, that is, a pair of oppositely charged electrons in exactly opposite directions, so that they can be detected in PET Scanning within a few millimeters of accuracy to indicate parts of a tumor that might contain dangerously live cells

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 survive, P = 14%. If 0.1 cells survive on average, P = 90%. This curve is steepest at the probability of 37%


Planning treatment volume—larger than CTV to allow for setup and treatment-planning errors


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


Proportion of a dose fraction that is recoverable, the beta-only term, which is d/[α/β] or Gd/[α/β] as a proportion of the whole RE = (1 + d/[α/β]) or (1 + Gd/[α/β]). The “1” part of the RE is the non-recovering, fixed part, independent of time after irradiation


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


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


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

Red Shell

An annuloidal shell surrounding a PTV, during treatment planning, to delineate tissues at risk from late reactions when prescription doses exceed normal tissue tolerance of nearby organs, at mm distances before sufficient dose falloff has occurred (Yang et al. 2010)


Radiation Therapy Oncology Group, USA


Surviving fraction after irradiation, usually of cells


Simultaneous Internal Boost. The addition of a deliberate “hot spot” into a planned non-uniform tumor dose distribution to enhance the local effect; a form of ‘dose painting’ by IMRT


Stereotactic Body Radiatiotherapy. Very accurately guided beams, often of small diameter and usually delivered by only a few large dose fractions, to treat cancer in certain organs outside the brain


Stereotactic Radiosurgery: it usually means in radiotherapy the use of a single treatment fraction, often in brain. Originated from the precise localizations in brain physiotherapy research. Has sometimes been wrongly used for SBRT


Potential doubling time of cells in a population; before allowing for the cell loss factor. T pot 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


Cell doubling time in a tissue during radiotherapy; probably somewhat faster than T pot. Determined from gross tumor (or other tissue) results when overall time is altered


Kick-off or onset time: the apparent starting time of rapid compensatory repopulation 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 T k, then constant doubling each T p days until end of treatment at T days. Accelerating repopulation is discussed in Sect. 5.6


Tumor control probability


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© Springer-Verlag Berlin Heidelberg  2011

Authors and Affiliations

  1. 1.Human Oncology and Medical PhysicsMedical School of University of WisconsinMadisonUSA
  2. 2.Gray LaboratoryLondonUK
  3. 3.LondonUK

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