Linear quadratic model

The linear quadratic model (LQM) is a formula in which the radiotherapy the biological effect may be compared by different fractional doses. The formula gives the biologically equivalent conventionally fractionated dose for non-conventional fractionations. It uses two tissue-specific factors for a linear and a quadratic term, which on the one hand fits well with the test results and on the other hand can also be justified radiophysically.

In practice, the LQM has replaced the older Strandqvist (1944) and Ellis formulas ( nominal standard dose NSD, 1969).

The radiation doses that are necessary to destroy certain types of cancer, as well as the tolerance doses of normal tissue, are known from many years of experience with conventional fractionated radiation therapy. With conventional fractionation, 1.8–2.0 Gy are irradiated daily until the prescribed total dose is reached. In the case of seriously ill patients or for the treatment of particularly fast-growing tumors, it may be desirable to select shorter intervals than 24 hours ( acceleration ) or higher daily doses than 2 Gy ( hypofractionation ) - up to and including the entire target dose in a single session. The biological effect on the tumor and certain side effects increase considerably if the total treatment time is shortened by days or even weeks, and the so-called late side effects also depend to a great extent on the level of the daily individual fraction.

The strength of the early and late tissue reactions depends on the properties of the irradiated tissue, especially the cell turnover and the number of stem cells that can divide. For example, the top layer of cells in the mucous membrane is replaced every 2-3 days; Radiation damage to the stem cells quickly leads to epithelial loss and serious side effects. On the other hand, the mucous membrane can also recover quickly because of its high proportion of stem cells. A counterexample is bone tissue with low early reactions but limited long-term tolerance.

The quantified clinical radiation reactions (tumor reduction, inflammation, scarring, etc.) are proportional to the dose D and thus to the logarithm of the killed portion S of the irradiated cells S ₀ , because the cell death follows an exponential relationship with the base e:

Surviving cell fraction , and radiation effect (effect) ${\ displaystyle S / S_ {0} = e ^ {- D / D_ {0}}}$ ${\ displaystyle E \ approx D / D_ {0} = - \ ln (S / S_ {0})}$

D ₀ is a fractionation, tissue and effect dependent sensitivity constant that is not known. The formula cannot therefore be used in clinical practice to calculate the expected effect of a dose. Strandqvist and later Ellis therefore tried to add additional terms and exponents for the effects of new fractionation schemes that they had determined in animal experiments, e.g. B. Frank Ellis (1969)

{\ displaystyle NSD = D \ times N ^ {0 {,} 24} \ times T ^ {0 {,} 11}}

with NSD ( nominal standard dose ) = conventionally fractionated dose with the same effect, D = irradiated dose, N = number of fractions, T = total treatment time in days. However, it soon became apparent that the estimates obtained in this way did not apply to all tissues and did not capture the differences between the early and late tissue reactions. In addition, the formula could not be justified from radiation biology.

In 1974 Fowler found a striking correspondence between the results of fractionation experiments on cell cultures and function

${\ displaystyle S / S_ {0} = e ^ {- (\ alpha \ cdot d + \ beta \ cdot d ^ {2})}}$

with d = height of a single fraction and α, β = tissue constants.

The exponent in Fowler's formula has a linear and a quadratic term. Kellerer and Rossi coined the term linear-quadratic model. and suspected a mechanistic explanation ( dual target hypothesis ). Barendsen described the basic applicability for early and late reacting tissues. Although the linear-square model was originally discovered purely empirically or intuitively, it can be justified in terms of radiation biology for the effect of rays with a low LET such as photons and electrons. α · d could represent the probability of an irreparable double-strand break in DNA caused by a single photon or electron, β · d the probability of a single strand break that can in principle be repaired, and β · d ² the probability of two such single strand breaks in a narrow temporal and spatial manner Coincidence that cannot be repaired anytime soon.

For a treatment with several fractions, with = number of fractions, provided that there is enough time for recovery between the fractions (empirically: at least six hours). The tissue factors α and β, like D _{0, are} not known. However, by taking the logarithm and transforming ${\ displaystyle E = n \ cdot (\ alpha \ cdot d + \ beta \ cdot d ^ {2})}$ ${\ displaystyle n}$

{\ displaystyle E / \ alpha = n \ cdot d \ cdot \ left (1 + {\ frac {d} {\ alpha / \ beta}} \ right)}

receive. By definition, the effect E should be constant. E / α is independent of the temporal irradiation pattern, n · d is the total dose of the irradiation series. With the same effect, there is therefore an inverse proportionality between the total dose n · d and the dose per fraction d . The proportionality factor (α / β) ⁻¹ or the ratio α / β can be determined experimentally for different tissues and tumors by plotting dose-effect curves on logarithmic paper.

The less the tissue is subject to the fractionation effect, the higher the α / β. H. the less the risk of long-term damage can be reduced by fractionating the total dose. Typical values are α / β = 2 - 4 for normal tissue that reacts late (lungs, kidneys) and 10-20 for normal tissue that reacts earlier (mucous membrane, bone marrow) and tumors.

If the α / β ratio is known, a dose with the same effect as conventional fractionation (5 × 2 Gy per week, = 2.00 Gy) with regard to the long-term effects can be calculated with a different individual fraction : ${\ displaystyle d_ {0}}$ ${\ displaystyle d_ {1}}$

{\ displaystyle D = D _ {\ text {conv}} \ cdot {\ frac {d_ {0} + \ alpha / \ beta} {d_ {1} + \ alpha / \ beta}}}

For example, bone metastases are conventionally treated with 18 × 2 Gy = 36 Gy. If the single dose is increased to 3 Gy and the number of fractions reduced to 10, then the irradiated 30 Gy is equivalent to 32.5 Gy on the tumor and bone marrow (α / β = 10) and 36 Gy on the basic bone tissue (α / β = 3). This means that with the same long-term effects as with conventional treatment, the treatment shortened from four to two weeks results in a somewhat lower anti-tumor effect.

Individual evidence

^ The New Treatments in Radiotherapy - Altered Fractionation. Gray Annual Report, 1993.
^ F. Ellis: Dose, time and fractionation: a clinical hypothesis. In: Clin Radiol. 20, 1969, pp. 1-7.
^ JF Fowler: The linear-quadratic formula and progress in fractionated radiotherapy. In: Br J Radiol. 62, 1989, pp. 679-694.
↑ AM Kellerer, HH Rossi: A generalized formulation of dual radiation action. In: Radiat Res. (75), 1978, pp. 471-488.
^ GW Barendsen: Dose fractionation, dose rate and isoeffect relationship for normal tissue response. In: Int J Radiat Oncol Biol Phys . (8), 1982, pp. 1981-1997.

literature

Thomas Herrmann, Michael Baumann: Clinical radiation biology. Fischer, Jena 1997, ISBN 3-437-31140-9 .

Web links

Biological equivalent dose calculator (German)

[1] The New Treatments in Radiotherapy - Altered Fractionation. Gray Annual Report, 1993.

[2] F. Ellis: Dose, time and fractionation: a clinical hypothesis. In: Clin Radiol. 20, 1969, pp. 1-7.

[3] JF Fowler: The linear-quadratic formula and progress in fractionated radiotherapy. In: Br J Radiol. 62, 1989, pp. 679-694.

[4] AM Kellerer, HH Rossi: A generalized formulation of dual radiation action. In: Radiat Res. (75), 1978, pp. 471-488.

[5] GW Barendsen: Dose fractionation, dose rate and isoeffect relationship for normal tissue response. In: Int J Radiat Oncol Biol Phys . (8), 1982, pp. 1981-1997.