Radiation Oncology/Radiobiology/Linear-Quadratic

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Linear-Quadratic Formalism

Linear-quadratic model

A model which describes cell killing, both for tumor control and for normal tissue complications
Most common underlying biological rationale is that radiation produces a double strand DNA break (DSB) using a single radiation track
Individual DSB can be repaired, with first order kinetics and half-life T_1/2
If more than one unrepaired DSB is present in the cell at the same time (arising from two separate radiation tracks), a misjoining can produce a lethal lesion (e.g. dicentrics)
The two separate DSB can happen at different times during treatment, allowing for repair of first DSB prior to misjoining with the second DSB
A single radiation track can also give rise to a lethal lesion by itself (e.g. point mutation in vital gene, deletion eliminating vital gene, induced apoptosis, etc)
In the LQ formalism, the yield of lethal lesions is the sum of lethal lesions produced from a single radiation track (which are linearly related to dose, αD) and lethal lesions produced from two radiation tracks (which are quadratically related to dose, βD²)
- Y = αD + βD²
Because the two separate DSB can be repaired prior to resulting in a lethal event, the second component is modified by the Lea-Catcheside time factor (G) to show dependence on dose protraction. For single fractions, G=1
- Y = αD + GβD²
Lethal lesions are thought to follow Poisson distribution from cell to cell. Therefore, the surviving fraction (SF) is
- SF = exp -(Y)
This leads to the standardized LQ equation
- SF = exp -(αD + GβD²)

Protracted Radiation

$SF=\exp -(\alpha D+\beta D^{2})$

SF = surviving fraction

First proposed by Douglas and Fowler in 1972 (PMID 1265229 - Douglas BG and Fowler JF. The effect of multiple small doses of X-rays on skin reactions in the mouse and a basic interpretation. Radiat Res 66, 401-26, 1976.)

E = -ln SF

E = biological radiation effect

$ETD=E/\alpha =D[1+D(\beta /\alpha )]=D\times RE$

ETD = extrapolated tolerance dose

D = total dose (Gy)

RE = relative effectiveness per unit dose

For fractionated treatments:
$RE=1+dn(\beta /\alpha )$

d = dose per fraction (Gy)

n = the number of total fraction

For protracted irradiation (constant dose rate):
$RE=1+(2R/\mu )(\beta /\alpha )\left\{1-(1/\mu )T\left[1-\exp(-\mu T)\right]\right\}$

R = dose rate, LDR (Gy/hr)

\mu

= sublethal damage repair exponential time constant (1/hr).

{\mbox{also, }}\mu ={\frac {\ln 2}{T_{\tfrac {1}{2}}}}{\mbox{, where }}T_{\tfrac {1}{2}}{\mbox{ is the half life of sublethal damage repair}}

T = treatment time (hr)

is approximately the same as,

RE=1+(2R/\mu )(\beta /\alpha )\left\{1-(1/\mu )T\right\}

,

for values of T: 10 hr > T > 100 hr.

Glasgow; 1998 PMID 9572622 -- "The linear-quadratic transformation of dose-volume histograms in fractionated radiotherapy." (Wheldon TE, Radiother Oncol. 1998 Mar;46(3):285-95.)
- Radiobiological transformation of physical DVH to incorporate fraction size effects
- Outcome: "hot spots" and "cold spots" are further from mean than physical distributions indicate; particularly important in plans with significant dose heterogeneity
- Conclusion: LQ-DVH should be computed in parallel with conventional DVHs

LQ and High Fractional Dose

Duke; 2008 PMID 18725110 -- "The linear-quadratic model is inappropriate to model high dose per fraction effects in radiosurgery." (Kirkpatrick JP, Semin Radiat Oncol. 2008 Oct;18(4):240-3.)
- Counterpoint argument to PMID 18725109.
- LQ model does not reflect vascular and stromal damage produced at high doses per fraction, it also ignores impact of radioresistant subpopulations of cells such as cancer stem cells

Columbia; 2008 PMID 18725109 -- "The linear-quadratic model is an appropriate methodology for determining isoeffective doses at large doses per fraction." (Brenner DJ, Semin Radiat Oncol. 2008 Oct;18(4):234-9.)
- Point argument to PMID 18725110
- Linear quadratic model is reasonably well validated for doses up to 10 Gy/fraction, and could be reasonably used to about 18 Gy/fraction

Extended LQ Models

Ohio State; 2010 PMID 20610850 -- "A generalized linear-quadratic model for radiosurgery, stereotactic body radiation therapy, and high-dose rate brachytherapy." (Wang JZ, Sci Transl Med. 2010 Jul 7;2(39):39ra48.)
- Generalized LQ model (gLQ) developed. Compared to in vitro data. Able to extrapolate up to 11-13 Gy from low dose data

UT Southwestern; 2008 PMID 18262098 -- "Universal survival curve and single fraction equivalent dose: useful tools in understanding potency of ablative radiotherapy." (Park C, Int J Radiat Oncol Biol Phys. 2008 Mar 1;70(3):847-52.)
- Hybridization of two classic radiobiologic models: LQ model and multi-target model. LQ model good for conventionally fractionated therapy; multi-target model good for high (ablative) fractional doses seen in SBRT
- Allows for easier conversion of doses

References

PMID 8631555 - Liu WS et al. Determination of the appropriate fraction number and size of the HDR brachytherapy for cervical cancer.Gynecol Oncol. 1996 Feb;60(2):295-300.