# 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, βD$^2$)
** Y = αD + βD$^2$
* 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 + GβD$^2$
* 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 + GβD$^2$)

## 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.