Radiation Oncology/Radiobiology/Linear-Quadratic

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 T1/2
 * 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