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 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.