## Linear-quadratic modelEdit

- 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}

- Y = αD + βD
- 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}

- Y = αD + GβD
- 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})

- SF = exp -(αD + GβD

## Protracted RadiationEdit

*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*= extrapolated tolerance dose*D*= total dose (Gy)*RE*= relative effectiveness per unit dose

For fractionated treatments:

*d*= dose per fraction (Gy)*n*= the number of total fraction

For protracted irradiation (constant dose rate):

*R*= dose rate, LDR (Gy/hr)- = sublethal damage repair exponential time constant (Liters/hr).
*T*= treatment time (hr)

is approximately the same as,

- ,
- 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 DoseEdit

**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 ModelsEdit

**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

## ReferencesEdit

- 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.