Radiation Oncology/Radiobiology/Classic

Linear-quadratic model

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

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

dn = dose per fraction (Gy)

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

R = dose rate, LDR (Gy/hr)

$\mu$ = sublethal damage repair exponential time constant (Liters/hr).

$\text{also, }\mu =\frac{\ln 2}{T_{\frac{1}{2}}}\text{, where }{T}_{\frac{1}{2}}\text{ is the half life of sublethal damage repair}$

T = treatment time (hr)

is approximately the same as,

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

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

Universal Survival Curve

• 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.