Basics of linear quadratic model

1. α/β:Where does it stand today??

2. Cell Kill
• Radiation causes damage to the “target”
• Target can be single or multiple
• A lethal event for the cell can occur by the following ways
– Single or multiple hits of single or multiple targets
• A target in a cell could be the nucleus or the cytoplasm

3. Targets

4. Mathematical Models
• These depict the survival patterns of studied cells exposed to
a range of radiation doses.
• Two basic models
– Target model
– Repair model
• Most widely used in clinical practice
– Linear-Quadratic model

5. Target models
• Single hit - single target
• Single hit – multi target
• Two hit – single target [Quadratic ]
• Two component model
Single hit-single target + two hit-single target [Linear-
Quadratic]
Single hit-single target + single hit-multiple target [Multi-
Target]

6. Single hit on a target
S = e-D/D0 = e-αD
Single hit and multiple hits on targets
S = e-αD-βD2
Multiple hits on a target
S = e-βD2
Single hit/target +
Single hit on multiple targets
S=e-D/1D0[1-(1-e-D/D0)n]

7. Linear cell survival and Quadratic cell
survival

8. Concept of α & β
• α : Component of killing which is proportional to the dose
• β : Component of killing which is proportional to square of the
dose.
• S = e -αD- βd
2
and α/β ratio is the dose at which linear and
quadratic components of killing are equal
• E/α = BED = (1 + d / (α/β)) * D = RE * D
Relative Effectiveness (RE)

9. LQ MODEL IS APPROPRIATE
• Assumptions:
– Cell killing is primarily a result of DNA damage (DSBs)
– For multifractionated treatment, the fractions are well
separated in time
– Irradiation time for EBRT is short and with a constant dose
rate
– α/β for tumour is 10 Gy, and for late complications is 3 Gy
– No tumour repopulation occurs within 2 weeks, and
– LQ modelling is valid up to 23 Gy per fraction

10. LQ MODEL: Hypo fractionated
Treatments
 In vitro studies: LQ Model fits well between 0-16 Gy
*Garcia LM et al Phys Med Biol 2006;51:2813–2823
 In Vivo studies :Also shows that LQ model reliably predicts
dose response between 2-20 Gy
Barendsen GW et al. Int J Radiat Oncol Biol Phys 1982;8:1981–1997
Van der Kogel AJ. Radiat Res Suppl 1985;8:S208–S216
Peck JW, Gibbs FA. Radiat Res 1994;138:272–281

11. α/β values: fixed or changing??
• Traditional concept:
– α/β for rapidly proliferating tissues and tumor
– α/β for slowly proliferating tissues and late responding
tissues
• Recent concept:
– α/β values depend on the end points chosen and also proliferating
capacity of tumors
– α/β =3.5 Gy for AV malformations and for <3 cm 4.6-6.4 Gy*
– α/β =1.5-2 Gy $ for for biochemical failure in prostate cancers and
another study reported as α/β >30 Gy #
– α/β =8.5 for well oxygenated and 15.5 for hypoxic colonogens of
prostate cancer@
*Kocher M et al. Radiotherapy and oncology 2004;71:109-114
# Richard Shaffer et al. Int. J. Radiation Oncology Biol.Phy. 2011;79;4:1029-36
$ Bentzen M Soren et al. Radiotherapy and oncology. 2005;76:1-3
@ Nahum A E et al. Int. J. Radiation Oncology Biol.Phy. 2003;57;2:391-401

12. So What`s the problem??

13. LQ Model: Application in SRT and
SBRT
 Leith JT et al*. calculated the radiation doses required to
control metastatic brain lesions using data from in vitro
survival curves.
– Found that the calculated dose required to obtain
a high tumour control probability was at least 25
to 35 Gy Vs 15-20 Gy (clinical doses)
 Karlson et al**. Found in the treatment of Arterio-venous
malformations that 42 Gy/12# was more effective than the
equivalent dose of 15 Gy/1# by LQ model
*Acta Neurochir Suppl 62:1827,1994.
**Neurosurg 57:42-49, 2005

14. What happens when LQ Equation is
used to calculate dose equivalence at
high dose per fraction

15.  Radiation Effect on Vasculature and supporting cells:
 Ionizing radiation can damage supporting tissue such as microvasculature,
mainly at high doses
 Vascular endothelial damage is triggered above 10 Gy per fraction secondary to
the activation of acid sphingomyelinase*,#
*Garcia-Barros M et al. Science 300:1155-1159, 2003
# Fuks Z et al. Cancer Cell 8:89-91, 2005
 Radiation Effects on tumor stem cells:
 Relatively radio resistant Stem cell play a role in malignant gliomas and other
tumors
 Stem cells have a radiation threshold after which they die (Usually >17-18
Gy)**
**Chang HJ, et al. Nat Med 11:484-490,2005

16. Modifications of LQ Model
 LQ-L Model: Accounts for high dose per fraction #
# Guerrero M et al. Phys Med Biol 49;4825-4835;2004
 Biophysical Models: To take in to account the dynamic
interaction of radiation and tissues*
– Γ-LQ Model (Kinetic model),
*Scheidegger S, et al. Z Med Phys .2011 Sep;21(3):164-73. 2011
 LQ-R Model: Accounts Redistributions and Re-oxygenation**
** Brenner D J et al,Int. J. Radiation Oncology Biol. Phys.1995;32;2:379-390

17. Modifications of LQ Model (Cont..)
 Modified LQ Model : Accounts for increased overall treatment
time
E = - ln S = n * d (α + βd) - γT
Including T ("kick off time") which allows for a time lag before
the tumor switches to the fastest repopulation time:
 LQ Model for Low dose rate brachytherapy:
Where g represents how much less effective the repairable
component has become

18. Survival curve with LQ-L Model

19. • Adds up the LQ and the
Multi target model.
• Gives Equivalent
functions: SFED (Single
fraction equivalent
dose),BED and SED
(standard effective dose)
for comparison of various
SBRT regimens

20. LQ Model: High LET vs. Low
LET Radiation
• Assumes high LET radiation
affects α co-efficient of damage
more than β co-efficient.
• Introduced a new parameter
RBEM
αH=αLRBEM

21. LQ Model: Conclusion I
 It is a mechanistic, biologically-based model.
 It has sufficiently few parameters to be practical
 It is reasonably well validated, experimentally and
theoretically, up to about 10 Gy /fraction, and would be
reasonable for use up to about 18 Gy per fraction
 To date, there is no evidence of problems when LQ has been
applied in the clinic within above constraints.

22. LQ Model: Conclusion II
 It does not take in to account the non-DNA targets of radiation.
 It overestimates the BED at doses used in SBRT and SRT
treatments.
 It can not be used to equate doses between high LET and low
LET radiations
 It does not take in to account the cell kinetics and interaction
of radiation with other factors.

23. Thanks
Do we have an
alternative to LQ Model
:No
So we have to stay with
it??