Examining the evidence behind the use of the linear quadratic formalism in extreme hypofractionation and high dose radiotherapy

1. To use or not to use the LQ model
at “high” radiation doses per
fraction
Dr Santam Chakraborty
Assistant Professor, Tata Memorial Hospital, Mumbai

2. Why are we afraid?
● Singh et al 1970s treated 24 patients with
stage III Ca Cx with 5.8 Gy in 5 fractions
one week apart.
○ Equivalent to NSD of 2 Gy in 20 fractions
○ Equivalent acute toxicity
○ Equivalent tumor control
● All patients developed RVF starting 7
months after completion of RT.

3. Probability Theory
The probability of two mutually
independent events occurring together
is the product of two individual
probabilities.
Probability of getting a “dui chokka” is
1/6 x 1/6 = 1/36

4. Basics of LQ Model
Cell Kill leads to
biological effects
DNA damage primary
effect of Radiation
Modelling DNA damage mechanism can
model radiation effect
Biggest utility in designing new dose fractionation schedules in a rational way

5. Mechanistic Basis of LQ Model
DSB
1 2
P ∝ D
Pair of DSB in both strands
of chromosome (Dicentric)
Failure of
Repair

6. Mechanistic Basis ...
● Probability of the two double strand breaks interacting in a
chromosome to produce lethal events (e.g. Dicentric) ∝ D x D = D2
● A single radiation track producing lethal lesion
● The probability of this occurrence is ∝ D
Thus,
● Probability of Lethal lesions ∝ D2
+ D
● Number of Lethal Lesions (Y) = αD + βD2

7. Mechanistic basis ...
● The number of lethal lesions due to radiation in a group of cells is
randomly distributed.
● This probability distribution follows a Poisson distribution.
○ Discreet & independent events
○ Randomly distributed
○ Take place in a certain time interval
○ Expected number of events is constant through the time period.
Survival = e-Y
Survival = e-(αD + βD2)

8. A typical cell survival curve
The initial linear portion of the cell survival
curve corresponds to the αD component.
The curved portion (representing “lost dose”)
represents the quadratic or the βD2
component.

9. LQ Model and the 5 R’s of Radiotherapy
1. Based principally on REPAIR of SLD
a. Quadratic component points a wasted dose as damage gets repaired
2. Which other R’s of radiotherapy are linked to the LQ model? =>
INTRINSIC RADIOSENSITIVITY.
3. Does not account for Repopulation, Reassortment and
Reoxygenation.

10. Advantages of the LQ model
1. It is a mechanistic model
2. Based on sound biological plausibility
3. It is also practical as it has just two parameters
4. Performs as well as other mechanistic models of cell killing
5. Well validated in doses upto 2 - 6 Gy and backed by reasonable
amount of clinical evidence.

11. What are the key assumptions
1. Assumes that DNA repair is the primary mechanism that explains
the cell survival following radiation.
2. Assumes that pair wise chromosomal aberrations are the only
mechanism mediating cell death
3. Assumes lethal lesions have a Poisson distribution
4. Repair rates reduce at higher dose / fraction
5. Assumes a constant radiosensitivity across all dose fractions
6. Assumes homogenous dose and dose rate

12. A “Real” Cell Survival Curve
● There is marked deviation of
the observed data from the
line predicted by the
equation at higher doses
● Model predicts higher cell
death than what is actually
observed.
● Direct use “underestimates”
of the dose equivalence

14. Cytotoxicity at high doses per fraction
High dose Radiation ( > 10 Gy)
Traditional dsDNA
Damage mediated
mitotic cell death
Stem Cell
Programmed
Cell Death
Vascular
Endothelial
Cell Death
Acid Sphingomyelinase (ASMase)
mediated ceramide generation
from cell membrane
ATM mediated direct
Ceramide synthesis from
cell membrane
Hypoxia*
*Acute hypoxia abrogates DNA repair and results in cell kill
1 2 3

15. Murine experiment (Song et al)
1. Irradiated Walker 256 rat tumors to 10 Gy
single fraction.
2. Removed the irradiated tumors at different
time points and plated
3. Survival linked to time at which tumor was
removed after irradiation
4. Excess loss in cell viability linked to
vascular endothelial damage.

16. Murine experiment (Song et al)

17. Murine Experiment (Lee et al)
● GK irradiation of C6/LacZ glioma cells in rats to a dose of 20 Gy +
TMZ or Thalidomide
● Both regimens - showed only apoptosis of endothelial cells at 24
hours.
● Both regimens - reduced microvascular density
● Significant reduction in tumor volume with Thalidomide -
suggesting synergistic anti-angiogenic property.

18. Proof in HPE
● Szeifert et al evaluated the histological
changes in 18 cerebral metastases treated
with Radiosurgery (14 - 20 Gy)
● Observed a brisk inflammatory response
in irradiated tumor volume
● Severe vasculopathy (progressive as a
function of time)

19. Computer Simulation
● Kocher fitted a computer simulation model to the effect of 20 Gy SRS
on brain metastases (90 patient data)
● Tumor size, oxygenation, cell division, vascularization and vascular
occlusion modelled
● Simulation could predict tumor control only if a vascular effect was
included in which 99% of the vessels surrounding the tumor were
obliterated
● Vascular effect contributed approximately 1/3rd of the effect of
single dose radiosurgery.

20. The story in AVMs
● Karlsson et al treated 28 AVMs with 3.5
Gy twice a week to a total dose of 41 -
50 Gy
● LQ-BED = 88.8 - 108.3 Gy3
● LQ-BED of 15 Gy SRS = 90Gy3
● Obliteration rate < 10%.
Obliteration rate of AVM as a function of the
marginal dose using SRS. Flickinger et al

21. Effect of High Doses on Stem Cells (Ch’ang)
● Gastrointestinal response to whole body radiation assayed in mice
● Between doses of 8 - 13 Gy stem cells regenerate the normal mucosa
despite endothelial injury
● Beyond 14 Gy death occurs due to gastrointestinal symptoms
● Deletion of Smpd1 gene (which produces ASMase) is protective till
17 Gy (blocks endothelial apoptosis)
● Blocking Ceramide synthase activity abrogates cell death response
at doses > 17 Gy

22. Effect of High Doses on Stem Cells (Ch’ang)

23. Effect on Immune Cells
M. A. Postow et al., Immunologic correlates of the abscopal effect in a patient with melanoma. N. Engl. J. Med. 366, 925–931 (2012).

24. In-situ Vaccination by RT
1. Immunogenic Cell Death is induced:
a. Extracellular release of ATP and HMGB1 (High mobility group protein B1)
b. Cell surface exposure of Calreticulin
2. High dose (> 10 Gy) results in activation of CD8+ T cells via activated
dendritic cells.
3. Expression of chemokines that attract effector T cells (CXCL16).
4. Upregulation of MHC Class I antigen on tumor

25. CD8+ Cells are required for RT ablation
Mice injected with B16
Melanoma cells and tumor
volume measured after 20 Gy
whole body RT
Nude mice injected with B16
Melanoma cells and tumor
volume measured after 20 Gy
whole body RT
Mice injected with B16
Melanoma cells and
tumor volume measured
after 20 Gy whole body
RT in presence of CD8
inhibition

26. Alternative Models
● The LQ model is a special case of a general multi-target model
● It became popular as the multi-target model predicted a zero cell kill
at low doses which is not correct.
● Several alternative models have been proposed to the LQ model.
○ Linear Quadratic Linear Model (Guerrero et al)
○ Universal Survival Curve (Park et al) - hybrid of LQ and multi-target models
○ Generalized Linear Quadratic Formula (Wang et al)

27. Generalized Linear Model
● Proposed specifically for the uncertainties in LQ model at high
doses.
● Accounts for the fact that amount of SLD is reduced at higher doses
as sublethal damage converts to lethal damage. - “Repair
Saturation Assumption”
● Also accounts for variable dose rate during treatment (e.g. SRS /
HDR)
● Unlike LQ model α/β estimates are not dose range dependant.

28. Generalized Linear Model
ɑ/β estimates from
low dose range

29. Back to the LQ Model ...
● LQ model fits the high dose data when LQ estimates are derived
from the entire dose range.
● LQ model can incorporate the dose rate independent mechanisms
of cytotoxic effects in the linear term. - can account for cell death
mechanisms like apoptosis.
● A higher ɑ/β ratio can “straighten” the LQ curve beyond a
threshold.

30. Back to the LQ Model ...
● Repair mechanisms may not get saturated at higher dose per
fraction.
● Utility of an empirical model diminishes with additional terms.
● The newer models also do not account for the additional cytotoxic
processes observed

31. Looking to the Future ...
1. Basal gene expression
profiles may determine the
LQ model parameters.
2. Seo et al investigated LQ
model parameters based
on basal gene expression
in 52 NCI-60 cancer cell
line panel.
LowɑHighβ

32. Conclusions
● LQ Model is a well tested and validated model
● Clinical data exists that shows that LQ model can predict the effect at
higher doses (esp 5 - 10 Gy)
● While newer models have higher accuracy they need clinical data to
support them
● The real need of the hour is to not to abandon the LQ model but to
integrate it with modern molecular biology
● Appreciate that the story does not end with understanding the
cancer cell response

33. Thank You