1. Fractionation and tumor control: applying mathematical models derived from radiobiology to the clinics Bleddyn Jones

2. Fractionation or bio-effective dose not mentioned!
Radiation Therapy
The successful radiation treatment of cancer depends vitally on knowledge of the precise amount and location of radiation given to a patient and the opportunity for therapists to exchange this information and the results achieved.
Fractionation or bio-effective dose not mentioned! 2

3. Total Dose=K.NxTy Power law: Emphasis on N SKIN : N to power 0.24
T to power 0.11
BRAIN/CNS
N to power 0.42
T to power 0.01

4. LQ: Emphasis on d (dose/fraction), based on BED equation
LQ model
0.3 K/day
/ 8 Gy
LQ model
0.01K/day
/ 2 Gy

5. Comparison of Power Law for TD=20.N0.32
and
LQ where BED=70 Gy8 [α/β=8Gy]
For iso-effect to 20 Gy in 1 single fraction

6.  LETHAL CHROMOSOME BREAK NUMBER E=d + d2
Douglas E Lea – pupil of Rutherford; author of ACTION OF RADIATIONS ON LIVING CELLS Cambridge University Press 1946
Showed good fit to linear and quadratic components to the accumulation of radiation induced chromosome breaks with increasing dose. Two parameter model.
Events for 37% survival fraction =e-1
Base damage>1000
Single strand breaks ~1000
Double strand breaks ~40
Chromo =1
E = d + d2 E d
Hall & Bedford: direct correlation between lethal type chromosome break accumulation and log of cell surviving fraction.
Lethal chromosome breaks
1:1 correlation
E = -ln(SF)
 LETHAL CHROMOSOME BREAK NUMBER E=d + d2

7. Poisson statistics
Background
Poisson considered very large numbers of equal opportunities for a small chance of “success”  in a set time interval or space.
Poisson equation, a much simpler equation, especially for chance of no event
1-P(0) = probability of any number of successes
e-1 = 0.37
“half way on a log scale”.

8. Applications in radiation biology
Cells surviving radiotherapy is a Poisson random variable (Munro. and Gilbert 1961, Porter 1980a, 1980b, Suit et al 1965, 1978)
For C cells before radiation, there are C×SF cells after radiation where SF is surviving fraction.
If E is the expected number of lethal events per cell: E=N(αd+βd2), then, Probability of there being no lethal events per cell is the probability of survival, so
Surviving number of cells Cx = C×SF =
Probability of cure TCP is probability of no surviving cell, or TCP = e- Cx
A double Poisson!

9. Some important contributory factors
Tumour radio-sensitivity variations (heterogeneity)…
Other causes of treatment failure
Other treatment modalities – surgery, chemotherapy etc

10. Full LQ equation with allowance for repopulation
The net surviving fraction is
A powerful equation - many applications & shows that smallest SF obtained with highest dose and highest radiosensitivities and longest doubling times in shortest overall time [See Fowler 1988 Progress in Fractionated Radiotherapy, Brit J Radiology]
The BED version of above is obtained by taking loge and multiplying by -1, then dividing by  to give.

11. 50 patients, each with different radiosensitivity and steep individual dose-response curves
50 patients, each with different radiosensitivity but overall dose reponse curve is shallower and reflects all individual curves

12. Dynamic changes in model parameters
Assumed to reflect average values during treatment
Average values can in some cases be found by integration if time course is predictable (e.g. repopulation where effective cellular doubling time might change slowly during treatment), or a step function is used. Jones & Dale Radiother & Oncol, 37, 136-9, 1995
Where continuum is broken and parameters change each fraction, a series expansion with appropriate truncation and simplification can be used, e.g. hypofractionation where dose and radiobiological parameters change with each fraction. Dale & Jones Radiother & Oncology 33, , 1994
Alternative is to use iterative computer and graphical methods.

13. Series expansion for n fractions
Assume dose increases each fraction according to a fctor Q, where f is time interval between fractions and z the link parameter.

14. Another variant…progressive repopulation
Assume cell loss factor () falls with tumour shrinkage and reoxygenation; also better blood supply for delivery of growth factors, vitamins etc.
if t = o exp[-zt] , where z is a parameter controlling cell loss and assumed connected with re-oxygenation and shrinkage rate.
Now effective doubling time Teff=Tpot/(1- )
Then, if value of  reduces during treatment between Repopluation correction will be given by integrating  between t=0 and t=t so that repopulation correction factor becomes

15. What about adaptive radiotherapy and missing part of a tumour on some fractions? Consider a spherical tumour, 10% of which can be under-dosed from 2 Gy down to 0.6 Gy for 0, 3, 5, 10 or 15 fractions. Compare this with a tumour, 10% of which can be under-dosed in 15 different sub-volumes on a random basis. Also a tumour, 40% of which can be under-dosed in 15 different sub-volumes on a random basis

16. same part of tumour under-dosed
Individual dose response curves
Collective dose response curves
Differences are reduced but remain significant –especially at 3“missed fractions” and above

17. Individual tumour
10% of tumour underdosed in 15 different parts
100 different tumours

18. Individual tumour
10% of tumour underdosed in 15 different parts
100 different tumours

20. Assumes greater significance when 40% of tumour randomly underdosed in 15 different sub-volumes

21. Volume of miss is important
Random misses appear to be more forgiving than repeated misses on same part of a tumour
Volume of miss is important
Increasing overall time (gaps)…may make matters worse or better (not modelled here)
Chemotherapy may reduce effect depending on altered position on dose response curve
Shape of dose response curve and position on it are important

22. Dynamic tumour regression
CV1, NTV1
CV2, NTV2
…..during treatment
CV3, NTV3

23. Tumour regression usually follows exponential decay kinetics

24. Beware the first few fractions – tumours can increase in volume!

25. Effect of tumour shrinkage on normal tissue volume included in field - if field size remains constant

26. The illusion of tumour volume shrinkage with time after treatment or after “bad treatment”

27. There will always be some surprises in store e. g
There will always be some surprises in store e.g. anatomical variants, haemorrhage, cystic expansion, infarction, exfoliation

28. Exponential growth
Cell division is binary. For any number of cells (N), rate of change in growth is proportional to number of cells present.
This means dN/dt N, so that dN/dt=kN, where k is a constant.
So,
When t=0, lnN0=C, so C is lnN0
Then - If w is the time required to double the cell population, 2=ekw, so ln2=kw, so that k=ln2/w; then
This is fractional increase in cell number which opposes the SF due to radiotherapy

29. Full LQ equation with allowance for repopulation
The net surviving fraction is
A powerful equation - many applications & shows that smallest SF obtained with highest dose and highest radiosensitivities and longest doubling times in shortest overall time [See Fowler 1988 Progress in Fractionated Radiotherapy, Brit J Radiology]
The BED version of above is obtained by taking loge and multiplying by -1, then dividing by  to give.

30. For BED
The BED version of is obtained by taking loge and multiplying by -1, then dividing by  to give.
Normally, we let
Now K is in units of
Thus K is the BED dose required counteract cellular one day of repopulation

31. Models of accelerated repopulation
Where tK is the time at which accelerated repopulation begins
This simplistic model is useful for t longer than tK, but is not useful at shorter times – since t-tK will then be positive.
One could use BED since the BED version of the above equation, obtained by dividing throughout by , will be
Where K is the BED required to oppose repopulation per day

32. Another variant…progressive repopulation
Assume cell loss factor () falls with tumour shrinkage and reoxygenation; also better blood supply for delivery of growth factors, vitamins etc.
if t = o exp[-zt] , where z is a parameter controlling cell loss and assumed connected with re-oxygenation and shrinkage rate.
Now effective doubling time Teff=Tpot/(1- )
The value of  reduces during treatment between Repopluation correction will be given by integrating  between t=0 and t=t so that repopulation correction factor becomes

33. ..and another approach
Assume different doubling times each week during treatment using a sliding scale for the doubling time included in the repopulation factor
In week 1 – use Tpot/(1-)
In week2 – use Tpot(1-0.8 )
In week3 – use Tpot(1-0.6 )
In week4 – use Tpot(1-0.4 )
In week 5 and onwards – use Tpot(1-0.2 )

34. E C Oxygen - cell survival curves Oxic = fully oxygenated
Hypoxic
Anoxic
Oxic
Dose
S.F. A B C D q r m E
Oxygen - cell survival curves
Oxic = fully oxygenated
Hypoxic = partially oxygenated
Anoxic = absence of oxygen
It follows that q=m/r, where m= maxOER, r is multiplier between anoxic and hypoxic ; q is multiplier between oxic and hypoxic (and the dose reduction factor)

35. Biological effective dose (BED) in hypoxia
In oxic conditions
For pure dose modification
If α and β changed by different hypoxia reduction factors. OER falls with dose
/ appears to be increased by the q factor

36. Analytical difficulties, re-oxygenation means that oxygen tension changes daily, so that a different value of R is required each day, or an average value over a time period – obtained by integration divided by elapsed time.

37. Reoxygenation rate (z) and initial hypoxic fraction (h)
A(h=5%), B(h=15%), C(h=30%)
A(z=1%), B(z=3%), C(z=7%)

38. For slow reoxygenation 1% per day
A=2 Gy per day x-rays, 5# per week
B= 1.4 Gy x-rays 10~ per week
C=C ions 2.1 Gy per fraction 5# per week
D=C ions 6 Gy per fraction 5# per week
Faster re-oxygenation, mean of 3% per day

39. Models of Tumour Hypoxia – iterative
DailyFlux of cells
Repopulating Oxic cells
Cell death
Quiescent Hypoxic cells
Radiosensitivities modified by hypoxia
Radiosensitivities not modified by hypoxia
Initial conditions and variables: hypoxic fraction, reoxygenation rate, OER, repopulation rates, radiosensitivities and mean inter-fraction interval. Model repeats every day until TCP > 0.05.
Modified from Scott (1988); alternative is to use analytical models with integration of effective OER with time to give average values. Results very similar.

40. Example of iterative loop in ‘Mathematica’
Nox = nox Exp[ -list d- list d^ f /list ]
Nhyp = nhyp Exp[ -listd/q- listd^2/q^2];
Ntot = nox + nhyp;
Tcp = Exp[-ntot];
n = n+1;
Reox = x nhyp;
ntot = nox + nhyp;
nhyp = nhyp – xnhyp - ynhyp;
Nox = nox + reox
Heterogeneity is included by having long lists of separate tumours each with different , , and w, the cell repopulation parameter.

41. Practical modelling to maintain tumour control
Treatment delivery errors: over-dosage or under-dosage of a tumour; known geographical miss. Jones B and Dale RG. Radiobiological compensation of treatment errors in radiotherapy. BJR, 81, , 2008.
2. Unintended treatment interruptions due to accelerator breakdown, patient illness etc.
Dale RG, Hendry JH, Jones B, Deehan C et al. Practical methods for compensating for missed treatment days in radiotherapy…. Clinical Oncology, 14, , 2002.
Jones B Hopewell JW & Dale RG. Radiobiological compensation for unintended treatment interruptions during palliative radiotherapy. BJR, 80, , 2007.

42. Mechanisms of sensitisation
Drugs that increase sublethal -> lethal damage….oxygen, mild cytotoxics e.g. Temazolamide…… increase Type B (β) damage > Type A (α) damage
Drugs that cause direct lethal injury…DNA strand cross links…Platinum, bifunctional alkylating agents (CCNU, BCNU)……..
increase Type A damage > Type B damage
High LET radiations example of later

43. Type A sensitised by A/Gy Type B damage by B/Gy
Consequences:
If A>B, sensitisation reduces with increasing dose per fraction
If B>A, sensitisation increases with increasing dose per fraction
If A=B, sensitisation is constant with dose per fraction [so called “pure dose modification”]

44. Simulation of Radiotherapy with or without Chemotherapy (+ P) in USA and UK
Jones B and Dale RG. The potential for mathematical modelling in the assessment of the radiation dose equivalent of cytotoxic chemotherapy given concomitantly with radiotherapy. Brit J Radiol 78, , 2005.

45. α/β (Gy)
Tpot (days)

46. LQ model with RBE limits and cell kinetic adaptation fit to data of Batterman - fast neutrons for human lung metastases, Eur J Cancer 1981

47. LQ model with RBE limits and cell kinetic adaptation fit to data of Batterman - fast neutrons for human lung metastases, Eur J Cancer 1981
RBE
Tpot (days)
Dose/# (Gy)

48. Medulloblstoma in a child
X-rays
X-rays
100 60 10
Proton particles
Proton particles

49. The inner ear (cochlea) is marked by the black outline
XX-rays
Protons

50. X-Ray Dose distributions for two opposed fields

51. INTERNAL DOSE GRADIENTS
Recurrent medulloblastoma

52. Medulloblastoma & CNS RBE
α/β=28 Gy !! rapid growth, expect low RBE of at Gy per fraction .
Many values in Paganetti et al data
Brain & spinal cord α/β = 2 Gy, RBE probably perhaps 1.15 or 1.2 (some values in Paganetti et al data)
A prescribed dose using RBE=1.1 might under-dose tumour & ‘over-treat’ CNS [by up to 5-10% in each case].
+Brain proton underdosing’ CSF space

53. Clinical Trials assume equal probability of success/failure in all patients, but patients are heterogenous for multiple parameters
Consider thought experiment: Trial is testing treatment A, B or C:
A and B are different fractionation schedules, A of higher dose, B of shorter time but to slightly lower dose. [tumours with lower radiosensitivities will do better with A, some with short doubling times with B, provided they are sufficiently radiosenstive]
C is a category where A or B is given as determined by good predictive assays & optimum dose per fraction applied

54. Some patients will do better with A, some better with B
Those given A will contain failures better treated by B
Those given B will contain failures better treated by A
Those given C will contain optimum numbers of good results and should far exceed results of A or B used indiscriminately.
For computer simulation of this scenario see Jones B, Dale RG. Radiobiological modelling and clinical trials. Int J Radiat Oncol Biol & Physics. 48, , 2000.

55. Drugs and ion beams
RBE is due mainly to  in  radiosensitivity parameter , the increase in  being small.
Drugs which sensitise  selectively may be useful …especially is tumour has “low RBE” due to poor repair capacity
Drugs which normalise blood vessels and reduce tumour progression…..
Ensure IB BED+ChemoBED > X-ray BED+ChemoRxBED in tumour
BUT that: IB BED+ChemoBED < X-ray BED+ChemoRxBED in NTissues

56. BED equivalent of changed tissue tolerance
Linear quadratic modelling of increased late normal tissue effects in special clinical situations. Int. J Radiat Oncol Biol & Physics,64:948-53, 2006.
Gave BED equivalents of surgery, age and previous chemotherapy from clinical data sets
The equivalent BED values were:
cyclophosphamide, methotrexate, and fluorouracil (CMF) chemotherapy (6.48 Gy3), surgery prior to abdominal radiotherapy (17.73 Gy3), and older age (3.61 Gy3).
* BED equivalent might include repopulation and radiosensitivity changes ….BED captures both.