Fractionation and tumor control: applying mathematical models derived from radiobiology to the clinics

Bleddyn Jones

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!

Radiation Therapy

BRAIN/CNSN to power 0.42T to power 0.01

SKIN : N to power 0.24T to power 0.11

Total Dose=K.NxTyPower law: Emphasis on N

LQ model0.3 K/day/ 8 Gy

LQ model0.01K/day/ 2 Gy

LQ: Emphasis on d (dose/fraction), based on BED equation

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

Douglas E Lea – pupil of Rutherford; author of ACTION OF RADIATIONS ON LIVING CELLS

Cambridge University Press 1946

Hall & Bedford: direct correlation between lethal type chromosome break accumulation and log of cell surviving fraction.

Showed good fit to linear and quadratic components to the accumulation of radiation induced chromosome breaks with increasing dose. Two parameter model.

E = d + d2

E

d

E = -ln(SF)

Lethal chromosome breaks

1:1 correlation

Events for 37% survival fraction =e-1

Base damage>1000

Single strand breaks ~1000

Double strand breaks ~40

Chromo =1

LETHAL CHROMOSOME BREAK NUMBER E=d + d2

Poisson statistics

• 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

m

m

m

m

mr

r

eP

emP

meP

eP

r

emP

1)0(1

2

!

2

)2(

)1(

)0(

)( 1-P(0) = probability of any number of successes

e-1 = 0.37

“half way on a log scale”.

Background

Applications in radiation biologyCells 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

EeSF )( 2ddNeSF Surviving number of cells Cx = C×SF = )( 2ddNCe

Probability of cure TCP is probability of no surviving cell, or TCP = e- Cx )2( ddNCeeTCP

A double Poisson!

Some important contributory factors

• Tumour radio-sensitivity variations (heterogeneity)…

• Other causes of treatment failure• Other treatment modalities – surgery,

chemotherapy etc

Full LQ equation with allowance for repopulation

pp T

tddn

T

t

ddn eeeSF.693.0.693.0 2

2

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.

tKd

DT

tdndBED

p

./

1.

.693.0

/1

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

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, 125-132, 1994

• Alternative is to use iterative computer and graphical methods.

Series expansion for n fractions

]1[]1[

.

Q

ddQ

k

ddBED

eQ fz

21

1

12;

1

11

]...1[].......1[

]1[]....1[]/1[

2

2

)12(42)1(2

)1()1(

.

seriesk

dseriesdBED

Q

Qseries

Q

Qseries

QQQk

dQQQdBED

k

dQdQ

k

dQdQkddBED

eQ

nn

nn

nn

fz

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

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

z

et

T

zt

pot

1693.00

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

same part of tumour under-dosedIndividual dose response curves

Collective dose response curvesDifferences are

reduced but remain significant –especially at 3“missed fractions” and above

Individual tumour

10% of tumour underdosed in 15 different parts

100 different tumours

Individual tumour

10% of tumour underdosed in 15 different parts

100 different tumours

Assumes greater significance when 40% of tumour randomly underdosed in 15 different

sub-volumes

• 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

Dynamic tumour regression

…..during treatment

CV1, NTV1

CV2, NTV2

CV3, NTV3

Tumour regression usually follows exponential decay kinetics

30

0

.

.zt

t

ztt

ell

eVV

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

Effect of tumour shrinkage on normal tissue volume included in field - if

field size remains constant

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

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

expansion, infarction, exfoliation

Exponential growthCell 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,

dtkN

dN.

ktt

ktt

t

eNN

eN

N

ktN

N

NktCktN

0

0

0

0

ln

lnln

Then - If w is the time required to double the cell population, 2=ekw, so ln2=kw, so that k=ln2/w; then

w

tt

w

t

t

eN

N

eNN.693.0

0

.693.0

0.

When t=0, lnN0=C, so C is lnN0

This is fractional increase in cell number which opposes the SF due to radiotherapy

Full LQ equation with allowance for repopulation

pp T

tddn

T

t

ddn eeeSF.693.0.693.0 2

2

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.

pT

tdndBED

.

.693.0

/1

For BEDThe BED version of is obtained by taking loge and multiplying

by -1, then dividing by to give.

pT

tdDBED

.

.693.0

/1

Normally, we let KTp

.

693.0

Now K is in units of

...

1 11

dayGraysDaysGy

KThus K is the BED dose required counteract cellular one day of repopulation

Models of accelerated repopulation

p

K

p

T

ttddn

T

tddn

eSF

eSF).(693.0

.693.0

2

2

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

)(/

1 KttKd

DBED

Where K is the BED required to oppose repopulation per day

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

z

et

T

zt

pot

1693.00

..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 )

Oxygen - cell survival curvesOxic = 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)

Biological effective dose (BED) in hypoxia

/1

avav q

d

q

DBED

/

1

q

d

qDBED

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

/1

dDBED

In oxic conditions

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.

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%)

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

For slow reoxygenation 1% per day

Faster re-oxygenation, mean of 3% per day

Models of Tumour Hypoxia – iterative

Quiescent Hypoxic cells

Repopulating Oxic cells

Cell death

Radiosensitivities modified by hypoxia

Radiosensitivities not modified by hypoxia

DailyFlux of cells

Modified from Scott (1988); alternative is to use analytical models with integration of effective OER with time to give average values. Results very similar.

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.

Example of iterative loop in ‘Mathematica’

Heterogeneity is included by having long lists of separate tumours each with different , , and w, the cell repopulation parameter.

Nox = nox Exp[ -list d- list d^2 + 0.693 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

Practical modelling to maintain tumour control1. 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, 323-326, 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, 382-393, 2002.Jones B Hopewell JW & Dale RG. Radiobiological compensation for unintended treatment interruptions during palliative radiotherapy. BJR, 80, 1006-1010, 2007.

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

Type A sensitised by A/GyType B damage by B/Gy

222SSuu dBAdMddN

Consequences:If A>B, sensitisation reduces with increasing dose per fractionIf B>A, sensitisation increases with increasing dose per fractionIf A=B, sensitisation is constant with dose per fraction [so called “pure dose modification”]

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, 939-944, 2005.

0 5 10 15 20 25 30 35-5

0

5

10

15

20

25

30

/ = 48.6/ Tpot

with 95% Confidence Intervals 41.4 & 55.8 with p-value < 2.39e-006

Tpot

(days)

/

(Gy

)

Tumour Data1/x regression95% Confidence

α/β (Gy)

Tpot (days)

101

102

103

0

1

2

3

4

5

6

7

8R

BE

Volume Doubling Times-(Days)

2 RBE models as a function of the Volume Doubling Time,VTD.

Data

(a) Radiobiological Model with p value= 2.13e-005

95% Confidence(b) Empirical Model with p value= 2.3492e-005

95% Confidence

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

0

10

20

30

01

23

45

1

2

3

4

5

6

7

8

Dose (Gy)Tpot

(days)

RB

E

2

3

4

5

6

7

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

Dose/# (Gy)

Tpot (days)

RBE

100

60

10

Medulloblstoma in a child

X-rays

Proton particles

X-rays

Proton particles

The inner ear (cochlea) is marked by the black outline

XX-rays Protons

X-Ray Dose distributions for two opposed fields

INTERNAL DOSE GRADIENTS

Recurrent medulloblastoma

Medulloblastoma & CNS RBE• α/β=28 Gy !! rapid growth, expect low RBE of

1.05-1.06 at 1.6 -1.8 Gy per fraction .• Many 1.03-1.06 values in Paganetti et al data• Brain & spinal cord α/β = 2 Gy,

RBE probably perhaps 1.15 or 1.2 (some 1.2-1.3 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

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

• 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, 259-265, 2000.

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

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.