mathematical modelling of radiotherapy: applying the lq model

Mathematical Modelling of Radiotherapy: Mathematical Modelling of Radiotherapy: Applying the LQ model.Applying the LQ model.

Helen McAneneyHelen McAneney1,21,2 & SFC O’Rourke & SFC O’Rourke22

11School of Medicine, Dentistry and Biomedical SciencesSchool of Medicine, Dentistry and Biomedical Sciences22School of Mathematics and PhysicsSchool of Mathematics and Physics

Background

• Radiation treatment is second

after surgery in battle against

cancer growths

• Success at killing cells, both

cancerous and normal cells

• Fractionated treatment schedules

• Treatment planning involves

– Localizing, Imaging,

Identifying, Optimizing,

calculations and reporting

Tumour pathologyStaging

Cure/palliationTreatment modalities

Tumour/normal tissue definitionPatient measurementsField shaping

Selection of techniqueComputation of dose distributionOptimization

Treatment verificationConfirmation of measurementsConfirmation of shields

Blocks/shieldsCompensators/bolusImmobilization devices

Verification of set-upVerification of equipment performanceDosimetry checksRecord keeping

Treatment toleranceTumour response

Tumour controlNormal tissue response

DIAGNOSIS

THERAPEUTIC DECISIONS

TARGET VOLUMELOCALIZATION

TREATMENT PLANNING

SIMULATION

FABRICATION OFTREATMENT AIDS

TREATMENT

PATIENT EVALUATIONDURING TREATMENT

PATIENT FOLLOW-UP

www.n-i.nhs.uk/medicalphysics

Constant Repopulation?

• Typically, the tumour sensitivity and repopulation are considered to be

constant during radiotherapy.

• Exponential re-growth has constant growth kinetics.

• Suggested that cell cycle regulation and anti-growth signals (hypoxia)

reduce response to radiotherapy

– S-phase of cell cycle,

– low levels of oxygenation

• Nutrient deprived cells are less apt at mitosis, therefore, as the tumour

shrinks and re-oxygenation occurs to areas previously deprived, the net

repopulation rate will increase

• Implies repopulation rate that is not constant throughout the course of

therapy

Non-constant repopulation

• One example of this may be

found in some human lung

cancers which have been

shown by Steel (1977, 2002)

to follow a Gompertzian

pattern of growth.

• It has been shown that larger

tumours have longer volume

doubling times than smaller

ones (Steel 1977, Spratt et al

1993).

Growth Laws

• Various Growth Laws for tumour

growth include

– Exponential

– Logistic (*)

– Gompertz (*)

• Does the nature of the re-growth

of tumour between treatments

effect outcome?

• Is prognosis similar or different

than when exponential re-growth

is considered?

0

0

10

00

0

2

0

lnwith

explogexp

:

1with

)exp(

:

/2lnwith

)exp(

:

NKgg

tgKNKN

Gompertz

KNgg

tgNKN

KNN

Logistic

tg

gtNN

lExponentia

g

g

l

l

LQ Model

• Linear-Quadratic model

• and characterise tissue’s response to radiotherapy. D is dose in Gy

)(exp 2DDfraction

survival

N

Nbeforet

aftert

Typical = 3 -10 Gy Adv. Head & neck = 20 GyNon-small-cell lung = 10 GyProstate = 1 Gy

5 R’s and AQ4N

• 5 R’s of radiotherapy

– Repair, repopulation, re-

distribution, re-oxygenation,

intrinsic radio-resistance

• AQ4N

– unique bioreductively

activated prodrug

– converted to a persistent

anticancer agent unaffected

by tumour re-oxygenation

– “most promising anticancer

bioreductive drug in

preclinical development”

British Journal of Cancer (2000) 83, 1589–1593. doi:10.1054/bjoc.2000.1564 (Review)

Clinical Cancer Research 14, 1096-1104, 2008.

doi: 10.1158/1078-0432.CCR-07-4020 (Phase I trial)

Re-distribution and Re-oxygenation

• Re-distribution

– Asynchronous cycling cell population, preferentially spare cells

in resistant part of cell cycle

• ‘Split-dose’ expt., time between fractions increased by

– A few hours: SF increases as sublethal damage repaired

– Cell cycle time: SF decreases as cells re-distribute, killed on 2nd

exposure.

• Re-oxygenation

– Surviving hypoxic cells move to more sensitive (oxic) state

before next exposure

Re-sensitization

• ‘Post irradiation increase the sensitivity of cells that survive an initial partial exposure’

• Occurs when

– An early part of a radiation exposure leads to a decreased average radiosensitivity just after the dose is administered, ie kills the more radiosensitive cells of a diverse population.

– Subsequent biologically driven changes gradually restore the original population average radiosensitivity.

Hlatky 1994, Brenner et al 1995

LQR model

• Before irradiation, has Gaussian probability distribution, variance 2

• After irradiation, still Gaussian, variance 2, but average value decreased, as resistant cells are preferentially spared

• Averaging over subpopulations gives

• Increase in SF due to cell to cell diversity.

– Extra resistance of particular resistant cells ‘outweighs’ extra resistance of particularly sensitive cells.

Hlatky 1994, Brenner et al 1995

2221exp DDSF

2-compartment LQR model

• Heterogeneity of cells: Hypoxic cells, re-oxygenation etc.

• Two-compartment LQR model, assuming bi-variate Gaussian distribution

22 expexp DDfDDfSF hyeff

hyeff

hyoxeff

oxeff

ox

Ro

hypoxic

oxic

Horas et al Phys. Med. Biol. 50 (2005) 1689-1701

22 eff




• Proliferation of oxic cells, but not hypoxic,


hyeff

hyoxeff

oxeff

ox

hypoxic

oxic


22 eff




• Proliferation of oxic cells, but not hypoxic, yet region increases due to viable rim of nutrients


hyeff

hyoxeff

oxeff

ox

Ro

hypoxic

oxic


22 eff




• Treatment: radio-resistance of hypoxic cells,


hyeff

hyoxeff

oxeff

ox

hypoxic

oxic


22 eff




• Treatment: radio-resistance of hypoxic cells, yet redistribution and re-oxygenation occurs.


hyeff

hyoxeff

oxeff

ox

Ro

hypoxic

oxic


22 eff

Local sensitivity

• Table 1, proposed expressions for

the local sensitivity for the three

studied models, whose

denomination comes from their

dependence with position r.

• α0 and β0 are parameters of each

model related with the

oxygenation level at the tumour

surface.


Overall radiosensitivity

• Ensemble average and volumetric average are interchangeable,

supposing that operating over sufficiently large volumes. Then


Overall radiosensitivity: two zones


• Viable rim r0 of 50 m for all tumour sizes. ‘Constant crust’ model.

• and ( and ) estimated by least squares fit to experimental data (Buffa et al) for spheroid with R = r0 = 50 μm in

oxic (hypoxic) conditions

• Oxic fraction given by


ox0 ox

0 h0 h

0

3

30

33

R

rRRf ox

Programming

• Fortran 90 language

• Lagrange Interpolation : Aiken

algorithm

• Relating R to N

volume of tumour

volume of cell

• Weighted averages of oxic and

hypoxic parameters to obtain

homogeneous parameters

• Repopulation: Exponential,

Logistic, Gompertz

• Treatment schemes: Uniform,

standardised, accelerated etc.

3

5

R

hhoxox ff hhoxox ff

A few results

Accelerated treatment - LQR

Linear local sensitivity

A few results

Accelerated treatment - LQ


A few results

Accelerated treatment - % dif


A few results

Parameter values: R0=375 m, D=2 Gy, t2=80 days, weekday treatments for 6 weeks. (left) Changing dynamics, proportions and therefore radio-sensitivity parameters of subpopulations within tumour throughout treatment schedule given different types of repopulation. (right) Fixed radio-sensitivity parameters at start of treatment schedule determined by weighted averages for different types of re-growth laws.

Quadratic Model of local sensitivity

LQR LQ

Questions

• Though cells more radio-resistant via hypoxia, less growth occurs also. Balances..? Dominate feature? Will hypoxia increase or decrease the effectiveness of radiotherapy?

• How effective is accelerated fractionation compared to standard fractionation on heterogeneous tumours?

• Does the level of heterogeneity of the tumour matter?

Acknowledgements

• Joe O’Sullivan

• Francesca O’Rourke

• Anita Sahoo

• Frank Kee - Director Centre of

Excellence for Public Health NI

• Leverhulme Trust

Publications1. H. McAneney and S.F.C. O’Rourke,

Investigation of various growth mechanisms of solid tumour growth within the linear-quadratic model for radiotherapy, Phys. Med. Biol. 52, (2007), 1039-1054.

2. S.F.C. O’Rourke, H.McAneney and T. Hillen, Linear Quadratic and Tumour Control Probability Modelling in External Beam Radiotherapy, J. Math. Biol. doi 10.1007/s00285-008-0222-y

3. S.F.C. O’Rourke, H.McAneney, C.Starrett and J.M. O’Sullivan, Repopulation Kinetics and the linear-quadratic Model, American Institute of Physics Conference Proceedings, accepted Aug 2008.

mathematical modelling of radiotherapy: applying the lq model

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