Mathematical Modelling of Mathematical Modelling of Radiotherapy: Radiotherapy: Applying the LQ model. Applying the LQ model. Helen McAneney Helen McAneney 1,2 1,2 & SFC O’Rourke & SFC O’Rourke 2 2 1 School of Medicine, Dentistry and Biomedical School of Medicine, Dentistry and Biomedical Sciences Sciences 2 School of Mathematics and Physics School of Mathematics and Physics
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Mathematical Modelling of Radiotherapy: Applying the LQ model.
Mathematical Modelling of Radiotherapy: Applying the LQ model. Helen McAneney 1,2 & SFC O’Rourke 2. 1 School of Medicine, Dentistry and Biomedical Sciences 2 School of Mathematics and Physics. Background. Radiation treatment is second after surgery in battle against cancer growths - PowerPoint PPT Presentation
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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
• ‘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
2-compartment LQR model
• Heterogeneity of cells: Hypoxic cells, re-oxygenation etc.
• Two-compartment LQR model, assuming bi-variate Gaussian distribution
• Proliferation of oxic cells, but not hypoxic,
22 expexp DDfDDfSF hyeff
hyeff
hyoxeff
oxeff
ox
hypoxic
oxic
Horas et al Phys. Med. Biol. 50 (2005) 1689-1701
22 eff
2-compartment LQR model
• Heterogeneity of cells: Hypoxic cells, re-oxygenation etc.
• Two-compartment LQR model, assuming bi-variate Gaussian distribution
• Proliferation of oxic cells, but not hypoxic, yet region increases due to viable rim of nutrients
22 expexp DDfDDfSF hyeff
hyeff
hyoxeff
oxeff
ox
Ro
hypoxic
oxic
Horas et al Phys. Med. Biol. 50 (2005) 1689-1701
22 eff
2-compartment LQR model
• Heterogeneity of cells: Hypoxic cells, re-oxygenation etc.
• Two-compartment LQR model, assuming bi-variate Gaussian distribution
• Treatment: radio-resistance of hypoxic cells,
22 expexp DDfDDfSF hyeff
hyeff
hyoxeff
oxeff
ox
hypoxic
oxic
Horas et al Phys. Med. Biol. 50 (2005) 1689-1701
22 eff
2-compartment LQR model
• Heterogeneity of cells: Hypoxic cells, re-oxygenation etc.
• Two-compartment LQR model, assuming bi-variate Gaussian distribution
• Treatment: radio-resistance of hypoxic cells, yet redistribution and re-oxygenation occurs.
22 expexp DDfDDfSF hyeff
hyeff
hyoxeff
oxeff
ox
Ro
hypoxic
oxic
Horas et al Phys. Med. Biol. 50 (2005) 1689-1701
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.
Horas et al Phys. Med. Biol. 50 (2005) 1689-1701
Overall radiosensitivity
• Ensemble average and volumetric average are interchangeable,
supposing that operating over sufficiently large volumes. Then
Horas et al Phys. Med. Biol. 50 (2005) 1689-1701
Overall radiosensitivity: two zones
Horas et al Phys. Med. Biol. 50 (2005) 1689-1701
• 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
Horas et al Phys. Med. Biol. 50 (2005) 1689-1701
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
Linear local sensitivity
A few results
Accelerated treatment - % dif
Linear local sensitivity
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.