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

Outline Background Repopulation & the LQ model The 5 R’s of radiotherapy –Repopulation, repair, re-oxygenation, re-distribution, and intrinsic radio-resistance. 2 compartment LQR Model Results

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

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?

Scaling of Growth Laws t 2 =60 days, N 0 =0.1 K, K=1 Phys Med Biol (2007)

LQ Model Linear-Quadratic model  and  characterise tissue’s response to radiotherapy. D is dose in Gy Typical  = Gy Adv. Head & neck  = 20 Gy Non-small-cell lung  = 10 Gy Prostate  = 1 Gy

LQ with Repopulation Exponential Logistic Gompertz Uniform treatment schedule: n treatments at intervals  t Phys Med Biol (2007)

Conclusions Gaps in treatment allow differences in growth laws to emerge Survival Fraction –Logistic and Exponential similar order of magnitude –Differences of larger for Gompertz No. of cells in tumour, means this has a larger potential to repopulate tumour Faster the doubling time, more apparent differences become Limited re-growth leading to poorer prognosis for tumour eradication is NOT intuitive Phys Med Biol (2007)

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: /bjoc (Review) Clinical Cancer Research 14, , doi: / CCR (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 2 nd 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

2-compartment LQR model Heterogeneity of cells : Hypoxic cells, re-oxygenation etc. Two-compartment LQR model, assuming bi-variate Gaussian distribution RoRo hypoxic oxic Horas et al Phys. Med. Biol. 50 (2005)

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, hypoxic oxic Horas et al Phys. Med. Biol. 50 (2005)

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 RoRo hypoxic oxic Horas et al Phys. Med. Biol. 50 (2005)

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, hypoxic oxic Horas et al Phys. Med. Biol. 50 (2005)

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. RoRo hypoxic oxic Horas et al Phys. Med. Biol. 50 (2005)

Local sensitivity Table 1, proposed expressions for the local sensitivity for the three studied models, whose denomination comes from their dependence with position r. α 0 and β 0 are parameters of each model related with the oxygenation level at the tumour surface. Horas et al Phys. Med. Biol. 50 (2005)

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)

Overall radiosensitivity: two zones Horas et al Phys. Med. Biol. 50 (2005)

Viable rim r 0 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 = r 0 = 50 μm in oxic (hypoxic) conditions Oxic fraction given by Horas et al Phys. Med. Biol. 50 (2005)

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.

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: R 0 =375  m, D=2 Gy, t 2 =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 LQRLQ

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 Publications 1.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), 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 /s 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.