Mathematical Modelling within Radiotherapy: Mathematical Modelling within Radiotherapy: The 5 R’s of Radiotherapy and the LQ model.The 5 R’s of Radiotherapy and the LQ model.

Helen McAneneyHelen McAneney11 and SFC O’Rourke and SFC O’Rourke1,21,2

11School Mathematics and Physics, Queen’s University Belfast, School Mathematics and Physics, Queen’s University Belfast, 2 2 Centre for Cancer Research and Cell BiologyCentre for Cancer Research and Cell Biology

IntroductionIntroductionRadiotherapy is still second after surgery in the fight against cancer as well as being much cheaper than chemotherapy. Modelling is therefore

crucial in advancing this form of treatment, with previous models assuming total population dynamics and general tumour types. Following earlier work1, presented here are specific calculations on a 2-compartment system of oxic and hypoxic cells with the LQ model extended to account for 4

R’s of radiotherapy (repopulation, repair, redistribution and re-oxygenation) under various treatment schedules for particular tumour types.

IntroductionIntroductionRadiotherapy is still second after surgery in the fight against cancer as well as being much cheaper than chemotherapy. Modelling is therefore

crucial in advancing this form of treatment, with previous models assuming total population dynamics and general tumour types. Following earlier work1, presented here are specific calculations on a 2-compartment system of oxic and hypoxic cells with the LQ model extended to account for 4

R’s of radiotherapy (repopulation, repair, redistribution and re-oxygenation) under various treatment schedules for particular tumour types.

References:References:1.H McA & SFC O’R, Phys. Med. Biol. 52 (2007) 1039-10542.JJ Kim & IF Tannock, Nature Cancer Review 5 (2005) 516-5253.DJ Brenner et al. Int. J. Radiat. Oncol. Biol. Phys. 32 (1995) 379-904.JA Horas, OR Olguin & MG Rizzotto, Phys. Med. Biol. 50 (2005) 1689-17015.FM Buffa et al. Int. J. Radiat. Oncol. Biol. Phys. 49 (2001) 1109-18

References:References:1.H McA & SFC O’R, Phys. Med. Biol. 52 (2007) 1039-10542.JJ Kim & IF Tannock, Nature Cancer Review 5 (2005) 516-5253.DJ Brenner et al. Int. J. Radiat. Oncol. Biol. Phys. 32 (1995) 379-904.JA Horas, OR Olguin & MG Rizzotto, Phys. Med. Biol. 50 (2005) 1689-17015.FM Buffa et al. Int. J. Radiat. Oncol. Biol. Phys. 49 (2001) 1109-18

ConclusionsConclusionsAlthough further investigation is required, for those results presented it should be noted that • Again the type of repopulation impacts on the success of treatment with Gompertzian re-growth being least favourable for the patient in both fixed and dynamic radio-sensitivity parameters• Logistic and exponential re-growth produces similar results, though one order of magnitude better in tumour eradication if cellular diversity is accounted for rather than fixed dynamics.• Thus radio resistance due to hypoxia and quiescent nature of tumour are important factors to consider.

Future questions to be addressed include• Does the level of heterogeneity play a significant role?• Does the choice of parameters significantly effect these conclusions?• Intrinsic radio-resistance of cells needs to be addressed.

ConclusionsConclusionsAlthough further investigation is required, for those results presented it should be noted that • Again the type of repopulation impacts on the success of treatment with Gompertzian re-growth being least favourable for the patient in both fixed and dynamic radio-sensitivity parameters• Logistic and exponential re-growth produces similar results, though one order of magnitude better in tumour eradication if cellular diversity is accounted for rather than fixed dynamics.• Thus radio resistance due to hypoxia and quiescent nature of tumour are important factors to consider.

Future questions to be addressed include• Does the level of heterogeneity play a significant role?• Does the choice of parameters significantly effect these conclusions?• Intrinsic radio-resistance of cells needs to be addressed.

The LQ model & RepopulationThe LQ model & RepopulationRepopulation of cancer cells during therapy can be an important cause of treatment failure, both for radiotherapy and chemotherapy2. This is often taken as exponential in nature, though there is evidence to suggest that growth is limiting, e.g. Gompertzian or logistic in nature (See figure 1).

Figure 1: (left) Human lung data showing Gompertzian pattern of growth, taken from Steel. (right) Illustration of types of growth laws

The linear-quadratic (LQ) model is the most commonly used approach to study the response of cells to radiation:

SF denotes surviving fraction, D the dose in Gys, G the Lea-Catcheside function (G1 for acute exposure) is characteristic of tissue type.

This model considers the effect of both irreparable damage and repairable damage susceptible to misrepair which ultimately leads to

mitotic cell death.

Inclusion of repopulation with treatment protocols led to following conclusions:

• Gaps in treatment allow differences in growth laws to emerge• Survival Fraction endpoint (See Table 1)

- Logistic and Exponential similar order of magnitude - Differences of 101-103 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

The LQ model & RepopulationThe LQ model & RepopulationRepopulation of cancer cells during therapy can be an important cause of treatment failure, both for radiotherapy and chemotherapy2. This is often taken as exponential in nature, though there is evidence to suggest that growth is limiting, e.g. Gompertzian or logistic in nature (See figure 1).

Figure 1: (left) Human lung data showing Gompertzian pattern of growth, taken from Steel. (right) Illustration of types of growth laws

The linear-quadratic (LQ) model is the most commonly used approach to study the response of cells to radiation:

SF denotes surviving fraction, D the dose in Gys, G the Lea-Catcheside function (G1 for acute exposure) is characteristic of tissue type.

This model considers the effect of both irreparable damage and repairable damage susceptible to misrepair which ultimately leads to

mitotic cell death.

Inclusion of repopulation with treatment protocols led to following conclusions:

• Gaps in treatment allow differences in growth laws to emerge• Survival Fraction endpoint (See Table 1)

- Logistic and Exponential similar order of magnitude - Differences of 101-103 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

2exp GDDSF

The 2-compartment LQR model & RepopulationThe 2-compartment LQR model & RepopulationExtending the LQ model to include:

• Redistribution - Asynchronous cycling cell population, preferentially spare cells in resistant part of cell cycle• Re-oxygenation - Surviving hypoxic cells move to more sensitive (oxic) state before next exposure

the LQR model accounts for cellular diversity through the dispersion about the mean radio-sensitivity . The average SF is obtained by assuming a Gaussian distribution of parameters, thus3

Extending the LQR model further to include 2-compartments of hypoxic and oxic tumour cells, Horas et al. obtained4

with

And comparing to experimental data of Buffa et al.5 of a human colon adenocarcinoma cell line obtained various tabulated radio-sensitivity

parameter values dependent on tumour size4.

Investigated are various forms of repopulation for the oxic portion of cells, conventional and accelerated treatment protocols, whilst observing

the changing proportions of the subpopulations and their dynamics throughout (changing R and therefore changing radio-sensitivity

parameters calculated using Lagrange interpolation).

Figure 2: 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.

The 2-compartment LQR model & RepopulationThe 2-compartment LQR model & RepopulationExtending the LQ model to include:

• Redistribution - Asynchronous cycling cell population, preferentially spare cells in resistant part of cell cycle• Re-oxygenation - Surviving hypoxic cells move to more sensitive (oxic) state before next exposure

the LQR model accounts for cellular diversity through the dispersion about the mean radio-sensitivity . The average SF is obtained by assuming a Gaussian distribution of parameters, thus3

Extending the LQR model further to include 2-compartments of hypoxic and oxic tumour cells, Horas et al. obtained4

with

And comparing to experimental data of Buffa et al.5 of a human colon adenocarcinoma cell line obtained various tabulated radio-sensitivity

parameter values dependent on tumour size4.

Investigated are various forms of repopulation for the oxic portion of cells, conventional and accelerated treatment protocols, whilst observing

the changing proportions of the subpopulations and their dynamics throughout (changing R and therefore changing radio-sensitivity

parameters calculated using Lagrange interpolation).

Figure 2: 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.

2221exp DDSF

Table1: Previous work1 were typical values are 3 -10 Gy, specifically Adv. Head & neck = 20 GyNon-small-cell lung = 10 Gy Prostate = 1 Gy

Note lack of eradication in Gompertz case

22 expexp DDfDDfSF hyeff

hyeff

hyoxeff

oxeff

ox

3

30

32 1 ,2

R

rRRff hyox

eff