Ruido admisible

3.5.1 Marsden TCP model

The Marsden TCP model developed by Nahum-Tait [31], Webb and Nahum [27] and Nahum and Sanchez-Nieto [32] is a quasi-mechanistic model based on LQ model of cell killing and Poisson statistics. In this model, the surviving fraction of cells after radiotherapy is expressed as a function of dose, characterised by two parameters α and β which are related to the initial slope and curvature of the cell survival curve. The surviving fraction of cells after uniform irradiation with dose D is given by equation 3.1.

SF = Ns

N0

=exp(−αD−βD2) (3.1)

The parameters α and β are proportional to the dose and square of the dose respectively and hence the name linear-quadratic model [129]. The number of cells surviving after irradiation of all the fractions is calculated by the formula given in equation 3.2. Ns =N0exp " −αD 1 + β αd !# (3.2)

where Ns is the number of cells that survive after irradiation, N0 is the initial

number of clonogenic cells, α is the cell radiosensitivity, D=nd in which n is the number of fractions, D is the total dose, d is the dose per fraction.

The Marsden model employs Poisson statistics given in equation 3.3 is used to calculate TCP. The Poisson equation gives the probability P of occurrence of exactly Y events when the mean number of events is N.

P(N,y) = e −N_Ny y! (3.3) TCP = 1 σα √ 2π Z ∞ 0 Y i exp " ρclViexp ( −αDi 1 + β αdi !)#! exp[−(α−α¯)2/2σ2]dα (3.4) In equation 3.4, i is the number of dose bins in the DVH, Di is the total dose in bin i in Gy, d is the dose per fraction in Gy, α and β are the clonogenic

radiosensitivities in the LQ expression, ¯αis the mean of the radiosensitivity within the patient population, σα is the distribution of radiosensitivity over the patient population, ρcl is the clonogenic cell density in clonogens/cm3. Another term

γ(T-Tk) is sometimes included in the square brackets of equation 3.4 to account for proliferation or repopulation of the tumour cells over the treatment period; this is valid only when T>Tk. Here, γ=ln 2/Td, Td is the average doubling time, T is the overall treatment time, Tk is the time at which proliferation or repopulation starts. Vi is the volume of the GTV corresponding to the dose bin

di in the GTV, Di of the ith dose voxel and is derived from the dDVHs of the GTVs.

3.5.2 Marsden TCP model parameters

NSCLC with tumour local control as the end point

The parameters for the Marsden TCP model were derived by Nahum et al. [130] by fitting data sets to clinically observed tumour control data for four radiother- apy regimens. The Marsden TCP model is characterised by six main parameters (see equation 3.4) out of which the mean radiosensitivity ¯α and distribution of radiosensitivity over a population of patients σα, delay before repopulation Tk, doubling time Td were obtained by fitting. The clonogenic cell density ρcl and

α/β ratio were kept constant. These parameters were derived by fitting to the Marsden quasi-mechanistic LQ and population based model using DVH data sets to clinical outcome data of the control and trial arms of the study done on con- tinuous hyper accelerated radiation therapy (CHART) by Mount Vernon Cancer Centre published by Saunders et al. [131], our own clinical experience at Clat- terbridge Cancer Centre and UMCC data of the work published by Martel et al. [37]. The clinically observed tumour control probabilities reported in these studies were 12%, 18%, 35% and 43% respectively. The parameters and their values used in this study are shown in table 3.1.

NSCLC SABR with tumour local control as the end point

have reported clinically observed TCPs (with an end point of 3 year primary tu- mour control) as high as 97.6% in patients with early stage medically inoperable non-small cell tumours measuring less than 5 cm diameter treated with hypofrac- tionated doses of 18 Gy in 3 fractions. However, the validity of the LQ model is in doubt when large doses per fraction are delivered. It is suggested that LQ model fails at larger dose per fraction and the cell survival curve tends to straighten out at daily doses larger than 7 Gy exhibiting a linear-quadratic-linear (LQL) relationship. Guerrero and Li [133] proposed a method to extend the LQ model for large fraction doses such as those delivered in SABR treatments. Carlone et al. [134] suggested a way to derive the Guerrero and Li LQL model mechanisti- cally. Alternatively, Fowler [135] has suggested an α/β value of 20 Gy instead of 10 Gy for NSCLC tumours.

In our study, we derived a separate TCP parameter set for NSCLC SABR treatments in order to address the issue of validity of the LQ model in large dose per fractions. We propose to use a lower ¯α value instead of using a higher α/β

ratio (greater than 20 Gy). In order to do this, ¯α and σα values were lowered by keeping the same ¯αtoσα ratio of NSCLC TCP parameter sets derived by Nahum

et al. until the TCP value calculated using a SABR DVH dropped to around 90% from the highest 100%. The rest of the Marsden TCP model parameters were kept the same as the original. The new ¯α to σα values are 0.14 Gy−1 and 0.017 Gy−1 respectively which are used to estimate the TCPs of SABR NSCLC patient plans as reported in chapter 6.

NPC with tumour local control as the end point

The Marsden model parameter set for NPC derived by Selvaraj et al. [136] is used to predict the TCPs reported in chapter 7. The mean radiosensitivity, ¯α

and variation in the radiosensitivity in a patient population,σα were obtained by adjusting these two parameters iteratively to match theγ50 value of 2.8 reported

by Steel [137] for NPC. The ¯αand σα values hence derived for NPC are 0.3 Gy−1 and 0.048 Gy−1 _{respectively.}

Table 3.1: Marsden TCP model parameters for NSCLC 3DCRT, NSCLC SABR and NPC with local control as the end point.

TCP parameters NSCLC 3DCRT NSCLC SABR NPC IMRT ¯ α[Gy−1_{] 0.307 0.140 0.300} σα [Gy−1] 0.037 0.017 0.048 ρcl [clonogens/cm3] 107 107 107 α/β[Gy] 10 10 10 Tk [days] 20.9 21.0 21.0 Td [days] 3.7 3.0 3.0

3.5.3 Effect of model parameters on the TCP curve

As explained in section 3.5.1, the Marsden TCP model is a four parameter model, the parameters being mean value of the intrinsic radiosensitivity over a patient population ¯α, statistical uncertainty on the intrinsic radiosensitivity over a patient population σα, time at which repopulation kicks in Tk, and tumour doubling time Td. The shape and slope of the predicted TCP curve depends upon the derived model parameters. A higher ¯αvalue indicates that the patient population responds to a lower dose or in other words, the TCP curve shifts towards the left hand side on the plot where absolute total dose along is specified along the x-axis and probability of local control along the y-axis. The σα value governs the slope of the curve, the lower the value the steeper is the TCP curve. The effect of varying ¯α for a given σα and vice-versa on the TCP curve by keeping the other parameters constant is shown in figures 3.4(a) and 3.4(b).