2.7 Statistical data analysis
2.7.1 Error estimation for survival experiments
The PE (see Eq. 2.2) of unirradiated control samples was determined in every survival experiment and it defines the 100% survival level for a specific experiment. All irradi-ated samples of this specific experiment have been normalized to this value according to Eq. 2.3. The ratio of sample numbers of unirradiated controls to irradiated samples was in all experiments described smaller or equal to 1:4.
In the following paragraph the error estimation for survival experiments is presented within two different cases and explained. In the first part, the error estimation for complete survival curves is presented while in the second part, the error calculation for isolated sur-vival values or experiments, which are done once is shown. The error calculation for the survival experiments with multiple samples has been carried out as followed. The average survivalSof the survival values S1, S2..., Sncorresponding to one dose was calculated as well as their standard deviationσS and their standard error of the mean SEMS in which the number of samples is given byn.
S = 1 n
n
X
i=1
Si (2.4)
σS = v u u t
1 n−1
n
X
i=1
(Si−S)2 (2.5)
SEMS = σS
√n (2.6)
From the linear-quadratic fit to the survival data the parametersαandβwere determined with their averagesαandβas well as their standard deviationsσα andσβ.
σα = v u u t
1 n−1
n
X
i=1
(αi−α)2 (2.7)
σβ = v u u t
1 n−1
n
X
i=1
(βi−β)2 (2.8)
For the radio-resistance parameterα/βthe relative errorσα/β was performed according to Eq. 2.9.
∆α/β = σα/β α/β =
s (σα
α )2+ (σβ
β )2 (2.9)
The error estimation for survival experiments which were done once was developed [136].
The survival within the linear-quadratic model is given by Eq. 2.10, andEis the associated effect as defined in Eq. 2.11.
S =e−(αD+βD2) (2.10)
E =αD+βD2 (2.11)
The error is on the one hand influenced by the fluctuation of the data points around the individual survival curve (ES) and on the other hand changed by the variation of the en-semble of the curves (Eα,β). The error contributions ofEsum up to∆Ein Eq. 2.12.
∆E = q
∆ES2 + ∆Eα,β2 (2.12)
The first factor contributing to the error is caused by the fluctuation of survival levels around their corresponding linear-quadratic curve. The associated error∆ESis calculated as the standard deviation of the measured effect levelsEi of unirradiated samples.
∆ES = v u u t
1 n−1
n
X
i=1
(Ei−E)2 (2.13)
Inserting Eq. 2.11 in the definition of the standard deviation resulting in Eq. 2.14.
∆Eα,β = q
σα2D2−2σα,β2 D3 +σβ2D4 (2.14)
The factorsσαandσβ were determined according to Eqs. 2.7 and 2.8 at which beam and energy dependent meanαandβ values were determined toα andβ. For the calculation of the curve fluctuation Eα,β it was assumed that α andβ are inversely correlated. The values were joined in Eq. 2.15 to calculate the third parameterσα,β.
σα,β = v u u t− 1
n−1
n
X
i=1
[(αi−α)(βi−β)] (2.15)
2.7 Statistical data analysis
Table 2.2:The table shows the error percentage of survival for selected doses calculated with error propagation for x-rays and two carbon ion energies.
Cell line Dose [Gy] 250 kVp x-ray 12C 100 MeV/u 12C 11.4 MeV/u
RAT-1 0 9.3 9.3 9.3
1 10.6 10.2 9.4
3 17.8 17.4 10.1
6 41.3 64.1 14.6
IEC-6 0 12.0 12.0 12.0
1 13.3 14.8 12.0
3 19.3 27.4 12.5
6 32.8 62.6 15.3
The upper and the lower error limits of the survival were calculated according to Eqs. 2.16 and 2.17. The error limits have almost equal distances to the mean survival. Henceforth, the error interval was regarded to be symmetric with respect to the mean survival values.
S =e−(E±∆E) (2.16)
∆S = Supper−Slower
2 = e−(E−∆E)−e−(E+∆E)
2 (2.17)
In table 2.2 the error percentage of survival for selected doses is shown. Both cell lines show in principle similar error values. The values increased with increasing doses due to higher uncertainties and variances in the survival levels to higher doses. This biological effect is expressed by the termσβ2D4in Eq. 2.14. Overall, the errors of 11.4 MeV/u irra-diation are smaller than the other irrairra-diation error since theβ containing terms converge to zero.
2.7.2 Calculation of RBE
The effect of different beam qualities on cells can be compared with the relative biological effectiveness (RBE). The RBE value calculation for RAT-1 cells and IEC-6 cells differs since the radio-resistance of RAT-1 cells stays stable and the radio-resistance of IEC-6 cells against x-rays increases clearly with cultivation time as discussed in section 4.2.
This causes changing α values for x-ray irradiation and higher doses for 10% survival
levels. It was not possible to detect this behavior also for carbon ions irradiation due to the limited amount of data. In the following, the calculation of errors for both cell lines is described.
RAT-1 cells
For every survival curve the parameters α and β as well as their average α andβ were determined for different energies and beams. Likewise, the standard deviationσαandσβ were calculated. The RBEα was determined according to Eq. 1.5 and the relative error of RBEαwas done according to Eq. 2.18.
∆RBEα = σRBEα RBEα =
rσα α
2 ion
+σα α
2
x−ray (2.18)
The RBE for 10% survival (RBE10) was calculated according to the same formalism as used for RBEαwith the doses for 10% survival.
∆RBED10= σRBE10 RBE10 =
sσD10 D10
2 ion
+σD10 D10
2 x−ray
(2.19)
IEC-6 cells
For carbon ion survival curves the parameterαwas determined with its means and stan-dard deviations as described for RAT-1 cells. For x-ray survival curves theαvalues were plotted versus passage number and fitted by a line, see figure 2.8. The error ofαwas set to be the 68% confidence interval. At the mean passage number of the performed carbon ion experiments (17.67) the α value with its deviation was evaluated (0.354± 0.021). The RBE for 10% survival was calculated along the lines of RBEα with the doses for 10%
survival. The determinedD10at passage number 17.7 was 5.0±0.13 Gy.
2.7.3 Analysis of chromosome samples
For each chromosome sample 300 metaphases were analyzed according to their chromo-some numbers. Furthermore, the mitotic index in 6000 cells per sample was counted.
From the received data a weighted mean m and a weighted standard deviation σm, for chromosome numbers or mitotic index, were calculated according to the following Eqs., where x is the data and w the weights withP
iwi = 1.
m =
Pxiwi
Pwi (2.20)
2.7 Statistical data analysis
(a)αvalues (b) D10values
Figure 2.8:αvalues and D10 of IEC-6 cell x-ray survival curves. The vertical line indicates the mean passage number of performed carbon ion experiments. Closed circles: serum batch I, open circles: serum batch II
σm = sP
x2iwi Pwi
−m2 (2.21)