Applied Statistics for Biological Dosimetry Part 2 Lecture Module

Applied Statistics for Biological Dosimetry Part 2 Lecture Module 8 IAEA International Atomic Energy Agency

Starting with appropriate dose response curve Radiation induces Chromsome Damage The yield of damage depends on dose, dose rate and radiation type IAEA 2

Y C D D2 Dicentrics per cel 2,00 1,60 1,20 y 0,80 yu yl 0,40 DL 0,00 0 1 2 3 D DU 4 5 Dose (Gy) Final goal in biological dosimetry is to convert an observed frequency of chromosomal aberrations, like dicentrics, into a dose IAEA 3

How many of patient’s cells to analyze? (1) To produce a dose estimate with statistical uncertainty small enough to be of value, large number of cells usually needs to be scored Decision is compromise based on case, available labour and quality of preparations IAEA 4

How many of patient’s cells to analyze? (2) There is no recommended single number of cells to be analyzed applicable in all cases For lower doses, where number of available cells is not limiting factor, dose estimate could be based on about 500 cells For a low or zero dicentric yield, confidence limits resulting from 500 scored cells are usually sufficient Decision to extend scoring beyond 500 to 1000 or more cells depends on evidence of a serious overexposure As a general rule it is suggested that 500 cells or 100 dicentrics should be scored in order to give reasonably accuracy IAEA 5

There is no difficulty in deriving dose from measured yield of dicentrics Dicentric (or dicentric plus centric ring) frequency is converted to absorbed dose by referring to appropriate in vitro calibration curve produced in the same laboratory with a comparable radiation quality 2 D 4 C Y 2 Procedure, which is simply solving this quadratic equation, provides an estimate of the averaged whole body absorbed dose IAEA 6

Difficulty comes when one wishes to determine the uncertainty on dose estimate There are a number of different ways in which the uncertainty on the yield can be derived Aim is to express uncertainty in terms of a confidence interval and it is standard practice to calculate 95% limits 95% confidence limits define interval that will encompass true dose on at least 95% of occasions IAEA 7

There is no absolute method for deriving confidence limits – it is always approximation Difficulty in computation of confidence limits arises because there are two components to uncertainty Uncertainties on calibration curve are distributed as normal probability function Uncertainties on measured aberration yield are usually Poissonian or overdispersed IAEA 8

Simplest method – Example 1 PARAMETERS OF THE DOSE-EFFECT CURVE Available from the output of the curve fitting software FROM THE PATIENT Five hundred cells were analysed and 25 of them were observed each to contain one dicentric. This gives a yield (in the formula,Y) of 0.05 dicentrics/cell. In this case the dispersion index and the u test were 0.95 and 0.78 respectively indicating that the cell distribution follows a Poisson Solving D 0.73 Gy Whole-body dose IAEA 9

Simplest method – Example 2 From observed yield of dicentrics and assuming the Poisson distribution, calculate yields corresponding to lower and upper 95% confidence limits on patient’s dicentric yield (YL and YU) IAEA 10

Simplest method – Example 3 Calculation of YU and YL Obs: 25 dic in 500 cells YL 16.768/500 0.034 YU 36.03/500 0.072 IAEA 11

Simplest method – Example 4 If u test statistic is higher than 1.96, YU and YL should be corrected by multiplying by a factor Where CL is Poisson confidence limit taken from standard table, X number of dicentrics observed, and σ2/y observed dispersion index Using same example, if instead of 25 cells with one dicentric, 19 cells with one dicentric and three cells with two were observed then the σ2/y will be 1.19, and the u value 3.19. In his case YU and YL are: IAEA 12

Simplest method – Example 5 To calculate dose at which YL intercepts upper curve. This is lower confidence limit (DL) on dose estimate. To calculate dose at which YU intercepts lower curve. This is upper confidence limit (DU). IAEA 13

Simplest method – Example 6 YL 16.768/500 0.034 upper curve YU 36.03/500 0.072 lower curve DL 0.51Gy Du 0.97Gy Calculation of point where YL and YU intercept upper and lower confidence curve, which are DL and DU, can be done by iteration IAEA 14

With well established calibration curves based on large amount of scoring, variance due to curve is small compared with variance on observed yield from subject and can be ignored. A simpler approximate estimate of DL and DU may be obtained directly from calibration curve, by considering where YL and YU cross solid line At 0.73 Gy the error associated with the present curve is 0.002. This value is obtained by inserting 0.73 Gy for D in the last term the following equation IAEA 15

Dose calculations for more complex exposures Situations that biodosimetry laboratories regularly encounter Low dose overexposures Partial body exposures Protracted and fractionated exposure and thankfully, rarely Critically accidents IAEA 16

Low dose overexposure cases The low dose detection limit depends on: 1. The background frequency of dicentrics 2. The uncertainties on the coefficients, particularly 3. The number of cells analyzed from the patient Generally for low-LET radiation the detection limit is around 100-200 mGy Because the ICRP recommended annual occupational dose limit is 20 mSv, there is often pressure on cytogenetics to try to resolve suspected low overdoses, perhaps pushing the method beyond its capabilities IAEA 17

Partial or whole body exposure? Whole body irradiation is the simplest scenario to describe mathematically but, normally heterogeneous exposures are more likely Y 0.0128 0.021D 0.0631D After whole body dose of 3 Gy gamma and using this curve total of 129 dic are expected to be found in 200 cells Expected Poisson dic distribution is shown in this table DI is close to 1 and u-test value lies between 1.96 IAEA 2 cells with X dicentrics 0 105 1 68 2 22 3 5 4 1 5 0 6 0 7 0 8 0 9 0 10 0 11 0 12 0 13 0 cells 200 dic 129 y 0,64 Var 0,65 DI 1,01 u 0,05 dic dicentrics; y frequency of dicentrics, Var variance; DI dispersion index (VAR/MEAN); u Papworth’s u 18

How to calculate u-test Frequency Variance y X N 0.64 cells with 0 dic 0 y 2 cells with 1 dic 1 y 2 . cells with n dic n y 2 VAR N 1 Dispersion Index U test X the total number of dicentrics, N the total analyzed cells DI N 1 u DI y 1 2 1 1 X IAEA VAR y 0.65 1.01 0.05 19

Partial irradiation This ‘dilutes’ dicentric frequency with undamaged cells cells with X dicentrics 0 105 1 68 2 22 3 5 4 1 5 0 6 0 7 0 8 0 9 0 10 0 11 0 12 0 13 0 cells 200 dic 129 y 0,64 Var 0,65 DI 1.01 u 0.05 Contaminated 305 68 22 5 1 0 0 0 0 0 0 0 0 0 400 129 0,32 0.43 1.33 4.61 Contamination with 200 unirradiated cells Same number of dicentrics in more cells. Frequency is lower, and consequently, if a whole body estimation is done, dose will be lower u value 1.96 indicates significant overdispersion IAEA 20

Unpicking a part-body exposure Two methods have been developed for resolving overdispersed distribution into its two components; sizes of unirradiated and irradiated fractions and dose to latter Qdr IAEA Contaminated Poisson 21

Contaminated Poisson (Dolphin) Method 1 YF X y N no 1 e YF is mean yield of dicentrics in irradiated fraction e–Y represents number of undamaged cells in irradiated fraction X is number of dicentrics observed N is total number of cells n0 is number of cells free of dicentrics Previous example of whole body exposure was 0.64 dic/cell which from dose response curve gives 3 Gy. By ‘diluting’ dicentrics with 200 undamaged cells mean frequency dropped to 0.32 dic/cell. However contaminated Poisson method restores dose estimate to irradiated fraction to 3 Gy IAEA 22

Contaminated Poisson (Dolphin) Method 2 YF can then be used to calculate the fraction, f, of cells scored which were irradiated Fraction obtained using example is 0.50 From 400 cells scored, 200 were non-irradiated It is also possible to estimate initial fraction of irradiated cells, representing fraction of body irradiated. This calculation takes into account a correction for effects of interphase death and mitotic delay Value p, fraction of irradiated cells that reach metaphase, is estimated D is estimated dose, and there is experimental evidence of D 0 values between 2.7 and 3.5. IAEA 23

Contaminated Poisson (Dolphin) method Limitations 1. Method assumes that exposure to irradiated fraction is homogeneous 2. It derives fraction of lymphocytes irradiated which can only be related to fraction of body irradiated by making the simplifying assumption that lymphocytes are uniformly distributed throughout body 3. It requires sufficiently high local dose so that there are number of cells observed with two or more dicentrics This is necessary for best-fit calculation of irradiated, but undamaged, cells 4. Method assumes minimal delay between irradiation and blood sampling, so that dicentric yield is not significantly diluted by newly formed undamaged cells entering circulation IAEA 24

Qdr Method 1 Y1 N Q dr N u 1 e Y1 Y2 This method considers yield of dicentrics and rings only in those cells that contain unstable aberrations and assumes that these cells were produced at time of accident Method therefore circumvents problems of dilution by undamaged cells from an unexposed fraction of body or post-irradiation replenishment from stem cell pool It also does not require presence of heavily damaged cells containing two or more aberrations. Qdr is expected yield of dicentrics and rings among damaged cells IAEA 25

Qdr Method 2 Y1 N Q dr N u 1 e Y1 Y2 X is number of dicentrics and rings, and Y1 and Y2 are yields of dicentrics plus rings and of excess acentrics, respectively. As Y1 and Y2 are known functions of dose and are derivable from in vitro dose–response curves, Qdr is function of dose alone and hence permits dose to irradiated part to be derived There also are some simplifying assumptions with this method: 1.It assumes, as does the contaminated Poisson method, that exposure to irradiated fraction is uniform 2.It assumes that excess acentric aberrations also have Poisson distributions, but this is not borne out by data from in vitro experiments – they tend to show overdispersion even from uniform irradiation IAEA 26

Protracted and fractionated exposure (1) Protraction or fractionation of a low LET exposure produces lower chromosome aberration yield than same acute dose For high LET radiation, where dose–response relationship is close to linear, no dose rate or fractionation effect would be expected For low LET radiation, however, effect of dose protraction is to reduce dose squared coefficient, β of linear-quadratic equation This term represents those aberrations, possibly of two track origin, which can be modified by repair mechanisms that have time to operate during protracted exposure or in periods between intermittent acute exposures IAEA 27

Protracted and fractionated exposure (2) Number of studies have shown that decrease in frequencies of aberrations appears to follow single exponential function with mean time of about 2 h Time dependent factor known as G function was proposed to enable modification of dose squared coefficient (β) and thus allow for effects of dose protraction For brief intermittent exposures, where interfraction intervals of more than six hours are involved, exposures may be considered as number of isolated acute irradiations for each of which induced aberration yields are additive IAEA 28

Protracted and fractionated exposure (3) G-function G function modifying β coefficient t is time over which irradiation occurred t0 is mean lifetime of breaks (about 2 h) IAEA 29

Protracted and fractionated exposure (4) G-function example Using curve parameters If 25 dicentrics are scored in 500 cells for 24 hours accidental exposure of person: y 24 12 2 G 0.153 G 0.96E 2 Solving Y C D GD 2 D 1.41 Gy This protracted dose is higher than what would be estimated for same dicentric frequency due to acute exposure, 0.73 Gy IAEA 30

Criticality accident Body is irradiated by both fission neutrons and gamma rays If ratio of neutron to gamma ray doses is known (this information is usually available from physical measurements) it is possible to estimate separate neutron and gamma ray doses by iteration The iteration process is: (1) Assume that all aberrations are attributable to neutrons, and from measured yield of dicentrics estimate dose from neutron curve (2) Use estimated neutron dose and supplied neutron to gamma ray ratio to estimate gamma ray dose (3) Use gamma ray dose to estimate yield of dicentrics due to gamma rays (4) Subtract this calculated gamma ray yield of dicentrics from measured yield to give new value for neutron yield (5) Repeat steps 1 to 4 until self-consistent estimates are obtained IAEA 31

Criticality accident – Example After critically accident 120 dicentrics have been observed in 100 cells (1.2 dic/cell). Neutron to gamma ratio, from physics, is 2:3 in absorbed dose. Using two dose effect curves for neutrons and gamma rays respectively: Neutrons Y 0.005 8.32x10-1D γ- rays Y 0.005 1.64x10 D 4.92x10 D -2 -2 2 dic/cell 1,200 0,934 (1) 1.20 dic/cell is equivalent to 1.442 Gy neutrons 1,033 (2) 1.442 Gy x (3/2) 2.163 Gy γ- rays 0,999 (3) 2.163 Gy γ- rays are equivalent to 0.266 dic/cell 1,011 (4) 1.20-0.266 0.934 dicentric yield attributable to neutrons (5) 0.934 dic/cell is equivalent to 1.122 Gy neutrons IAEA 1,006 1,008 1,007 1,008 Iteration 1 2 3 4 5 6 7 8 9 Dose N Dose γ Dose N Dose γ Dose N Dose γ Dose N Dose γ Dose N Dose γ Dose N Dose γ Dose N Dose γ Dose N Dose γ Dose N Dose γ Doses 1,442 2,163 1,122 1,683 1,240 1,861 1,200 1,800 1,214 1,821 1,209 1,814 1,211 1,816 1,210 1,815 1,210 1,816 dic equivalent to γ-rays 0,266 0,167 0,201 0,189 0,194 0,192 0,193 0,192 0,192 Repeating stage 2, 1.122 x3/2 1.683 Gy γ-rays. After a few iterations estimated doses are 1.21 Gy for neutrons and 1.82 Gy 32 for γ-rays

Final comment Don’t panic about doing statistics and calculations It is more important that you understand principles of how to estimate dose uncertainties and how to deal with complications such as inhomogeneity, dose protraction and mixed radiations For example, if you see overdispersion don’t just give the averaged dose User friendly, freely available software packages: CABAS and Dose Estimate could assist IAEA 33