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Applied Statistics for Biological Dosimetry Part 2

Applied Statistics for Biological Dosimetry Part 2. Lecture Module 8. Starting with appropriate dose response curve. Radiation induces Chromsome Damage. The yield of damage depends on dose, dose rate and radiation type. Y = C +  D +  D 2. y u. y. y l. D U. D L. D.

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Applied Statistics for Biological Dosimetry Part 2

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  1. Applied Statistics for Biological DosimetryPart 2 Lecture Module 8

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

  3. Y = C+  D +  D2 yu y yl DU DL D Final goal in biological dosimetry is to convert an observed frequency of chromosomal aberrations, like dicentrics, into a dose

  4. 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

  5. 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

  6. 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 Procedure, which is simply solving this quadratic equation, provides an estimate of the averaged whole body absorbed dose

  7. 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

  8. 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

  9. Simplest method – Example 1 • C = 1.28E-3 • α = 2.10E-2 • β = 6.31E-2 1941 • var C = 2.22E-07 • var α = 2.66E-05 • var β =1.61E-05 • covar (C, α) = -9.95E-07 • covar (C, β) = 4.38E-07 • covar (α,β) = -1.512E-05 PARAMETERS OF THE DOSE-EFFECT CURVE Availablefromthe 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-bodydose

  10. 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)

  11. 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

  12. 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:

  13. 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).

  14. Simplest method – Example 6 YL=16.768/500=0.034 upper curve DL = 0.51Gy lower curve YU = 36.03/500= 0.072 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

  15. 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

  16. 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

  17. The low dose detection limit depends on: Low dose overexposure cases 1. The backgroundfrequency of dicentrics 2. The uncertaintieson 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

  18. 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+0.0631D2 • 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 dic = dicentrics; y = frequency of dicentrics, Var= variance; DI= dispersion index (VAR/MEAN); u= Papworth’s u

  19. How to calculate u-test Frequency 0.64 X the total number of dicentrics, N the total analyzed cells Variance 0.65 DispersionIndex 1.01 U test 0.05

  20. Partial irradiation This ‘dilutes’ dicentric frequency with undamaged cells 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

  21. 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 Contaminated Poisson

  22. Contaminated Poisson (Dolphin) Method 1 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

  23. 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 D0 values between 2.7 and 3.5.

  24. Contaminated Poisson (Dolphin) method Limitations • Method assumes that exposure to irradiated fraction is homogeneous • 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 • 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 • Method assumes minimal delay between irradiation and blood sampling, so that dicentric yield is not significantly diluted by newly formed undamaged cells entering circulation

  25. QdrMethod 1 • 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

  26. QdrMethod 2 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: It assumes, as does the contaminated Poisson method, that exposure to irradiated fraction is uniform 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

  27. 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

  28. 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

  29. Protracted and fractionated exposure (3) G-function G function modifying β coefficient t is time over which irradiation occurred t0is mean lifetime of breaks (about 2 h)

  30. Protracted and fractionated exposure (4) G-function example • C = 1.28E-3 • α = 2.10E-2 • β = 6.31E-2  to be corrected Using curve parameters If 25 dicentrics are scored in 500 cells for 24 hours accidental exposure of person: Solving 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

  31. 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

  32. 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 + 1.64x10-2D + 4.92x10-2D2 (1) 1.20 dic/cellisequivalentto 1.442 Gy neutrons (2) 1.442 Gy x (3/2) = 2.163 Gy γ- rays (3) 2.163 Gy γ- rays are equivalentto 0.266 dic/cell (4) 1.20-0.266=0.934 dicentric yield attributable to neutrons (5) 0.934 dic/cellisequivalentto 1.122 Gy neutrons Repeating stage 2, 1.122 x3/2 = 1.683 Gyγ-rays. After a few iterations estimated doses are 1.21 Gy for neutrons and 1.82 Gy for γ-rays

  33. 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: CABASandDose Estimate could assist

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