Applied Statistics for Biological Dosimetry Part 1

1. Applied Statistics for Biological DosimetryPart 1 Lecture Module 8

2. Radiation induces Chromosome Damage The yield of chromosome damage depends on dose, dose rate and radiation type Dose estimation is made using a calibration curve and any laboratory intending to carry out biological dosimetry should establish its own dose–response data

3. How to produce dose-effect calibration curve • Lymphocytes should be irradiated in vitro to approximate as closely as possible the invivo situation • Freshly taken blood specimens in lithium heparin tubes should be used and irradiated as whole blood at 37°C • After irradiation they should be held for a further 2 h at 37oC • Forlow LET radiation, 10 or more doses should be used in the range 0 – 5.0 Gy. However, if the laboratory is capable of obtaining data at doses below 0.25 Gy, this is very desirable • For high LET radiation a maximum dose of 2.0 Gy is suggested

4. Physics considerations • The preparation of a dose–response curve must be • supported by reliable and accurate physical dosimetry • The irradiation should be uniform • There must be sufficient material surrounding the • blood to provide charged particle equilibrium

5. Biological Considerations • Stimulate the lymphocytes with a mitogen (PHA) • Culture for 48-50h in presence of BrdU • Stain with FPG MI MII • To restrict the analysis to guaranteed first division cells

6. Number of cells to be analyzed • At higher doses, scoring should aim to detect • 100 dicentrics at each dose • At lower doses it is difficult to achieve 100 • dicentrics, and instead several thousand cells • per point should be scored; a number between • 3000 and 5000 is suggested • In all cases, the actual number of cells scored • should be dependent on the number of dose • points in the low dose region, with the focus on • minimizing the error on the fitted curve

7. Which aberrations to score ? Chromosome-type aberrations: dicentrics and rings

8. For low LET radiation there is very strong evidence that the yields of chromosome aberrations or micronuclei (Y) are related to dose (D) by the linear quadratic equation Linear quadratic function Y = C+  D +  D2 Forhigh LET radiation, the α-term becomes large and eventually the β-term becomes biologically less relevant and also statistically ‘masked‘ and the dose response is approximated by the linear equation Y = C+  D

9. Y = C+ αD + βD2 Linear term αD corresponds to one-track action, and quadratic term βD2 corresponds to two-track action. C is background frequency of dicentrics αD :dicentric formed by one track βD:dicentric formed by two tracks Dicentric, and other aberrations, formation, is believed to start with radiation induction of DNA double-strand breaks (DSB). A key assumption is that any DSB is made by traversal of one track but that track can continue on and produce second track elsewhere

10. Implications of linear quadratic model (1) • Ratio α/β delineates dose at which two terms contribute equally to number of dicentrics formed • For low-LETradiation at moderate or high doses, where there are many tracks per cell nucleus, but each track has only small probability of making one DSB and very small probability of making more than one DSB, quadratic term, βD2, dominates linear term

11. Implications of linear quadratic model (2) • For high-LETradiation at low doses, where typically there are very few radiation tracks per cell nucleus, each track typically makes a number of DSBs, linear term, αD, dominates • At sufficiently low doses of any type of radiation, when the average number of events per cell is less than one, the function αD + βD2 is reduced to αD • Close to the origin the slope is linear

12. Poisson or non-Poisson • Objective of curve fitting is to determine those values of coefficients C, α and β which best fit data points • For dicentrics, irradiation with X or gamma rays produces a distribution of damage which is very well represented by Poisson distribution • Key assumption is that variance (σ2) equals mean (y) • In contrast, neutrons and other types of high LET radiation produce distributions which display overdispersion, where variance exceeds the mean • For micronuclei data tend to overdispersion at all doses even with photon irradiation

13. However, it must be pointed out that since early radiobiology it was accepted that cellular damage induced by radiation is generally not a simple Poisson process Particles traversing cell nucleus follow a Poisson process Interation with DNA is a random process Repair–misrepair is another type of random process

14. Poisson tracks nuclei Poisson-distributed events: Tracks intersecting cell nuclei

15. Poisson distribution is discrete probability distribution that expresses probability of number of events occurring in fixed period of time if these events occur with known average rate and independently of time since last event • If expected number of occurrences in this interval is λ, then probability that there are exactly n occurrences (n being a whole number, n = 0, 1, 2, ...) is equal to:

16.  = frequency of dicentrics per cell  =0.25  =0.50  =1  =4 n Probability to have a cell (x axis) with n number of dicentrics. As the  increases the shape of the distribution trends to the normal distribution

17. Test data for conformity to Poisson • Because curve fitting methods are based on Poisson statistics, dicentric cell distribution should be tested for compliance with Poisson distribution for each dose point used to construct calibration curve

18. How to test? Parameter λ is not only the mean number of occurrences, but also its variance. Then the dispersion index would be 1

19. A normalized unit of DI is Papworth’s u test N indicates the number of cells analyzed and X the number of dicentrics detected u values higher than 1.96 indicate overdispersion (with a two-sided significance level, α = 0.025)

20. After low-LET radiation exposure only one dose shows overdispersion. Whereas after high-LET exposure three doses showed u values >1.96

21. Fitting dose response data to curve Objective of curve fitting is to find values for C, a and b for which curve is closest to observed data points

22. Fitting X ray data Dosecellsdicvar/mean 0. 5000 8 1 0.1 5002 14 1 0.25 2008 22 1 0.5 2002 55 1 0.75 1832 100 1 1.0 1168 109 1 1.5 562 100 1 2.0 332 103 1 3.0 193 108 1 4.0 103 103 1 5.0 59 107 1 Conforms with Poisson 10 doses, in Gy, equally distributed Observed frequency

23. Y = C+  D +  D2 maximizing likelihood of observations by the method of iteratively reweighted least squares Assuming the Poisson distribution

24. A common criterion for closeness is the sum of squares differences Iteratively Reweighted Least Squares

25. More accurate are data points, because more aberrations have been scored, closer curve should lie to the point Accuracy = SE/Y0 (100) less accuracy more accuracy

26. Common approach is to minimize SSD with weights of data points by inverse of their variance Because for Poisson distribution variance is equal to mean, weight used is the inverse of mean Iteratively Reweighted Least Squares

27. Handling overdispersion when curve fitting • For overdispersed (non-Poisson) distributions, as obtained after high LET radiation, weights must take into account the overdispersion • If data show a statistically significant trend of σ2/y with dose, then that trend should be used • Otherwise, the Poisson weight on each data point should be divided by the average value of σ2/y

28. First coefficients are obtained by minimising the equation • w (yo-yf)2 where yo is the observed yield, yf the expected yield from a linear-quadratic model and w = 1/(yo/N) is the weighting factor. N is the number of cells at each dose. Then the coefficients are recalculated using as weighting factor w = 1/(yf/N), obtaining new coefficients and new expected frequencies y’f for each dose. This procedure is repeated with new weighting factors, w = 1/(y’f/N) and so on, until the coefficients do not vary. Iteratively Reweighted Least Squares

29. Finally • Goodness of fit of curve and significance of fitted α and β coefficients should then be tested, for instance using Chi-squared (2) test and appropriate form of F- test (e.g., F-test, z-test or t-test) respectively • If there is evidence of lack of fit (i.e. 2 is greater than degrees of freedom (DF)), then standard error should be increased by (2/DF)1/2 • Additionally, as most of programs calculate the SE values based on sum of squares, instead of Poisson estimate of variance, it can be considered good practice to increase the SE by (DF/2)1/2

30. p values shown indicate that the fitted data points were not statistically different from the observed ones confirming a good fit Moreover the significance of the linear and quadratic coefficients was also confirmed by the F-test; for each coefficient the F value was higher than 3.44 (the cut off value for F.05 [8, 8]) and the z value was higher than 1.96 .

31. Several programs can be used to obtain coefficients Some of them can only be used for fitting While others are more user friendly and versatile with many other biodosimetry applications