Applied Statistics for
Biological Dosimetry
Part 1
Lecture
Module 8
IAEA
International Atomic Energy Agency
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
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How to produce dose-effect calibration
curve
Lymphocytes should be irradiated in vitro to approximate as
closely as possible the in vivo 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
For low 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
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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
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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
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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
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Which aberrations to score ?
Chromosome-type aberrations: dicentrics and rings
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Linear quadratic function
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
Y = C +  D +  D2
For high 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
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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
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Implications of linear quadratic model (1)
• Ratio α/β delineates dose at which two terms
contribute equally to number of dicentrics formed
• For low-LET radiation 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
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Implications of linear quadratic model (2)
•For high-LET radiation 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
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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
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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
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Repair–misrepair is
another type of random
process
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Poisson
tracks
nuclei
Poisson-distributed events: Tracks intersecting cell nuclei
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• 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:
f n;    
e
e
0!
1!
0

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
n

e
n!
e
e
e
2!
3!
4!
2

3

4

15
 = 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
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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
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How to test?
f n;    
n

e
n!
Parameter λ is not only the mean number of
occurrences, but also its variance.

2


2
Then the dispersion index would be 1
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
1
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A normalized unit of DI is Papworth’s u test
u  (
2
N 1
y  1)
2(1  1 )
X
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)
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g-rays (Cobalt-60)
dose (Gy)
0.000
0.100
0.250
0.500
0.750
1.000
1.500
2.000
3.000
4.000
5.000
20 MeV 4He particles
dose (Gy)
0.000
0.051
0.104
0.511
1.010
1.536
2.050
2.526
3.029
N
5000
5002
2008
2002
1832
1168
562
332
193
103
59
N
2000
900
1029
1136
304
142
137
144
98
X
8
14
22
55
100
109
100
103
108
103
107
0
4992
4988
1987
1947
1736
1064
474
251
104
35
11
cell distribution of dicentrics
1
2
3
4
5
6
8
14
20
1
55
92
4
99
5
76
12
63
17
2
72
15
2
41
21
4
2
19
11
9
6
3
0
1997
881
1004
960
217
75
63
66
47
cell distribution of dicentrics
1
2
3
4
5
6
3
19
23
2
154
21
1
69
15
3
40
25
2
44
16
12
2
34
25
14
3
2
16
17
17
0
X
3
19
27
199
108
96
120
148
108
D
u
1.00
1.00
1.08
0.97
1.03
1.00
1.06
1.14
0.83
0.88
1.15
-0.07
-0.13
2.61
-0.86
0.79
-0.02
1.08
1.82
-1.64
-0.84
0.81
D
u
1.00
0.98
1.12
1.07
1.09
0.98
1.20
1.40
1.56
-0.04
-0.44
2.84
1.60
1.15
-0.20
1.65
3.40
3.93
7
1
After low-LET radiation exposure only one dose shows overdispersion.
Whereas after high-LET exposure three doses showed u values >1.96
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Fitting dose response data to curve
2,5
2
1,5
1
0,5
0
0
1
2
3
4
5
Objective of curve fitting is to find values for C, a and b for which
curve is closest to observed data points
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Fitting X ray data
10 doses, in Dose cells
5000
Gy, equally 0.
5002
distributed 0.1
0.25 2008
0.5
2002
0.75 1832
1.0
1168
1.5
562
2.0
332
3.0
193
4.0
103
5.0
59
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dic var/mean
8
1
14
1
22
1
55
1
100 1
109 1
100 1
103 1
108 1
103 1
107 1
Conforms
with Poisson
Observed
frequency
22
Y = C +  D +  D2
maximizing likelihood of
observations by the method of
iteratively reweighted least squares
Assuming the Poisson distribution
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2,5
2
1,5
1
0,5
0
0
1
2
3
4
5
A common criterion for closeness is the sum of squares differences
SSD   Y0  Y f 
n
2
i 1
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Iteratively Reweighted Least Squares
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More accurate are data points, because more aberrations
have been scored, closer curve should lie to the point
Accuracy = SE/Y0 (100)
DOSE
cells
Aberrations
Accuracy
0
5000
8
35.4
0.1
5000
14
26.7
0.25
2008
22
21.3
0.5
2002
55
13.5
0.75
1832
100
10.0
1
1168
109
9.6
1.5
562
100
10.0
2
332
103
9.9
3
193
108
9.6
4
103
103
9.9
5
59
107
9.7
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less accuracy
more accuracy
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Common approach is to minimize SSD with
weights of data points by inverse of their variance


Y

Y

Y  Y ( )
SSD

 
SSD  
n
2
n
i
0
f
i
ii
2
i i11
2
2
1
1
w
 2
w
2
i
Because for Poisson distribution variance is
equal to mean, weight used is the inverse of mean
Iteratively Reweighted Least Squares
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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
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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
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Finally
• Goodness of fit of curve and significance of fitted α and β
coefficients should then be tested, for instance using Chisquared (2) test and appropriate form of F- test (e.g., Ftest, 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
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p values shown indicate that the fitted data points were not statistically
different from the observed ones confirming a good fit
g-rays (Cobalt-60)
C  SE
 (Gy-1) SE
 (Gy-2) SE
0.00128  0.00047
t = 2.72, p < 0.03
0.02103  0.00516
t = 4.08, p<0.005
0.06307  0.00401
t = 15.73, p<<0.01
2
DF
6.61
8
P = 0.58
20 MeV 4He particles
C  SE
 (Gy-1) SE
 (Gy-2) SE
0.00143  0.00093
t = 1.54, p = 0.17
0.32790  0.02875
t = 11.41, p<0.001
0.02932  0.01636
t = 1.79, p = 0.19
0,00193  0,00097
t = 1,99, p = 0.09
0,37290  0,01787
t = 20,87, p<<0.01
2
df
7.40
7
p = 0.39
10,91
7
P=0.14
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 .
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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
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