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
yu
yl
0,80
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
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
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+0.0631D2
• 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
IAEA
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 1dic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 D0
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
Qdr 

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
Qdr 

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
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:
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
dic/cell
1,200
Iteration
1
0,934
2
(1) 1.20 dic/cell is equivalent to
1.442 Gy neutrons
1,033
3
(2) 1.442 Gy x (3/2) = 2.163 Gy γ- rays
0,999
4
(3) 2.163 Gy γ- rays are
equivalent to 0.266 dic/cell
1,011
5
1,006
6
1,008
7
1,007
8
1,008
9
γ- rays Y=0.005 +
1.64x10-2D
+
4.92x10-2D2
(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
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
dic equivalent to γ-rays
1,442
2,163
0,266
1,122
1,683
0,167
1,240
1,861
0,201
1,200
1,800
0,189
1,214
1,821
0,194
1,209
1,814
0,192
1,211
1,816
0,193
1,210
1,815
0,192
1,210
1,816
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