Treatment of Uncertainties
in Radiation Dosimetryy
Michael G.
G Mitch,
Mitch Ph.D.
Ph D 1
Larry A. DeWerd, Ph.D.2
Ronaldo
o do Minniti,, Ph.D.
. .1
Jeffrey F. Williamson, Ph.D.3
1Physics
Laboratory, National Institute of Standards and Technology (NIST)
2Deptartment
3Department
Of Medical Physics, University of Wisconsin-Madison
of Radiation Oncology, Virginia Commonwealth University
Why is Uncertainty Analysis Important?
1.
Assessment of the quality of a measurement or calculation
2.
Quantitative comparison of results from different investigators
3.
Critical analysis of measurement or calculation method
“Have I thought about all possible factors that influence the result of
my measurement or calculation?”
calculation?
.
Dw = 14.28
14 28 mGy / s
.
Dw = 14.28
14 28 mGy / s
.
Dw = (14.28 ± 0.12) mGy / s
.
Dw = 14.28
14 28 mGy / s
.
Dw = (14.28 ± 0.12) mGy / s
Uncertainty Component
Heat defect
Reproducibility of measurement groups
Beam attenuation from glass wall
Beam attenuation from calorimeter lid
Field size
Vessel positioning
Thermistor calibration
Water density
Quadratic sum
Type A
(%)
Type B
(%)
0.30
0.15
0.10
0.05
0.23
0.02
0.01
0 02
0.02
0.16
0.39
Relative combined standard uncertainty
0.42 %
Relative expanded uncertainty (k = 2)
0.84 %
Error vs. Uncertainty
Error = Difference between a measured or calculated value of a quantity and
the “true” value (unknowable)
Uncertainty = An interval about the average value of a series of measurements
or calculations
l l ti
which,
hi h within
ithi a certain
t i level
l l off confidence,
fid
is
i believed
b li d
to contain the “true” value of a quantity
Error vs. Uncertainty
Error = Difference between a measured or calculated value of a quantity and
the “true” value (unknowable)
Uncertainty = An interval about the average value of a series of measurements
or calculations
l l ti
which,
hi h within
ithi a certain
t i level
l l off confidence,
fid
is
i believed
b li d
to contain the “true” value of a quantity
NOTE: A measurement or calculated result with a low uncertainty is not
necessarily a result of high quality.
Method of Classifying Uncertainties
Type A Uncertainty = calculated by statistical methods
Type B Uncertainty = evaluated by other means
1981 – CIPM (Comité International des Poids et Mesures)
1993 – GUM (Guide to the Expression of Uncertainty in
Measurement),
) ISO
SO ((International
i lO
Organization
i i for
f
Strandardization)
1994 – NIST (National Institute of Standards and Technology)
Technical Note 1297
Method of Classifying Uncertainties
Type A Uncertainty = calculated by statistical methods
Type B Uncertainty = evaluated by other means
Random Effect = the variation in the results of measurements or calculations
that averages to the (true value ± bias) over many iterations
Systematic Effect = an error that is constant for each iteration = bias (unknown)
Precision = random effects only
Accuracy = random and systematic effects
Method of Evaluating Uncertainties
Type A Uncertainty = standard deviation of the mean, uA = s
n
 1

s
( zi  z ) 2 

 n(n  1) i 1

1/ 2
z
1 n
 zi
n i 1
Type B Uncertainty = scientific judgment, uB
1 instrument manufacturer’s
1.
manufacturer s specifications
2. investigator’s knowledge and experience
uB 
a-
a+
a  a
2 3
Combined Standard Uncertainty, uc
y  f ( x1 , x2 ,..., x N )
 N  f
u c   
 i 1  xi
2
N 1 N

 2
f f
 u ( xi )  2 
u ( xi , x j ) 
i 1 j i 1x i x j


u 2 ( xi )  u A2 ( xi )  u B2 ( xi )
u ( xi , x j ) 
1 n
 ( xik  xi )( x jk  x j )
n  1 k 1
1/ 2
Combined Standard Uncertainty, uc
y  f ( x1 , x2 ,..., xN )
 N  f
u c   
 i 1  xi
2
N 1 N

 2
f f
 u ( xi )  2 
u ( xi , x j ) 
i 1 j i 1x i x j


1/ 2
u 2 ( xi )  u A2 ( xi )  u B2 ( xi )
u ( xi , x j ) 
1 n
 ( xik  xi )( x jk  x j )
n  1 k 1
THE LAW OF PROPAGATION OF UNCERTAINTY
Combined Standard Uncertainty, uc
If all variables xi are independent, then u(xi, xj) = 0
 N  f
u c   
 i 1  xi
2

 2
 u ( xi )


1/ 2
Sums and differences
Products and quotients
y  x1  x2
y  x1 x2
y  x1  x2
y  x1 x 21
u c  u A ( x1 )   u B ( x1 )   u A ( x 2 )   u B ( x 2 ) 
 
 
 

 
y  x1   x1   x 2   x 2 

2

u c  u A2 ( x1 )  u B2 ( x1 )  u A2 ( x 2 )  u B2 ( x 2 )

1/ 2

2
2
%u c  %u A2 ( x1 )  %u B2 ( x1 )  %u A2 ( x 2 )  %u B2 ( x 2 )

2 1/ 2
1/ 2



Interpretation of y ± uc
• For a normal distribution with mean  and standard deviation ,
the interval  ±  contains 68.27 % of the distribution.
• Assuming
A
i that
h the
h di
distribution
ib i associated
i d with
i h the
h results
l from
f
our measurements
or calculations is approximately normal (and we perform enough iterations), then
the interval y ± uc contains about 68 % of the distribution, and we state that the
“true”
true value is believed to lie within this interval with a 68 % level of confidence.
confidence


Expanded Uncertainty, V
• When the results of measurements and calculations are to be used where
health and safety are a concern (such as in medical physics), an expanded
uncertainty is used.
V = kuc
k is the coverage factor
• NIST primary standards for all dosimetric quantities in medical physics
use k = 2, corresponding to an interval with a 95 % level of confidence.

2
Student’s t
• If the number of measurements is small, one should consider using the t value
to calculate a confidence interval.
y ± tuc
No. of Deg. of
meas. freedom 68.27 % 95.45 %
(n)
( = n – 1) (k = 1) (k = 2)
2
1
1.84
13.97
10
9
1 06
1.06
2 32
2.32
20
19
1.03
2.14
1.00
2.00
∞
Uncertainty Budget, NIST SK Standard for 125I seeds
 W  d 2
S K  K air (Q )d 2   
 e   air Veff

 K dr ( K ) M det ( K , Q ) K i  K j (Q )

i
j

Value
Net current, M det ( K , Q)
W /e
Air density, ρair
Aperture distance, d
Effective chamber volume, Veff
Decay correction, K1
Recombination K dr (K )
Recombination,
Attenuation in filter, K3(Q)
Air attenuation in WAFAC, K4(Q)
Source-aperture attenuation, K5(Q)
Inverse-square correction, K6
Humidity, K7(Q)
In-chamber photon scatter, K8(Q)
Source-holder scatter, K9
Electron loss, K10
Aperture penetration, K11(Q)
External photon scatter, K12(Q)
Combined standard uncertainty, uc
Expanded uncertainty, V
33.97 J / C
1.196 mg / cm3
T1/2 = 59.43 d
< 1.004
1 004
1.0295
1.0042
1.0125
1.0089
0.9982
0.9966
0.9985
1.0
0.9999
1.0
Type A (%)
Type B (%)
ssI
0.11
-
0.06
0.15
0.03
0.24
0.01
0.02
0 05
0.05
0.61
0.08
0.24
0.01
0.07
0.07
0.05
0.05
0.02
0.17
(s2 + 0.7622)1/2
2uc
AAPM TG-138:
Photon Brachytherapy Source Dosimetric Uncertainty Analysis
Larry DeWerd (Chair), Geoffrey Ibbott, Ali Meigooni, Michael Mitch,
Mark Rivard, Kurt Stump, and Bruce Thomadsen
1 Measurement
1.
M
t andd Monte
M t Carlo
C l uncertainties
t i ti
2. Uncertainty in TG-43 formalism parameters
3. Transfer of NIST SK standard to ADCLs
4. Uncertainty in clinical measurements
Measurement Traceability for Brachytherapy Sources
sources
ADCL
well-ionization
chambers
SK
secondary standard
verification for
treatment planning
Clinic
sources
Manufacturer
sources
SKClinic
Measurement Traceability for Brachytherapy Sources – Uncertainties
seed
SK (± 0.8 %)
ADCL
seed
Manufacturer
SK / IADCL (± 0.9 %)
Clinic
Measurement Traceability for Brachytherapy Sources – Uncertainties
ADCL
SKADCL ((± 1.1 %))
seedd
WIC
SKADCL / IClinic (± 1.2 %)
Clinic
Manufacturer
Measurement Traceability for Brachytherapy Sources – Uncertainties
ADCL
Manufacturer
WIC
seed (SKM)
Clinic
SKClinic (± 1.3 %)
Measurement Traceability for Brachytherapy Sources – Uncertainties
Step in
chain
Measurement Description
Quantity (Units)
Relative Propagated
Uncertainty (%)
1
NIST WAFAC calibration
SK (U)
0.80
2
ADCL well-ion chamber calibration
SK / IADCL (U / A)
0.94
3
ADCL calibration of seed from manufacturer
SKADCL (U)
1.06
4
ADCL calibration of Clinic well-ion chamber
SKADCL / IClinic (U / A)
1.17
5
Clinic measures seed air-kerma strength
SKClinic (U)
1.27
Expanded uncertainty (k = 2)
SKClinic (U)
2.54
Step in
chain
Measurement Description
Quantity (Units)
Relative Propagated
y ((%))
Uncertainty
(1)
NIST WAFAC calibration
SK (U)
0.80
6
Manufacturer well-ion chamber calibration
SK / IM (U / A)
0.94
7
Manufacturer calibration of QA seed
SKM (U)
1.06
8
Manufacturer calibration of QA well-ion
chamber
SKM / IM (U / A)
1.17
9
Manufacturer calibrates seed for Clinic
SKM (U)
1.27
10
Manufacturer places seed in 2 % bin
SKM (U)
1.40
Expanded uncertainty (k = 2)
SKM (U)
2.80
Does SKClinic Agree With SKM ?
SKClinic = (1.034 ± 0.026) U
SKM = (1.000 ± 0.028) U
1.08
1.06
SK (U)
1.04
1.02
1.00
Cli i
Clinic
0.98
0.96
Manufacturer
Degree of Equivalence
SKClinic – SKM <
0.08
0.04
0.02
SK
Clinic
C
- SK
M
(U)
0.06
0.00
-0.02
V 2Clinic + V 2M – V 2NIST
AAPM
Board of Directors
Science Council
Therapy Physics Committee
Brachytherapy SC
Calibration Laboratory Accreditation SC
Low Energy Brachytherapy Source Dosimetry WG
High Energy Brachytherapy Source Dosimetry WG
Brachytherapy Source Registry WG
Special Brachytherapy Modalities WG
Robotic Brachytherapy WG
ADCLs
AAPM
Board of Directors
Science Council
Therapy Physics Committee
Brachytherapy SC
Calibration Laboratory Accreditation SC
Low Energy Brachytherapy Source Dosimetry WG
High Energy Brachytherapy Source Dosimetry WG
Brachytherapy Source Registry WG
Special Brachytherapy Modalities WG
Robotic Brachytherapy WG
ADCLs
Recommendations of the Calibration Laboratory Accreditation SC:
New source
1. 5 sources are sent to NIST for SK calibration,
1
calibration well chamber
measurements (SK / I), and spectrum analysis
2. If (SK / I) for each source is within ± 1.00
2
1 00 % of average
average, 3 sources are
sent to the ADCLs, and 2 sources are returned to the manufacturer or
sent to a dosimetry investigator for measurement of D(r, )
.
3. If (SK / I) is out of tolerance for one or more sources, another set of 5
sources is sent by the manufacturer to NIST
ref:
f Med.
M d Ph
Phys. 31 (3)
(3), M
March
h 2004
2004, pp. 675
675-681.
681
Measurement Traceability for Brachytherapy Sources – New Source
SK
ADCL
5 sources
Manufacturer
Clinic
Measurement Traceability for Brachytherapy Sources – New Source
3 sources
ADCL1 ADCL2 ADCL3
SK
secondary standard
(SK / I)0
2 sources
Manufacturer
SK / I ?
ADCL calibration date
Clinic
Measurement Traceability for Brachytherapy Sources - Clinics
ADCL
Manufacturer
well-ionization
chambers
(SK / I)ADCL
Clinic
Measurement Traceability for Brachytherapy Sources - Clinics
ADCL
Manufacturer
verification for
treatment planning
(SK / I)ADCL
Clinic
sources (SKM)
SKClinic
Recommendations of the Calibration Laboratory Accreditation SC:
QA for sources with established NIST SK standard
1. 3 sources sent to NIST (preferably within 6 months but not exceeding
1
1 year) for SK calibration and (SK / I) evaluation
2. If (SK / I) for each source is within ± 2.00
2
2 00 % of established
(SK / I) at NIST or the ADCLs, no action needs to be taken
3. If (SK / I) is out of tolerance
3
tolerance, the cause should be investigated
investigated,
and another set of 3 sources is sent by the manufacturer to NIST
and the ADCLs
4. If (SK / I) remains out of tolerance for the second set of source
measurements, discrepancies among the ADCLs and NIST should
be resolved quickly
Measurement Traceability for Brachytherapy Sources – Annual QA
3 sources
SK
ADCL1 ADCL2 ADCL3
3 sources
(SK / I)t
± 2.00 %
vs.
(SK / I)0
Clinic
3 sources
Manufacturer
Well-ionization Chambers
Note that due to the use of well chambers of different designs by NIST
and the 3 ADCLs, discrepancies in tolerance level achievement do occur.
Control Chart, I / SK, seed “E”
I / SK (pA
A / U)
5.9
5.7
5.5
5.3
5.1
49
4.9
Jul02
Aug02
Oct04
Jan05
Oct05
May06
Sep06
Nov06
May07
Jan08
Control Chart, I / SK, seed “E”
I / SK (pA
A / U)
5.9
5.7
Jul02
Aug02
Oct04
Jan05
Oct05
May06
Sep06
Nov06
May07
Jan08
May07
Jan08
5.5
5.3
5.1
49
4.9
Manufacturer vs. NIST (SKM / SKNIST)
1.05
M
S K / SK
NIST
1.04
1.03
1.02
1.01
1.00
0.99
Oct04
Jan05
Oct05
May06
Sep06
Nov06
Fluorescence K / Decay K, seed “E”
F K  / D K  x 100
1.0
Jul02
Aug02
Oct04
Jan05
Oct05
May06
Sep06
Nov06
May07
Jan08
May07
Jan08
0.8
0.6
0.4
0.2
00
0.0
Manufacturer vs. NIST (SKM / SKNIST)
1.05
M
S K / SK
NIST
1.04
1.03
1.02
1.01
1.00
0.99
Oct04
Jan05
Oct05
May06
Sep06
Nov06
Control Chart, I / SK, seed “E”
I / SK (pA
A / U)
5.9
5.7
Jul02
Aug02
Oct04
Jan05
Oct05
May06
Sep06
Nov06
May07
Jan08
May07
Jan08
5.5
5.3
ADCL
reset
5.1
49
4.9
Manufacturer vs. NIST (SKM / SKNIST)
1.05
M
S K / SK
NIST
1.04
1.03
1.02
1.01
1.00
0.99
Oct04
Jan05
Oct05
May06
Sep06
Nov06
Control Chart, I / SK, seed “E”
I / SK (pA
A / U)
5.9
5.7
Jul02
Aug02
Oct04
Jan05
Oct05
May06
Sep06
Nov06
May07
Jan08
May07
Jan08
5.5
5.3
ADCL
reset
5.1
49
4.9
Manufacturer vs. NIST (SKM / SKNIST)
1.05
SKM / SKNIST
1.04
1.03
1.02
1.01
1.00
0.99
Oct04
Jan05
Oct05
May06
Sep06
Nov06
Manufacturer vs. NIST (SKM / SKNIST)
1.05
1.04
1.03
1.02
1.01
1.00
0 99
0.99
0.98
0.97
0.96
0.95
3.0 %
45%
4.5
3.0 %
3%
5%
Uncertainty of SKM from calibration certificate
103Pd
125I
Overall Average = 1.001,  = 0.008
Source manufacturers have ggenerallyy been successful in transferringg the NIST SK standard to their
facilities. However, there is much variation with respect to the magnitude and precision of reported
uncertainties on calibration certificates, if uncertainties are reported at all.
Uncertainty in Secondary Standards based on
Well-Ionization
Well
Ionization Chamber Measurements
To minimize uncertainty:
• Maintenance of secondary standards at ADCLs (AAPM recommendations)
1) NIST receives a batch of 3 seeds of each design annually
2) NIST characterization measurements detect normal
manufacturing variability and anomalous sources
To quantify uncertainty:
• Utilize control charts for results of characterization measurements
1) Calculate
C l l standard
d d deviation
d i i (s)
( ) andd range off values
l
off I / SK for
f
seeds with a significant calibration history at NIST (includes 3 103Pd
k = 1 uncertainty component
and 8 125I source models), s
2) Study variations in measured spectra and anisotropy (A) to
distinguish normal manufacturing variability from design change
Standard Deviation and Range of (I / SK)
1 4%
1.4%
103Pd
1.2%
max = 1.3 %
125I
1.0%
 , I / SK
0 8%
0.8%
0.6%
min = 0.5 %
0.4%
0 2%
0.2%
0.0%
1
6.0%
2
3
Model #
4
5
6
103Pd
5.0%
7
8
9
10
11
125I
4.0%
Range, I / SK
±2%
AAPM
tolerance
level
3.0%
2.0%
1.0%
%
0.0%
1
2
3
Model #
4
5
6
7
8
9
10
11
Range of (Ag K / ) and (I / SK)
30%
Note wide variation
in admixture of Ag
fluorescence x rays
causing range of
I / SK to exceedd
AAPM tolerance
level (Model # 11)
125I
25%
20%
Range, Ag K / 
15%
10%
This
hi seedd model
d l is
i
no longer produced
5%
0%
1
6 0%
6.0%
2
3
Model #
4
5
6
103Pd
5.0%
7
8
9
10
11
125I
4.0%
Range, I / SK
±2%
AAPM
tolerance
level
3.0%
2.0%
1.0%
0.0%
1
2
3
Model #
4
5
6
7
8
9
10
11
American Association of Physicists in Medicine (AAPM)
Board of Directors
Science Council
Therapy Physics Committee
Brachytherapy SC
Calibration Laboratory Accreditation SC
Low Energy Brachytherapy Source Dosimetry WG
High Energy Brachytherapy Source Dosimetry WG
Brachytherapy Source Registry WG
Special Brachytherapy Modalities WG
Robotic Brachytherapy WG
ADCLs
American Association of Physicists in Medicine (AAPM)
Board of Directors
Science Council
Therapy Physics Committee
Brachytherapy SC
Calibration Laboratory Accreditation SC
Low Energy Brachytherapy Source Dosimetry WG
High Energy Brachytherapy Source Dosimetry WG
Brachytherapy Source Registry WG
Special Brachytherapy Modalities WG
Robotic Brachytherapy WG
ADCLs
Low Energy Brachytherapy Source Dosimetry WG
TG-43 Report (1995)
1. Dosimetry formalism introduced
2. Consensus datasets for 2 125I, 1 103Pd, and 1 LDR 192Ir seeds
TG-43U1 (2004)
1. Dosimetry formalism updated (includes NIST WAFAC SK standard)
2 Consensus
2.
C
datasets
d t t for
f 6 125I andd 2 103Pd seeds
d
3. Recommended dosimetry methodology (TLD, Monte Carlo)
TG-43U1S1 (2007)
(
)
1. Consensus datasets for 7 125I and 1 103Pd seeds
2. Interpolation and extrapolation methods
TG-43U1S2 (in preparation)
1. Consensus datasets for 2 125I, 2 103Pd, 1 131Cs seeds…
2. Experimental method evaluation (TLD powder in water, photon
spectrometry radiochromic film)
spectrometry,
Update of AAPM Task Group No. 43 Report:
A Revised AAPM Protocol for Brachytherapy Dose Calculations
(TG 43U1)
(TG-43U1)
Mark Rivard, Bert Coursey, Larry DeWerd, William Hanson, Saiful Huq,
Geoffreyy Ibbott, Michael Mitch, Ravinder Nath, and Jeffreyy Williamson
• Specification of measurement and Monte Carlo calculation methodologies
includes a comprehensive uncertainty analysis
• Good practice for Monte Carlo calculations includes:
Type A uncertainty component ≤ 2 % at r ≤ 5 cm for dose rate
Type A uncertainty component ≤ 1 % for air-kerma
TG-43 Formalism
D ( r0 ,  0 )  S K  
Dose rate in water
Geometry Function
G (r ,  )
D (r , )  S K    L
 g L (r )  F (r ,  )
G L (r0 ,  0 )
G L ((rr ,  ) 

Lr sin 
G L (r ,0)  (r 2  L2 / 4) 1
Dose rate constant (NIST-traceable SK)
D (r0 ,  0 )

SK
r0 = 1 cm
0 =  / 2
Radial Dose Function
g X (r ) 
D (r ,  0 ) G X (r0 ,  0 )
D (r0 ,  0 ) G X (r ,  0 )
2D Anisotropy Function
D (r , ) G L (r ,  0 )
F (r , ) 
D (r ,  0 ) G L (r ,  )
Uncertainty of Dose Rate Constant
EXP
Component
Repeated TLD measurements
MC
Type A
(%)
Type B
(%)
1.3
C
Component
Statistics
Type A
T
(%)
Type B
T
(%)
0.2
TLD calibration (inc. linac cal.)
1.8
Photon cross sections
0.7
Absorbed dose energy dep. and
PMMA-to-liquid water conv.
0.7
Seed geometry
0.75
Seed and TLD positioning
1.2
Source energy spectrum
0.2
Intrinsic energy dep. corr.
5
Combined std. unc., uc
NIST-traceable SK meas.
Combined std. unc., uc
1.1
1
5.7
Dolan and Williamson, 2006
Uncertainty of Consensus Dose Rate Constant
EXP
CON

EXP
= (0.980 ± 0.056) cGy h-1 U-1
(5.7 %)
= (0.950 ± 0.010) cGy h-1 U-1
(1.1 %)
  MC 
2
MC
u2  u2

2
uc   EXP 2 MC  u BIAS

2


1/ 2
CON
= (0.965 ± 0.028) cGy h-1 U-1
(2.9 %)
(without bias term)
u BIAS 
EXP
  MCC 
2 3
CON
= (0.965 ± 0.030) cGy h-1 U-1
(including bias term)
(3.1 %)
Summary
• Uncertainty analysis is a critical element of the science of metrology
• All factors that could possibly influence the result of a measurement
or calculation should be considered
• An uncertainty budget quantifies Type A and Type B components
• Expanded uncertainties (k = 2) should be used in clinical dosimetry