8/2/2013
Mean Squared
Error:
Radiobiological rationale for accuracy considerations in RT
Root Mean
Squared Error:
Probability density
Statistics of radiation delivery
Søren M Bentzen, Ph.D., D.Sc.
precision, 
bias
0
Departments of Human Oncology; Medical Physics; Biostatistics and Medical Informatics,
Deviation from planned dose [%]
University of Wisconsin Carbone Cancer Center, Madison, Wisconsin, USA
/SMB 8/13
bentzen@humonc.wisc.edu
Dosimetric precision in vivo
Probability density
Comparing radiation dose deliveries
Root
Mean
Square
Errors
Entrance and exit
Si diode measurements
in 11 patients with
HNSCC
0.32%
x = -0.5%
SD = 4.3%
2.04%
1.50%
Deviation from planned dose [%]
 Leunens,…, van der Schueren, R&O 25: 242 (1992)
/SMB 8/13
/SMB 8/13
Steepness of DR curves for HNSCC
The normalized dose‐response gradient
3.0
2.5
Response probability
P(D)
Larynx
Head&Neck
Supraglottic
Pharynx
Neck nodes
2.0
P
1.5
D
1.0
0.5
0.0
elm
Hj
an
-H
n
se
n
n es
n
r
n
s
n
s
s
s
s
in ard
in
in
so
so
so
ze
he lev
iem
ilip ame aylo ame ame sse sse ame
S erga Jack Jack ham Gr Jack ent Ph
T
Co
B
T s&
Th Gho Gho
Th
Th
Th
&
&
Ov rt & rt &
ch
rt
ar
a
n
a
a
k
e
ew
ew tew
To
Mo
St
St
S
Dose (Gy)
/SMB 8/13
 Bentzen R&O 32: 1 (1994)
/SMB 8/13
1
8/2/2013
50 vs. local ‐value
Steepness of normal‐tissue dose‐response curves
50
50%
fixed dose/F
fixed no. F
Local steepness, x
7
6
5
4
3
2
1
0
Influenced by dose inhomogeneity (?)
HNSCC
•  varies with position on the dose‐
response curve, i.e. with the response level.
r
a
te
ia
d
ia
rly
sis
la
oi
lde
as
as
ea
em
ro
m
ct
ct
g,
ou
g,
ed
ig
fib
un
gie
gie
-s
sh
al
s
un
L
n
n
o
n
e
u
t
L
la
la
c
e
ng
eo
oz
Te
Te
Re
ry
an
Fr
La
ut
bc
u
S
 Bentzen R&O 32: 1 (1994)
• The curve is still parameterized in terms of D50 and 50
• However, in most situations the local ‐value should be applied
Response level, x
/SMB 8/13
/SMB
 D2   2pop   F2   2pop   2f / N
‐value for dose‐per‐fraction escalation
Uncertainty components
The delivered dose can be decomposed as:
 At a reference dose per fraction of dr the relationship between N and d is
Effect of BIAS on outcome
Dˆ  DP  b  
EXAMPLE:
Assume / = 2 Gy dr = 2 Gy ï N = 1.5d
dr = 6 Gy ï N = 1.75d
∆
∙
where DP is the planned (intended or acceptable to the physician) dose,
b is the bias, and  is a random error
The variance of  is
 Two asymptotic results are
2
2
2
 2   course
  Fx
  course
  2fx / N
where course is the patient-to-patient variability that does not vary
between fractions, f is the fraction-to-fraction variability, N is the
number of fractions
/SMB 8/13
 Bentzen Acta Oncol 44: 825 (2005)
Estimating the clinical effect of imprecision
 Bentzen, IAEA report (in preparation)
/SMB 8/13
Second derivative of dose‐response function
 Logistic dose‐response curve: D50=60 Gy, 50=1.8
The expectation value of P(D) in the presence of variability in dose is
,
∙
,
;
P’’(D)
P(D)21%
= MAXIMUM INCREASE IN NTCP
The change in P(D) due to imprecision in dose delivery is
,
,
∆
Dose, D
The Taylor expansion of P(D) is
∙
∙
∙
⋯
The first order term cancels out in the convolution integral due to the symmetry of the p.d.f.
MAXIMUM LOSS OF TCP =
/SMB 8/13
P(D)79%
 Bentzen, IAEA report (in preparation)
/SMB 8/13
2
8/2/2013
Increase in NTCP with reduced precision
Loss of TCP with reduced precision
Around NTCP=21%
Increase in NTCP, %
Around TCP=79%
Loss of TCP, %
50=3
50=2
50=1
50=6
50=4
50=2
0
0
Precision, x
Precision, x
/SMB 8/13
/SMB 8/13
So, what’s the accuracy target then?
So, what’s the accuracy target then?
Tumor control
Tumor control
• GOAL: lack of accuracy results in <5% loss of tumor control probability
• GOAL: lack of accuracy results in <5% loss of tumor control probability
• ASSUMPTION: 50,N=3
• What is  at the 80% level where the precision requirement is tightest?
• What is  at the 80% level where the precision requirement is tightest?
Local steepness, x
• ASSUMPTION: 50,N=3
50%
• Answer: 80,N2.1
50=3
• Accepting a maximum loss due to bias of 3% ï biasd1.5%
80=2.1
Response level, x
• This caps the loss due to imprecision at 2%
/SMB 8/13
Inflation of sample size needed in RCT
/SMB 8/13
EPILOGUE: Accuracy in a (humbling) perspective
% of patients who lost >3 Gy due to poor compliance
80
73 75
70
OVERALL:
dose=9%
64
60
AT THE PATIENT LEVEL!!!
%
50
41 44
40
*
28
30
*
20
23
27
16
*
10
4
0
22811
22851
Conventional
 Bentzen (in preparation)
/SMB 8/13
22791
Altered
PMH
CHART
Altered + Miso
 Khalil et al. IJROBP 55: 568 (2003)
/SMB
3
8/2/2013
EPILOGUE: Accuracy in a (humbling) perspective
Variation in Mean Equivalent Lung Dose
 18 patients with NSCLC receiving chemo‐RT
 Average MELD=10.2 Gy
 CVMELD=42%
/SMB 8/13
4