Review of last week: Introduction to Nuclear Physics and Nuclear Decay
Decay of Radioactivity
Robert Miyaoka, PhD
rmiyaoka@u.washington.edu
Nuclear Medicine Basic Science Lectures
https://www.rad.washington.edu/research/Research/groups/irl/ed
ucation/basic-science-resident-lectures/2011-12-basic-scienceresident-lectures
September 12, 2011
Nuclear shell model – “orbitals” for protons and neutrons – and
how the energy differences leads to nuclear decay
Differences between isotopes, isobars, isotones, isomers
Know how to diagram basic decays, e.g.
Conservation principles: Energy (equivalently, mass); linear
momentum; angular momentum (including intrinsic spin);
and charge are all conserved in radioactive transitions
Understand where gamma rays come from (shell transitions in
the daughter nucleus) and where X-rays come from.
Understand half-life and how to use it
Fall 2011, Copyright UW Imaging Research Laboratory
The Math of Radioactive decay
1.
2.
3.
4.
5.
Activity, Decay Constant, Units of Activity
Exponential Decay Function
Determining Decay Factors
Activity corrections
Parent-Daughter Decay
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Radioactive decay
Radioactive decay:
-is a spontaneous process
-can not be predicted exactly for any single nucleus
-can only be described statistically and probabilistically
i.e., can only give averages and probabilities
The description of the mathematical aspects of
radioactive decay is today's topic.
Physics in Nuclear Medicine - Chapter 4
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Fall 2011, Copyright UW Imaging Research Laboratory
Activity
Decay Constant
• Average number of radioactive decays per unit time (rate)
• or - Change in number of radioactive nuclei present:
A = -dN/dt
• Depends on number of nuclei present (N). During decay of a
given sample, A will decrease with time.
• Measured in Becquerel (Bq):
1 Bq = 1 disintegration per second (dps)
• {Traditionally measured in Curies (Ci):
1 Ci = 3.7 x 1010 Bq (1 mCi = 37 MBq)
Traditionally: 1 Ci is the activity of 1g 226Ra.}
• Fraction of nuclei that will decay per unit time:
 = -(dN/dt) / N(t) = A(t) / N(t)
• Constant in time, characteristic of each nuclide
• Related to activity: A = λ * N
• Measured in (time)-1
Example: Tc-99m has λ= 0.1151 hr-1, i.e., 11.5% decay/hr
Mo-99 has λ = 0.252 day-1, i.e., 25.2% decay/day
• If nuclide has several decay modes, each has its own λi.
Total decay constant is sum of modes: λ = λ1 + λ2 + λ3 +
• Branching ratio = fraction in particular mode: Bri = λi / λ
Example: 18F: 97% β+, 3% Electron Capture (EC)
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Fall 2011, Copyright UW Imaging Research Laboratory
Answer
Question
Example: N(0) = 1000 and λ = 0.1 sec-1
N(0) = 1000, λ = 0.1 sec-1
A(0) = 100; N(1) = 900; A(1) = 90; N(2) = 810
What is A(0) = ?
What is N(2 sec) = ?
Both the number of nuclei and activity decrease by λ with time
Fall 2011, Copyright UW Imaging Research Laboratory
Fall 2011, Copyright UW Imaging Research Laboratory
Exponential Equations
• Can describe growth or decrease of an initial sample
• Describe the disappearance/addition of a constant fraction of
the present quantity in a sample in any given time interval.
• Rate of change is greater than linear.
• The number of decays is proportional to the number of
undecayed nuclei.
General math for exponentials:
- inverse is natural logarithm: ln: ln(e-λt) = -λt
- additive exponents yield multiplicative exponentials:
e-(λ t+λ t+λ t+…) = e-λ t * e-λ t * e-λ t * …
! "t
- for small exponents (-λt): approximate: e # 1 ! "t
1
2
3
1
2
Exponential Decay Equation
The number of decaying and remaining nuclei is proportional
to the original number: dN/dt = -λ * N
=>* N(t) = N(0) * e-λt
This also holds for the activity (number of decaying nuclei):
A(t) = A(0) * e-λt
Decay factor: e -λt
= N(t)/N(0): fraction of nuclei remaining
or
= A(t)/A(0): fraction of activity remaining
3
*Mathematical derivation:
!
dN
1
= " # !dt ! $ dN = $ " #dt
N
N
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Half-life
Radioactive decay shows disappearance of a constant fraction of
activity per unit time
Half-life: time required to decay a sample to 50% of its initial
activity: 1/2 = e –(λ*T1/2)
! T1/ 2 = ln 2 / "
! = ln 2 / T1/ 2
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Some Clinical Examples
Radionuclide
T1/2
Flourine 18
110 min
λ
0.0063 min-1
Technetium 99m 6.2 hr
0.1152 hr-1
Constant in time, characteristic for each nuclide
Iodine 123
13.3 hr
0.0522 hr-1
Convenient to calculate the decay factor in multiples of T1/2:
Molybdenum 99
2.75 d
0.2522 d-1
Iodine 131
8.02 d
0.0864 d-1
$
(ln 2 ! 0.693)
t '
t
& "0.693 # T ()
A(t )
1
1/2
= e(!t ) = e%
= ( ) T1/2
A(0)
2
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λ ≈ 0.693 / T1/2
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Average Lifetime
Average time until decay varies between nuclides from 'almost
zero' to 'almost infinity'.
This average lifetime  is characteristic for each nuclide and
doesn't change over time:
! = 1/ " = T1/2 / ln 2
τ is longer than the half-life by a factor of 1/ln2 (~1.44)
Average lifetime important in radiation dosimetry calculations
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Determining Decay Factors (e-λt)
Table of Decay factors for Tc-99m
Minutes
Hours
0
15
30
45
0
1.000
0.972
0.944
0.917
1
0.891
0.866
0.844
0.817
3
0.707
0.687
0.667
0.648
5
0.561
0.545
0.530
0.515
7
0.445
0.433
0.420
0.408
9
0.354
0.343
0.334
0.324
Read of DF directly or combine for wider time coverage:
DF(t1+t2+..) = DF(t1)*DF(t2)*…
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Other Methods to Determine DF
Other Methods to Determine DF
• Graphical method: DF vs. number of half-lives elapsed
• Graphical method: DF vs. number of half-lives elapsed
Determine the time as multiples
Determine the time as multiples
of T1/2.
of T1/2.
Read the corresponding DF
Read the corresponding DF
number off the chart.
number off the chart.
Example:
Example:
DF (3.5) =
DF (3.5) =
0.088
• Pocket calculator with exponential/logarithm functions
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Fall 2011, Copyright UW Imaging Research Laboratory
Answer
Examples
Tc-99m:
Hours
Tc-99m:
Minutes
0
15
30
45
0
1.000
0.972
0.944
0.917
1
0.891
0.866
0.844
0.817
3
0.707
0.687
0.667
0.648
5
0.561
0.545
0.530
0.515
7
0.445
0.433
0.420
0.408
9
0.354
0.343
0.334
0.324
What is the DF after 6 hrs? (Remember, exponentials are multiplicative)
Minutes
0
15
30
45
0
1.000
0.972
0.944
0.917
1
0.891
0.866
0.844
0.817
3
0.707
0.687
0.667
0.648
5
0.561
0.545
0.530
0.515
7
0.445
0.433
0.420
0.408
9
0.354
0.343
0.334
0.324
Hours
What is the DF after 6 hrs?
DF(6hr) = DF(3hr) * DF(3hr) = 0.707 * 0.707 = 0.500
DF(6hr) = DF(5hr) * DF(1hr) = 0.561 * 0.891 = 0.500
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Examples
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Answer
C-11 has a half-life of 20 minutes. Initial sample has 1000
nuclei.
C-11 has a half-life of 20 minutes. Initial sample has 1000
nuclei.
Q: How many are left after 40 minutes?
After 60 minutes?
When is less than 1 left?
Q: How many are left after 40 minutes?
After 60 minutes?
When is less than 1 left?
A: After 2 half-lives (40 min) 1/22 = 1/4 of the initial activity is
left (25%).
After 3 half-lives (60 min) 1/8th is left (12.5%)
To have less than one left, one needs a DF < 1/1000.
This happens after 10 half-lives because (1/2)10 = 1/1024, so
after 200 minutes (3 hrs 20 min).
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Fall 2011, Copyright UW Imaging Research Laboratory
Question
Image Frame Decay Correction
If the acquisition time of an image frame Δt is not short compared to the
half-life of the used nuclide, then the nuclide is decaying measurably during
the acquisition time.
In this situation one needs to adjust the decay factor (i.e., DF(t)) to account
for decay of activity during Δt, because a0 = DF(t)*Δt ≠ ad.
⇒ DFeff ( t , !t ) = DF (t )* ad / a0 = DF (t ) * [(1 " e" x ) / x ]
x = ln 2 * !t / T1/ 2
DF(t)
Approximations for small x:
study − image frame is 30 - 45 sec after injection.
How much difference does it make to use or not to use
the exact effective DF, and is approximation OK?
15O
What about if the image frame is from 30-154 seconds
after injection?
The half-live of 15O is 124 seconds.
DFeff ≈ 1/2 * [DF(t)+DF(t+Δt)]
a0
DFeff ≈ DF(t+(Δt/2))
DFeff ≈ DF(t) * [1 − (x/2)]
t
ad
t+Δt
t
Fall 2011, Copyright UW Imaging Research Laboratory
Answer - Effective Decay Factor
T1/2(15O)=124
sec. and Δt = 15 sec.
DFeff(t,Δt) = DF(t) * [(1-exp(-x))/x],
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Answer - Effective Decay Factor
T1/2(15O)=124 sec. and Δt = 124 sec.
x = ln 2 * Δt/T1/2
DFeff(t,Δt) = DF(t) * [(1-exp(-x))/x],
x = ln 2 * Δt/T1/2
DF(t) = DF(30sec) = exp(-ln
exp(-ln 2 * t / T1/2) = 0.846
DF(t) = DF(30sec) = exp(-ln
exp(-ln 2 * t / T1/2) = 0.846
Params: x = ln 2*(Δt/ T1/2) = 0.084, [1-exp(-x
)]/x
/x ≈ 0.959
[1-exp(-x)]
Params: x = ln 2*(Δt/ T1/2) = 0.693, [1-exp(-x
)]/x
/x ≈ 0.721
[1-exp(-x)]
• Exact Solution:
⇒ DFeff = DF(30 sec) * [1-exp(-x
)]/x
/x = 0.811
[1-exp(-x)]
Decay correction factor for this frame: 1/0.811 = 1.233
• Exact Solution:
⇒ DFeff = DF(30 sec) * [1-exp(-x
)]/x
/x = 0.610
[1-exp(-x)]
Decay correction factor for this frame: 1/0.610 = 1.64
• With approximation: DFeff = DF(t) * [1 - (x/2)] = 0.810
⇒ Approximation is very good in this case.
• With approximation: DFeff = DF(t) * [1 - (x/2)] = 0.654
⇒ Approximation is NOT very good in this case.
Fall 2011, Copyright UW Imaging Research Laboratory
Fall 2011, Copyright UW Imaging Research Laboratory
Effective Decay Factor
Specific Activity
x = 0.693 * !t / T1/ 2
Taylor series expansion: DFeff ( t , !t ) " DF (t ) # [1 $ ( x / 2)]
DFeff is accurate to within 1% when x<0.25
Midpoint of frame:
DFeff ( t , !t ) " DF (t ) # [ t + (!t / 2)]
DFeff is accurate to within 1% when x<0.5
• Ratio of total radioisotope activity to total mass of the
element present
• Measured in Bq/g or Bq/mole
• Useful if radioactive and non-radioactive isotopes of the
same element are in mixture (sample with carrier)
• Maximum Specific Activity in carrier-free samples:
Carrier Free Specific Activity (CFSA)
CFSA [Bq/g] ≈ 4.8 x 1018 /(A x T1/2) for T1/2 given in days, A: atomic
[ DF ( t ) + DF ( t + !t )]
2
is accurate to within 1% when x<0.35
Average of DF's:
DFeff
DFeff ( t , !t ) "
weight
CFSA(Tc-99m) = 4.8 x 1018 / (99 * 0.25) = 1.9 x 1017 Bq/g
= 5.3 x 106 Ci/g
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Question
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Answer
What has a higher Carrier Free Specific Activity (CFSA)
What has a higher Carrier Free Specific Activity (CFSA)
Carbon-11 or Fluorine-18?
Carbon-11 or Fluorine-18?
CFSA(C-11) = 4.8 x 1018 / (11 * 0.014) = 3.1 x 1019 Bq/g
= 8.3 x 108 Ci/g
CFSA(F-18) = 4.8 x 1018 / (18 * 0.076) = 3.5 x 1018 Bq/g
= 9.45 x 107 Ci/g
Fall 2011, Copyright UW Imaging Research Laboratory
Fall 2011, Copyright UW Imaging Research Laboratory
Decay of Mixed Radionuclide Sample
For unrelated radionuclides in a mixture, the total Activity is
the sum of the individual activities:
( !0.693* t / T1 /2 ,1 )
At ( t ) = A1(0) * e
( !0.693* t / T1/ 2 ,1 )
+ A2 (0) * e
+ ...
Biologic and Physical Decay
The effective half-life of a radiopharmaceutical with a physical
half-life of T1/2,physical and a biological half-life of T1/2, biological is:
1 / T1/2 , eff = (1 / T1/2 , phy sical ) + (1 / T1/2 , bio log ical )
or
At(t) always eventually follows the
slope of the longest half-life nuclide.
T1 /2, eff =
(T1/ 2, phy sic al ! T1/2 , bio log ical )
(T1/ 2, phy sic al + T1/2 , bio log ical )
or
This can be used to extrapolate back
to the contributors to the mix.
!eff = ! phy sical + !bio log ical
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Parent-Daughter Decay
Secular Equilibrium
Parent-daughter decay is illustrated below:
Parent
T1/2 = Tp
Parent-Daughter Decay Special Case:
Half-life of parent very long, decrease of Ap negligible during
observation, λp ≈ 0.
Example: Ra-226 (Tp ≈ 1620yr) → Rn-222 (Td ≈ 4.8 days)
Daughter
T1/2 = Td
Equation for activity of parent is:
Ap (t ) = e
Bateman Approximation: Ad(t) ≈ Ap(0) (1− exp(-λdt)) * BR
!"pt
After one daughter half-life Td:
Ad(t) ≈ 1/2 Ap(t)
After 2*Td: Ad(t) ≈ 3/4 Ap(t)
Equation for activity of daughter is described by the
Bateman Equation:
*, $
!d
"! t
Ad ( t ) = + & Ap (0)
# e
" e"!
!
d " !p
-, &%
(
) (
Figure from Physics in Nuclear Medicine (Cherry, Sorenson and Phelps)
p
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dt
'
.
))) # B.R.,/ + A (0)e
(
0,
d
"!dt
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Eventually Ad ≈ Ap , asymptotically
approaches equilibrium
Figure from Physics in Nuclear Medicine (Cherry, Sorenson and Phelps)
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No Equilibrium
Transient Equilibrium
If Td > Tp, no equilibrium is possible.
Parent Half-life longer than Daughter’s, but not ‘infinite’
99Mo (T
99mTc (T
Example:
1/2 = 66hr) →
1/2 = 6 hr)
Example:
Ap decreases significantly over the observation time, so λp≈0.
The daughter activity increases,
exceeds that of the parent,
reaches a maximum value,
then decreases and follows the
decay of the parent. Then the
ratio of daughter to parent is
constant: transient equilibrium.
equilibrium
Ad/Ap = [Tp/(Tp - Td)] * BR
Figure from Physics in Nuclear Medicine (Cherry, Sorenson and Phelps)
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Summary
Mathematical description of decay only yields probabilities
and averages.
Decay equation: N(t) = N(0) ⋅ e−λ⋅t = N(0) ⋅ e−0.693⋅t /T
Decay constant :
Half-life:
Average lifetime:
1/2
λ = A(t)/N(t)
T1/2= ln 2 / λ = 0.693 / λ
τ=1/λ
Decay factor (DF): e−λ⋅t = e-0.693⋅t/T
1/2
Image frame decay correction - effective decay factors
Specific Activity
Mixed, and parent-daughter decays
NEXT WEEK: Dr. Hunter - Interactions with Matter
Reading: Physics in Nuclear Medicine - Chapter 6
Fall 2011, Copyright UW Imaging Research Laboratory
131mTe
(T1/2 = 30hr)
131I
(T1/2 = 8 days)
Buildup and decay of daughter shows maximum again.
When parent distribution is gone, further daughter decay is
according to the daughter’s
decay equation only.
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