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Home , Exponential decay, Nuclide, Radioactive decay, Radionuclide, Specific activity

Review of last week: Introduction to and Nuclear Decay

Decay of Radioactivity – “orbitals” for and – and how the differences to nuclear decay Differences between , isobars, , isomers Know how to diagram basic decays, e.g. Robert Miyaoka, PhD

[email protected] Basic Lectures Conservation principles: Energy (equivalently, ); linear https://www.rad.washington.edu/research/Research/groups/irl/ed momentum; angular momentum (including intrinsic spin); ucation/basic-science-resident-lectures/2011-12-basic-science- and charge are all conserved in radioactive transitions resident-lectures Understand where gamma rays come from (shell transitions in September 12, 2011 the daughter nucleus) and where -rays come from. Understand half-life and how to use it

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The Math of Radioactive decay

1. Activity, Decay Constant, Units of Activity Radioactive decay: 2. Function -is a spontaneous process 3. Determining Decay Factors -can not be predicted exactly for any single nucleus 4. Activity corrections -can only be described statistically and probabilistically i.e., can only give averages and probabilities 5. Parent-Daughter Decay The description of the mathematical aspects of radioactive decay is today's topic. Physics in Nuclear Medicine - Chapter 4

Fall 2011, Copyright UW Imaging Research Laboratory Fall 2011, Copyright UW Imaging Research Laboratory Activity Decay Constant

• Average number of radioactive decays per unit (rate) • Fraction of nuclei that will decay per unit time: • or - Change in number of radioactive nuclei present:  = -(dN/dt) / N(t) = A(t) / N(t) A = -dN/dt • Constant in time, characteristic of each • Depends on number of nuclei present (N). During decay of a • Related to activity: A = λ * N given sample, A will decrease with time. • Measured in (time)-1 • Measured in (Bq): Example: Tc-99m has λ= 0.1151 hr-1, i.e., 11.5% decay/hr 1 Bq = 1 disintegration per second (dps) Mo-99 has λ = 0.252 -1, i.e., 25.2% decay/day • {Traditionally measured in Curies (Ci): • If nuclide has several decay modes, each has its own λi. 10 1 Ci = 3.7 x 10 Bq (1 mCi = 37 MBq) Total decay constant is sum of modes: λ = λ1 + λ2 + λ3 + 226 Traditionally: 1 Ci is the activity of 1g Ra.} • Branching ratio = fraction in particular mode: Bri = λi / λ Example: 18F: 97% β+, 3% Capture (EC)

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Question Answer N(0) = 1000, λ = 0.1 sec-1 Example: N(0) = 1000 and 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 Exponential Decay Equation

• Can describe growth or decrease of an initial sample The number of decaying and remaining nuclei is proportional • Describe the disappearance/addition of a constant fraction of to the original number: dN/dt = -λ * N the present quantity in a sample in any given time interval. =>* N(t) = N(0) * e-λt • Rate of change is greater than linear. This also holds for the activity (number of decaying nuclei): • The number of decays is proportional to the number of A(t) = A(0) * e-λt undecayed nuclei. Decay factor: e -λt General math for exponentials: = N(t)/N(0): fraction of nuclei remaining - inverse is : ln: ln(e-λt) = -λt or - additive exponents yield multiplicative exponentials: = A(t)/A(0): fraction of activity remaining -(λ t+λ t+λ t+…) -λ t -λ t -λ t e 1 2 3 = e 1 * e 2 * e 3 * … ! "t dN 1 - for small exponents (-λt): approximate: e # 1 ! "t *Mathematical derivation: ! = "#!dt ! $ dN = $ "#dt N N Fall 2011, Copyright UW Imaging Research Laboratory Fall 2011, Copyright UW Imaging Research Laboratory

Half-life Some Clinical Examples Radioactive decay shows disappearance of a constant fraction of activity per unit time T1/2 λ

Half-life: time required to decay a sample to 50% of its initial –( *T ) activity: 1/2 = e λ 1/2 Flourine 18 110 min 0.0063 min-1 ! T = ln 2 / " 1/ 2 99m 6.2 hr 0.1152 hr-1 ! = ln 2 / T1/ 2 (ln 2 ! 0.693) Constant in time, characteristic for each nuclide 123 13.3 hr 0.0522 hr-1 -1 Convenient to calculate the decay factor in multiples of T1/2: 99 2.75 d 0.2522 d

$ t ' t -1 "0.693 # Iodine 131 8.02 d 0.0864 d A(t) (!t ) %& T () 1 T = e = e 1/2 = ( ) 1/2 A(0) 2 λ ≈ 0.693 / T1/2

Fall 2011, Copyright UW Imaging Research Laboratory Fall 2011, Copyright UW Imaging Research Laboratory Average Lifetime Determining Decay Factors (e-λt)

Table of Decay factors for Tc-99m

Average time until decay varies between from 'almost Minutes zero' to 'almost infinity'. Hours 0 15 30 45 This average lifetime  is characteristic for each nuclide and 0 1.000 0.972 0.944 0.917 doesn't change over time: 1 0.891 0.866 0.844 0.817 ! = 1/ " = T1/2 / ln 2 3 0.707 0.687 0.667 0.648 τ is longer than the half-life by a factor of 1/ln2 (~1.44) 5 0.561 0.545 0.530 0.515

7 0.445 0.433 0.420 0.408 Average lifetime important in calculations 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)*… Fall 2011, Copyright UW Imaging Research Laboratory Fall 2011, Copyright UW Imaging Research Laboratory

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 Fall 2011, Copyright UW Imaging Research Laboratory Fall 2011, Copyright UW Imaging Research Laboratory Examples Answer

Tc-99m: Minutes Tc-99m: Minutes Hours Hours 0 15 30 45 0 15 30 45 0 1.000 0.972 0.944 0.917 0 1.000 0.972 0.944 0.917 1 0.891 0.866 0.844 0.817 1 0.891 0.866 0.844 0.817 3 0.707 0.687 0.667 0.648 3 0.707 0.687 0.667 0.648 5 0.561 0.545 0.530 0.515 5 0.561 0.545 0.530 0.515 7 0.445 0.433 0.420 0.408 7 0.445 0.433 0.420 0.408 9 0.354 0.343 0.334 0.324 9 0.354 0.343 0.334 0.324

What is the DF after 6 hrs? What is the DF after 6 hrs? (Remember, exponentials are multiplicative) 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 Answer

C-11 has a half-life of 20 minutes. Initial sample has 1000 C-11 has a half-life of 20 minutes. Initial sample has 1000 nuclei. nuclei.

Q: How many are left after 40 minutes? Q: How many are left after 40 minutes? After 60 minutes? After 60 minutes? When is less than 1 left? 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).

Fall 2011, Copyright UW Imaging Research Laboratory Fall 2011, Copyright UW Imaging Research Laboratory Image Frame Decay Correction Question 15 If the acquisition time of an image frame Δt is not short compared to the O study − image frame is 30 - 45 sec after injection. half-life of the used nuclide, then the nuclide is decaying measurably during How much difference does it make to use or not to use the acquisition time. the exact effective DF, and is approximation OK? In this situation one needs to adjust the decay factor (i.e., DF(t)) to account for decay of activity during Δt, because a = DF(t)*Δt ≠ a . 0 d What about if the image frame is from 30-154 seconds ⇒ DF (t, !t) = DF(t )* a / a = DF(t) *[(1 " e" x ) / x] x = ln2 * !t / T eff d 0 1/ 2 after injection?

Approximations for small x: The half-live of 15O is 124 seconds.

DFeff ≈ 1/2 * [DF(t)+DF(t+Δt)] DF(t) a0 DFeff ≈ DF(t+(Δt/2))

DFeff ≈ DF(t) * [1 − (x/2)]

t t+Δt t ad Fall 2011, Copyright UW Imaging Research Laboratory Fall 2011, Copyright UW Imaging Research Laboratory

Answer - Effective Decay Factor Answer - Effective Decay Factor

15 15 T1/2( O)=124 sec. and Δt = 15 sec. T1/2( O)=124 sec. and Δt = 124 sec.

DFeff(t,Δt) = DF(t) * [(1-exp(-x))/x], 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 2 * t / T1/2) = 0.846 DF(t) = DF(30sec) = exp(-ln 2 * t / T1/2) = 0.846

Params: x = ln 2*(Δt/ T1/2) = 0.084, [1-exp(-x)]/x ≈ 0.959 Params: x = ln 2*(Δt/ T1/2) = 0.693, [1-exp(-x)]/x ≈ 0.721

• Exact Solution: • Exact Solution:

⇒ DFeff = DF(30 sec) * [1-exp(-x)]/x = 0.811 ⇒ DFeff = DF(30 sec) * [1-exp(-x)]/x = 0.610 Decay correction factor for this frame: 1/0.811 = 1.233 Decay correction factor for this frame: 1/0.610 = 1.64

• With approximation: DFeff = DF(t) * [1 - (x/2)] = 0.810 • With approximation: DFeff = DF(t) * [1 - (x/2)] = 0.654 ⇒ Approximation is very good in this case. ⇒ 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 x = 0.693 * !t / T1/ 2

Taylor series expansion: DFeff (t, !t) " DF(t) # [1 $ (x / 2)] • Ratio of total radioisotope activity to total mass of the element present DF is accurate to within 1% when x<0.25 eff • Measured in Bq/g or Bq/mole • Useful if radioactive and non-radioactive isotopes of the same element are in mixture (sample with carrier) Midpoint of frame: DFeff (t, !t) " DF(t) # [t + (!t / 2)] • Maximum Specific Activity in carrier-free samples:

DFeff is accurate to within 1% when x<0.5 Carrier Free Specific Activity (CFSA) 18 CFSA [Bq/g] ≈ 4.8 x 10 /(A x T1/2) for T1/2 given in days, A: atomic [DF(t) + DF(t + !t)] weight Average of DF's: DF (t, !t) " CFSA(Tc-99m) = 4.8 x 1018 / (99 * 0.25) = 1.9 x 1017 Bq/g eff 2 = 5.3 x 106 Ci/g DFeff is accurate to within 1% when x<0.35

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Question Answer

What has a higher Carrier Free Specific Activity (CFSA) What has a higher Carrier Free Specific Activity (CFSA) -11 or -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 Biologic and Physical Decay

For unrelated in a mixture, the total Activity is The effective half-life of a with a physical

the sum of the individual activities: half-life of T1/2,physical and a biological half-life of T1/2, biological is:

(!0.693* t / T1 /2 ,1 ) (!0.693* t /T1/ 2 ,1 ) At (t) = A1(0) * e + A2 (0) * e + ... 1/ T1/2 ,eff = (1 / T1/2 , phy sical ) + (1 / T1/2 ,bio log ical ) or A (t) always eventually follows the t (T1/ 2, phy sical ! T1/2 ,bio log ical ) slope of the longest half-life nuclide. T1 /2, eff = (T1/ 2, phy sical + T1/2 ,bio log ical )

This can be used to extrapolate back or 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-Daughter Decay Special Case: Half-life of parent very long, decrease of A negligible during Parent Daughter p observation, λ ≈ 0. T1/2 = Tp T1/2 = Td p Example: Ra-226 (Tp ≈ 1620yr) → Rn-222 (Td ≈ 4.8 days) Equation for activity of parent is: Bateman Approximation: Ad(t) ≈ Ap(0) (1− exp(-λdt)) * BR !" p t Ap (t) = e After one daughter half-life Td: Equation for activity of daughter is described by the Ad(t) ≈ 1/2 Ap(t) Bateman Equation: After 2*Td: Ad(t) ≈ 3/4 Ap(t)

* . $ t ' Eventually Ad ≈ Ap , asymptotically , !d "! p "! d t , "! d t Ad (t) = +&Ap (0) # e " e ) # B.R./ + Ad (0)e ! " ! ( ) approaches equilibrium -,%& ( d p ) () 0,

Figure from Physics in Nuclear Medicine (Cherry, Sorenson and Phelps) Fall 2011, Copyright UW Imaging Research Laboratory Figure from Physics in Nuclear Medicine (Cherry, Sorenson and Phelps) Fall 2011, Copyright UW Imaging Research Laboratory Transient Equilibrium No Equilibrium

If Td > Tp, no equilibrium is possible. Parent Half-life longer than Daughter’s, but not ‘infinite’ 99 99m 131m 131 Example: Mo (T1/2 = 66hr) → Tc (T1/2 = 6 hr) Example: Te (T1/2 = 30hr) I (T1/2 = 8 days)

Ap decreases significantly over the observation time, so λp≈0. Buildup and decay of daughter shows maximum again. When parent distribution is gone, further daughter decay is The daughter activity increases, according to the daughter’s exceeds that of the parent, decay equation only. reaches a maximum value, then decreases and follows the decay of the parent. Then the ratio of daughter to parent is constant: transient equilibrium.

Ad/Ap = [Tp/(Tp - Td)] * BR

Figure from Physics in Nuclear Medicine (Cherry, Sorenson and Phelps) Fall 2011, Copyright UW Imaging Research Laboratory Fall 2011, Copyright UW Imaging Research Laboratory

Summary Mathematical description of decay only yields probabilities and averages.

−λ⋅t −0.693⋅t /T Decay equation: N(t) = N(0) ⋅ e = N(0) ⋅ e 1/2 Decay constant : λ = A(t)/N(t) Half-life: T1/2= ln 2 / λ = 0.693 / λ Average lifetime: τ = 1 / λ

t -0.693 t/T Decay factor (DF): e−λ⋅ = e ⋅ 1/2 Image frame decay correction - effective decay factors Specific Activity Mixed, and parent-daughter decays

NEXT WEEK: Dr. Hunter - Interactions with Reading: Physics in Nuclear Medicine - Chapter 6 Fall 2011, Copyright UW Imaging Research Laboratory