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Decay of Radioactivity the Math of Radioactive Decay Radioactive Decay Review of last week: Introduction to Nuclear Physics and Nuclear Decay Decay of Radioactivity 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. Robert Miyaoka, PhD [email protected] Nuclear Medicine Basic Science Lectures Conservation principles: Energy (equivalently, mass); 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 X-rays come from. Understand half-life and how to use it Fall 2011, Copyright UW Imaging Research Laboratory Fall 2011, Copyright UW Imaging Research Laboratory The Math of Radioactive decay Radioactive decay 1. Activity, Decay Constant, Units of Activity Radioactive decay: 2. Exponential Decay 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 time (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 nuclide • 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 Becquerel (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 day-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% Electron Capture (EC) Fall 2011, Copyright UW Imaging Research Laboratory Fall 2011, Copyright UW Imaging Research Laboratory 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 natural logarithm: 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 Radionuclide 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 Technetium 99m 6.2 hr 0.1152 hr-1 ! = ln 2 / T1/ 2 (ln 2 ! 0.693) Constant in time, characteristic for each nuclide Iodine 123 13.3 hr 0.0522 hr-1 -1 Convenient to calculate the decay factor in multiples of T1/2: Molybdenum 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 nuclides 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 radiation dosimetry 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 Fall 2011, Copyright UW Imaging Research Laboratory Fall 2011, Copyright UW Imaging Research Laboratory 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.
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