REVIEWERS
Jonathan Baldwin, BSRT CNMT Clinical Assistant Professor and Clinical Coordinator OUHSC MIRS Nuclear Medicine Oklahoma City, Oklahoma
Jeff L. Berry, MS, RT(R)(CT) Associate Professor, Radiography Program Director University of Oklahoma Health Sciences Center College of Allied Health Department of Medical Imaging and Radiation Sciences Oklahoma City, Oklahoma
Nicole Dhanraj, PhD Independent Researcher/Contractor Mangilao, Guam
Kerry Greene-Donnelly, MBA, RT (R)(M)(CT)(QM) Assistant Professor Upstate Medical University Syracuse, New York
Kristi Klein, MS Ed(R)(M)(CT) Program Director School of Health Education Madison Area Technical College Madison, Wisconsin
Diana E. Mishler, MBA-HM, RT(R)(S), RDMS Clinical Assistant Professor and Coordinator Medical Imaging Technology Program Indiana University Kokomo, Indiana
Robin Rayman, AAS in Nuclear Medicine, CNMT, NMTCB-CT, PET, RT(N), EMT
Nuclear Medicine Technologist Salem, South Dakota
We would like to thank the hard work and extraordinary efforts of the contributing authors of this text. Without their dedication and commitment to the project, it would not have been possible. It is also their talent in presenting information that allows the reader to transfer theory, facts, and data into comprehension through which they can practice nuclear medicine.
We would also like to thank and recognize the administrators, co-workers, families, and loved ones of all of those involved in the book whose names do not appear in writing but who endured the time commitment made by the authors and editors to ensure the delivery of quality material.
We cannot forget to thank our dedicated editors at Elsevier, as well as the production and design team, whose patience, support, guidance, and diligent work have made this project a success.
Lastly, we want to give a special acknowledgment to the man who has been involved in this book from the very first edition—Paul Christian! We thank you and salute you for really making this a reality for all of us in Nuclear Medicine.
From the first edition of this textbook, Paul Christian has been integral to its success. Paul’s first involvement with the book was as a co-author with the late Ed Coleman on the Skeletal chapter. When the second edition was started, he was invited by Don Bernier and Jim Langan to be a co-editor and by the fourth edition was the primary editor. A journey of several decades.
There are few in the field of nuclear medicine who do not know or are not familiar with the name Paul Christian. This is no surprise. He has given over 80 invited lectures, has provided PET Board Exam Review Courses, and has more than 100 publications to his name. He has served as a journal reviewer of four premier journals in the field, one of them continuously since 1976 and another since 1983. He has served on
advisory boards for major companies in the field of nuclear medicine (e.g., GE Medical Systems, Picker). He has served on the Board of Directors for the Joint Review Committee in Nuclear Medicine Technology (JRCNMT), the Nuclear Medicine Technologist Certification Board (NMTCB), and the Intersocietal Commission of the Accreditation of Nuclear Medicine Laboratories. He has been an active member of several professional organizations, most notably the Society of Nuclear Medicine (now the Society of Nuclear Medicine and Molecular Imaging) and the Technologist Section of this organization. His participation has included membership on numerous committees and task forces, and he has served as chair on several committees. His recognitions also include scientific exhibits at professional meetings through the years, several awards, and a patent to his name.
Paul’s professional career spanned decades at University of Utah, Salt Lake City, where he held a variety of positions and served on several committees. Paul’s appointments included but were not limited to Education Director and later Program Director of the Nuclear Medicine Technology School; Clinical Instructor of Nuclear Pharmacy; Assistant Research Instructor of Radiology in the College of Medicine; Director, Cyclotron Radiochemistry Laboratory and PET/ CT Imaging; and Associate Director-Operations, Molecular Imaging Program.
Under his direction this textbook has served as a required textbook of many nuclear medicine technology programs and a valued resource throughout the educational and clinical nuclear medicine community worldwide. Thank you, Paul, for all of your work in the field of nuclear medicine and your dedication to this textbook.
Kristen M. Waterstram-Rich David
Gilmore
CONTENTS
SECTION 1 Foundations
1 Mathematics and Statistics, 1
Maria Mackin and Helen Timberlake
Fundamentals, 2
Practical Applications, 10
Statistics, 22
2 Cell and Molecular Biology, 40
Maureen Ferran
Eukaryotic Cell Structure, 41
Control of Gene Expression, 44
The Cell Cycle, 47
Molecular Basis of Cancer, 50
3 General Chemistry and Biochemistry, 56
Leslie A. Bishop
Elements, Compounds, and Mixtures, 57
Laws of Constant Composition and Multiple Proportion, 63
Atomic Weights, Molecular Weights, and the Mole Concept, 64
Solutions and Colloids, 64
Chemical Reactions and Equations, 66
Acids and Bases, 68
Equilibriums and Equilibrium Constant, 70
The pH Concept, 70
Buffer Solutions, 71
Organic Compounds, 71
4 Radiochemistry and Radiopharmacology, 77
Sally W. Schwarz, Reiko Oyama, and Michele M. Beauvais
Production of Radionuclides, 78
Technetium Radiopharmaceuticals, 84
Gallium and Indium Radiopharmaceuticals, 90
Thallium Chloride, 92
Iodinated Radiopharmaceuticals, 92
PET Radiopharmaceuticals, 93
Therapeutic Radiopharmaceuticals, 97
Regulatory Issues: Radiopharmaceutical Quality Assurance, 98
Radiopharmaceutical Quality Control, 99
5 Radiation Safety in Nuclear Medicine, 108
Norman E. Bolus and Krystle Worthington Glasgow Radiation Safety Program, 111
Sources of Radiation Exposure, 129
Radiation Regulations, 130
Radiation Dose, 131
Biological Effects of Ionizing Radiation, 135
SECTION 2 Patient Care, Management, and Research
6 Patient Care, 140
Kathy Thompson Hunt and Donna C. Mars
Patient Care, 142
Patient Preparation, 142
Patient-Centered Care, 142
Age-Specific Care, 143
Pediatric Considerations, 145
Body Mechanics, 147
Medication Administration, 149
Contrast Media, 164
Infection Control, 168
Vital Signs and Patient Assessment, 171
Emergency Care, 173
Ancillary Equipment, 175
7 Department Administration, 181
Erin Beloin, Denise A. Merlino, and Mary Beth Farrell
Health Care Leadership, 182
Health Care Management, 182
Coding and Reimbursement, 183
Quality Measures and Improvement, 185
Credentials and Accreditation, 197
8 Clinical Research, 201
LisaAnn Trembath
Defining Clinical Research, 201
Clinical Trials and Studies, 202
Conclusion, 206
9 Health Informatics in Imaging, 207
Frances Keech
Background, 208
Computers in Health Care, 208
Computer Hardware, 209 Computer Software, 209
Image Acquisition, 209
Image Display and Processing, 212 Region of Interest, 215 Clinical Applications, 217 Health Information Systems, 217 Electronic Health Records, 218
Radiology Information System, 219 Standard Operating Procedures, 220 Future Advances, 220
SECTION 3 Physics and Instrumentation
10 Physics of Nuclear Medicine, 223
Patrick Byrne and Cybil Nielsen Matter, 224
Nucleus of an Atom, 224
Nature of Electromagnetic Radiation, 226 Mass Energy Equivalence, 228 Units, 228
Modes of Radioactive Decay, 229
Mathematics of Decay, 234 Units of Radioactivity, 235
Decay Factor and Precalibration Factor, 236
Effective Half-Life, 237
Parent-Daughter Radionuclide Relationships, 237
1
Mathematics and Statistics
Maria Mackin and Helen Timberlakea
OUTLINE
Fundamentals, 2
Scientific Notation, 2 Fractions and Percentages, 3
Algebraic Equations and Ratios, 3
Inverse Square Law, 4 Units, 5
Exponent Laws and Logarithms, 6
Numeric Accuracy: Significance and Rounding, 9 Calculators and Computer Programs, 10
Practical Applications, 10
Radioactive Decay, 10
Decay Factor Tables for Radioactive Decay, 12
Concentration-Volume Calculations, 13 99Mo-99mTc Radionuclide Generators, 14
OBJECTIVES
After completing this chapter, the reader will be able to:
• Use scientific notation in performing algebraic operations.
• Use the inverse square law to calculate the intensity of a radiation field at various distances.
• Perform radioactive dilution calculations.
• Define the units of radioactivity, radiation exposure, radiation absorbed dose, and radiation dose equivalent.
• Perform calculations with logarithms and exponents using a calculator.
• Discuss numeric accuracy, significant digits, and rounding.
• Calculate quantities of radioactivity using the general form of the decay equation and decay factors.
• Use tables of decay factors to calculate remaining radioactivity.
• Calculate concentration and volume and radioactivity for patient doses.
aThe authors wish to acknowledge Paul H. Brown for his previous contributions to this chapter.
Half-Life: Biological, Physical, and Effective, 15 Attenuation of Radiation, 15 Graphs, 17
Measurement of Effective Half-Life, 19
Least Squares Curve Fitting, 20
Other Graphs, 22 Statistics, 22
Mean and Standard Deviation, 22
Gaussian Distributions, 24
Poisson Distributions and Counting Statistics, 24
Chi-Square Tests, 26 t-Tests, 28
Medical Decision Making, 30
• Compute the concentration of 99Mo in 99mTc.
• Compute effective half-life and biological half-life.
• Calculate intensity with half-value layers.
• Diagram various types of graphs and graphing techniques.
• Discuss curve-fitting techniques.
• Define mean, standard deviation, and coefficient of variation.
• Discuss Gaussian and Poisson distributions.
• State the formula for standard deviation, and perform calculations in the presence of background.
• Explain the function of the chi-square test and interpretation of results.
• Interpret the results of a chi-square test using a probability table.
• Discuss the use and interpretation of t-tests.
• Describe the interpretation of sensitivity, specificity, prevalence, and accuracy.
Inverse Square Law
The radiation exposure from a radioactive point source is governed by a mathematical relationship called the inverse square law. This law states that the radiation exposure or intensity (I) at a distance (d) from a radioactive source is proportional to the inverse square of the distance.
The radiation dose emitted from a source is symmetrical with the photons traveling in all directions. As the distance from the source increases, the emissions diverge from one another; therefore, as you move further from the source, you have fewer interactions with emissions. Conversely, as you move closer to the source, you have increased interactions with emissions.
This law holds for a point source (a source that is very small compared with the distance from the source) that emits radiation not absorbed while traveling over the distances involved. Syringe source γ-rays at a distance of 1 meter (m) or more in air would qualify as a point source. Air is a weak absorber of x-rays or γ-rays in the energy ranges typically encountered in nuclear medicine. Note that a syringe source for exposure to the hands does not obey the inverse square law, because the distance from the syringe to the hands is not large compared with the syringe dimensions. Similarly, the exposure from standing near patients does not follow the simple inverse square law.
Mathematically, the inverse square law states:
where k is a proportionality constant that depends on the type of radioactive source and its activity. The symbol ∝ means “is proportional to.” The inverse square law results in the radiation intensity quadrupling if the distance from the source is halved, or the radiation intensity decreasing to one fourth of its value if the distance is doubled. Changing the distance by a factor of 3 results in a factor of 9 change in intensity (nine times more intensity for one third the distance, or one ninth the intensity for three times the distance). This is usually stated in a form relating the prior intensity (I2) at some prior distance (d2) to a new intensity (I1) at some new distance (d1):
I1(d1)2 = I2(d2)2
The intensity of radiation is usually measured in units of roentgens (R) or milliroentgens (mR) per hour (hr). For example, suppose a technologist working 2 m from a small vial of radioactivity results in an intensity or exposure level of 2.5 mR/hr at 2 m. If the technologist moves to a distance of 3 m from the source, what is the new radiation intensity?
I1 = X
d1 = 3m
I2 = 2.5mR/hr
d2 = 2m
Therefore:
I1 ×(3 m)2 =(2.5 mR/hr)×(2 m)2
or I1 ( 2.5 mR )×( 2 m)2 (3 m)2 hr = I1 ( 2.5 mR )× 4 m 2 9 m 2 hr =
The technologist’s radiation exposure is more than halved by moving 1 m farther from the source. Note that scientific calculators generally have an x2 key, which facilitates the calculation:
2.5 enter intensity value
× multiply key
2 enter old distance value
x2 squaring key
÷ divide key
3 enter new distance value
x2 squaring key
= (see result of 1.1 in display)
In another example, a technologist is working 4 feet (ft) from a point source that results in an exposure of 0.62 mR/ hr. Radiation safety is concerned about the exposure rate and suggests that the intensity should be maintained at no more than 0.25 mR/hr for a safe working environment. To what distance from the source should the technologist move to achieve an exposure level of 0.25 mR/hr?
0.25 mR/hr ×(d1)2 = 0.62 mR/hr ×(4 ft)2
0.25 mR/hr (0.62 mR/hr)×(4 ft)2 (0.62 mR/hr)× 16 ft2 (d1)2 = (d1)2 = (d1)2 = 39.7 ft2
0.25 mR/hr
(d1)2 = 39.7 ft2
d1 = 6.3 ft
Again, scientific calculators generally have a square root key (√ ), which facilitates calculation:
0.62 enter old intensity value
× multiply key
4 enter old distance value
x2 squaring key
÷ divide key
0.25 enter new intensity value
= results of division √ square root key (see result of 6.3 in display)
Notice that in solving this problem the left side of the equation contained (d1)2, but because the object was to solve for d1, the square root of both sides of the equation was taken. The squaring function and the square root function canceled on the left side of the equation because x2 = x. Functions that have this property are called inverse functions. The squaring function (x)2 and the square root function (√ ) are inverse functions.
In nuclear medicine and molecular imaging, there are a number of situations in which a solution of a known concentration is used to make another solution of a lower concentration.
8 enter exponent value x = (see result of 256 in display)
A special case arises when the exponent is zero. By mathematic definition, any number (except zero) raised to the zero power is equal to 1:
e0 = 1
100 = 1 20 = 1
Negative exponents provide a convenient form for representing small numbers:
10 4 = 1 104 = 0.0001
Note that this shows how a number in exponent notation can be moved from numerator to denominator simply by changing the sign of the exponent:
28 = 1 2 8 = 256
The algebra of exponents in equations follows certain rules.
Multiplication: add the exponents. Bx × By = Bx+y 102 × 103 = 105 104 × 10 – 5 = 10 – 1
To find the area of a rectangle, multiply width by height: Area = 20cm × 30cm = 600cm2
Note how this follows the rule for multiplying exponents: cm1 × cm1 = cm2
Division: subtract the exponents.
A practical problem involves the calculation of the mass attenuation coefficient μm, which is defined by the quotient of the linear attenuation coefficient μ (in units of cm−1 or 1/cm) divided by the density r (in units of g/cm3) for some substances such as human soft tissues.
μm =μ/ρ
For example: if µ= 0.121/cm, and ρ= 3.4 g/cm3, then
µ= 0.121/cm
The difficulty lies in how to evaluate the units. First, eliminate denominator units within each term of the equation— that is, write 1/cm as cm−1 and g/cm3 as g × cm−3, using the rule discussed previously for moving from denominator to numerator by changing the sign of the exponent. Then:
µ= 0.12 cm 1 3.4 g/cm 3 m
Now follow the previous rule for division of exponents with centimeter dimensions:
µ m === 0.035 0.12 cm
Alternatively, the units of μm might be written as:
µ m 1 = 0.035 cm × g 2
Another example is hertz (Hz), the measure of frequency expressed as the number of waves or cycles per second. For example, if three waves pass by a certain point in space in 1 second, then the frequency (v) is given by: v = 3/sec, or 3 sec , or 3 Hz 1
Taking the root of a number is the inverse of raising it to a power. A special case is the square root (√ ) of a positive number x, which is defined by:
√x × √x = x
e.g.,9 × 9 = 3 × 3 = 9
Note that finding the square root is the same as raising a number to the half power: √x = x1/2. As mentioned previously, the square root and squaring operation are inverses of each other because
xx 2 ()= and x ()= x 2
Whether the squaring or the square root is performed first, the inverse function always cancels the other operation and simply returns the number x. Other roots can be calculated as the nth root of a number, which is written as n √x . For example, 3 √64 = 4, because 4 × 4 × 4 = 64. Some scientific calculators have a root key such as x √y , or the calculator might have an inverse (INV) key which is pressed before pressing another function to get the inverse of that function. For example, to calculate 3 √64 on the calculator:
64 enter value y to find cube root
INV and Yx or root key x √y , which is same as INV − Yx
3 enter root value x
= (see display of result, 4)
The base e (= 2.718 ) is an irrational number called Euler’s number, named after Swiss mathematician and physicist Leonhard Euler (1707-1783). Calculations involving e pervade the mathematical, physical, and biological world: radioactive decay, absorption of radiation, growth of bacteria, and radiation damage to cells. Scientific calculators often have
Numeric
Accuracy: Significance and Rounding
The accuracy of mathematical calculations is governed by three concepts: significant figures, rounding, and significant decimal places. Significant figures refer to the number of digits required to preserve the mathematical accuracy in a number.
Numeric Accuracy Rule 1. For a number with no leading or trailing zeros, the number of significant figures is the number of digits.
Number Number of Significant Figures 3 1 3.45 3
Numeric Accuracy Rule 2. For a number with leading zeros, the leading zeros are not significant.
Number Number of Significant Figures
0.0015 2
0.0463 3
Notice that expressing a number like 0.0015 in scientific notation as 1.5 × 10−3 eliminates the leading zeros and therefore eliminates the need for rule 2.
Numeric Accuracy Rule 3. Trailing zeros in a number should be retained only if they are significant, which depends on the context of the problem. Thus a problem may state that a drug costs $37; the cost has two significant figures. Or the problem may state that the drug costs $37.00, which is interpreted as being accurate in both dollars and cents; it has four significant figures. A length expressed as 4 cm has one significant figure; the length was measured to the nearest centimeter. But a length expressed as 4.0 cm has two significant figures; the length was measured more accurately to the nearest millimeter.
Numeric Accuracy Rule 4. The accuracy of the result in multiplication or division is such that the product or quotient has the number of significant figures equal to that of the term with the smaller number of significant figures.
For example, 2 × 2.54 = 5; the correct answer has only one significant figure because the 2 in the calculation has only one significant figure. But 2.00 × 2.54 = 5.08; the result has three significant figures because it is presumed that the trailing zeros in the 2.00 are significant. In another example,
It is necessary to properly round off the result from the calculator before recording the result. For example:
0.061 1.233 = 0.0494728 = 0.049 whereas 0.061 1.232 = 0.0495130 = 0.050
Both answers have two significant figures, but the results are different because of rounding. The mechanics of rounding a number consist of carrying the mathematical calculations to several more digits than are needed in the final answer. The final result is rounded by the following rules.
Numeric Accuracy Rule 5. If the rightmost digits, beyond the significant figures in the final result, are less than 5000, then simply drop the rightmost digits. As an example, consider 0.061 / 1.233 = 0.0494728, which must be rounded to two significant figures (because 0.061 has two significant figures). The rightmost digits are 4728, which is less than 5000; therefore, the 0.049 is the correct, rounded, final answer.
Numeric Accuracy Rule 6. If the rightmost digits beyond the significant figures are greater than 5000, then increase the least significant figure by 1. As an example, consider 0.061 / 1.232 = 0.0495130, which again must be rounded to two significant figures (because of the 0.061). The rightmost digits are 5130, which is greater than 5000; therefore, the final answer needs to be rounded up by one least significant digit (0.049 + 0.001). The final answer is 0.050, rounded correctly to two significant figures.
Numeric Accuracy Rule 7 It may sometimes be necessary to round a number when the rightmost digit is exactly 5. The rule in this case is to round down the number if the digit to the left of the 5 is even, and round up the number if the digit to the left of the 5 is odd. For example 2.45 is 2.4, rounded to two significant figures; 1.5 is 2, rounded to one significant figure. This rounding scheme for numbers that end in 5 is arbitrary and results in averaging out rounding errors when a large number of calculations is performed.
Numeric Accuracy Rule 8. For addition and subtraction, the final result has the same number of significant decimal places (rather than significant figures) as the number in the problem with the least number of significant decimal places. For example:
0.123 + 3.42 = 3.54 (twosignificantdecimalplaces)
0 1 + 3.42 = 3.5 (onesignificantdecimalplace) 1 + 3.42 = 4 (zerosignificantdecimalplaces)
The result has only two significant figures. Note that a calculator display might show 0.0049433, but the proper answer to record as a result is 0.0049 with only two significant figures.
Sometimes addition and rounding must be used simultaneously, as in 0.125 + 3.42 = 3.545, which should properly be rounded to 3.55, with only two significant decimal places because the 3.42 has only two significant decimal places.
t12 0 693 0.693 0.1153 hr 1 6.01 hr === λ
Careful attention to the units is necessary to avoid errors. To make the units clearer, this might be restated as:
t12 0 693 0 1153 hr 1 =
To obtain the units of 1/hr out of the denominator, multiply both numerator and denominator by units of hours (essentially multiply by 1):
t hr hr hr 12 0 693 0 1153 1 =
The 1/hr and the hr cancel in the denominator, leaving: thr 12 0693 01153 =
The common radioactivity problem is to solve for one of the four variables (At, A0, t, t1/2) when the problem specifies three of them.
For example, on Monday at 8 am, a sample of 131I (t1/2 = 8.04 days) is calibrated for an activity of 10 μCi. What amount of radioactivity will be given to the patient if the dose is administered on the following Friday at 2 pm?
Given: A0 = 10 µCi, t = Mon 8 am → Fri 2 pm, t12 = 8.04 days
Solvefor :At
A digression to discuss units is necessary. Remember the good practice of always writing down the units associated with every number. Any problem involving the ex function must have a dimensionless number for the value of x, so it is absolutely necessary that identical units be used for both t and t1/2. Then the units cancel in the numerator and denominator of t/t1/2, making the calculation independent of the units chosen to measure time. In this problem, the time of decay (Mon 8 am → Fri 2 pm) is 102 hours, but t1/2 has already been given in days. It would be correct to express the time of decay t as 4.25 days (rather than 102 hours) with the t1/2 also in days, or it would be correct to express the t1/2 as 193 hours (rather than 8.04 days) with the t also in hours.
A = 10 µCi × e 0.693 ×(4.25 days/8.04 days) or
A = 10 µCi × e 0.693 ×(102 hr 193 hr)
A = 10 µCi × e 0.366 = 6.93 µCi
A quick check of the results of the calculator’s answer is also useful, based on the 100% → 50% → 25% → timeline for each half-life. In this example, the decay time of 4.25 days is less than one t1/2 (8.04 days), so the answer should be between 100% and 50% of the initial 10μCi activity. The calculator result of 6.93 μCi agrees with the mental check. Inadvertent calculator usage errors can be prevented with these mental checks.
A radionuclide is often calibrated for some activity level on a Friday, but it might have been administered to the patient on the previous Monday. This type of radioactivity problem can be solved using negative time values for times that precede the time of A0 calibration. For example, a radionuclide with a 2-day half-life is calibrated for Friday at noon to be 3 mCi. What activity level was present on the preceding Monday at noon?
Given: A0 = 3 mCi
t = 96 hr (–4 days),t1 2 = 2 days
Find :A = A0e0.693 ×(t t1 2)
A = 3 mCi × e 0.693 ×( 4days 2days)
A = 3 mCi × e+1.386 (notice [] times [] is [+])
A = 12mCi
This result is easy to check mentally because the time difference is exactly 2 half-lives; the answer should be that Monday noon has four times the activity of Friday noon, in agreement with the calculated result.
Sometimes a problem concerns only the fraction of remaining radioactivity (A/A0), rather than the actual remaining activity in mCi. For example, what fraction of radioactivity is left at a time equal to 3 half-lives? Here, t = 3t1/2 is given and the object is to find A/A0. The radioactive decay law can be algebraically rearranged (dividing both sides of the decay equation by A0) as:
A A0 = e 0.693 ×(t t1 2)
A A0 = e 0.693 ×(3t1 2 t1 2)
A A0 = e 0.693 × 3
AA0 = e 2.079
A A0 = 0.125
A quick mental check confirms the result: in 3 half-lives the radioactivity should decay 100% → 50% → 25% → 12.5%.
Remember that the radioactive decay equation always refers to the remaining activity, which is the same as 100% minus the decayed activity. It is important to determine whether the problem is stating the amount of radioactivity remaining, which the decay equation predicts, or the radioactivity that has decayed away. For example, the previous problem showed that only 12.5% of the original radioactivity remains after 3 half-lives. This problem could also state that 87.5% of the radioactivity decayed away in 3 half-lives.
Radioactive decay problems can also require the solving for the t or t1/2 value in the radioactive decay law. These values are contained in the decay law as part of the exponent function, so the exponent must be removed. An example might be to calculate the time necessary for 99.9% of a sample of 99mTc to decay away. Remember that the decay law works with the remaining radioactivity.
Given: A A0 = 0.1% = 0.001 t12 = 6.01 hr
Solve for t1 2in: A A0 = e (0.693)×(t t1 2)
0.001 = e 0.693 ×(t 6.01 hr)
How much saline should be added to the vial, and what will the concentration be once it is diluted? Using the following formula:
Volume (toaddtovial) = Volume (final) – Volume (initial)
Volume (toaddtovial) = 0 5 – 0 2 = 0 3ml
The technologist should add 0.3 ml of saline to the vial. What is the concentration in the new 0.5 ml volume?
Concentration = Activity/Volume
Concentration = 20mCi/0 5ml = 40mCi/ml
Often it is necessary to combine radioactive decay calculations with concentration-volume problems. The nuclear medicine department may obtain its radioactivity from a 99Mo-99mTc generator through an early-morning elution of the generator. This eluate decays throughout the day, resulting in a change in concentration (mCi/ml). For example, a generator is eluted at 7 am, yielding 900 mCi in 20 ml of eluate solution. What volume should be withdrawn from the eluate vial into a syringe to yield 15 mCi for a scan at 2 pm? First, calculate the radioactivity remaining in the eluate vial at 2 pm:
A = 900 mCi × e 0.693 ×(7 hr 6.01 hr)= 402 mCi
(A quick mental check confirms the reasonableness of this answer: slightly less than 50% remaining at a time slightly greater than t1/2.) The concentration (radioactivity per volume) in the eluate vial is then 402 mCi/20 ml at 2 pm. The volume needed to be withdrawn into the syringe for a 15-μCi dose at 2 pm can be calculated from the equation: Activity = Concentration × Volume.
= C × V
99Mo-99mTc Radionuclide Generators
Another common problem for radioactive decay is to calculate the ratio of 99Mo (t1/2 = 65.9 hours) activity to 99mTc activity in generator eluate. The problem is that some 99Mo is also eluted out along with the 99mTc in the morning elution. The 99Mo is a radionuclidic impurity that is limited by regulatory agencies to be less than 0.15 μCi 99Mo per mCi 99mTc at the time of injection into a patient. Consider a generator that is eluted at 7 am and yields an eluate vial containing 30 μCi 99Mo along with 250 mCi 99mTc. Can this eluate be used for a brain scan at the elution time of 7 am? Calculate the ratio of 99Mo to 99mTc activity at 7 am as:
30
In this example, you must solve for the decay of 99Mo and 99mTc and then calculate the 99Mo to 99mTc ratio. This eluate is less than the regulatory limit (0.15) and may be used.
Could this same eluate be used 6 hours later to prepare a lung scan? Now the ratio of activities at 1 pm is:
99Mo
99mTc
30 µCi × e 0.693 × (6 hr 65.9 hr)
250 mCi × e (0.693)× (6 hr 6.01 hr)
30 µCi × 0.939
250 mCi × 0.500
= 0.23 µCi99Mo mCi99mTc
Ci99Mo
mCi99mTc
This eluate cannot be used because the 99Mo/99mTc ratio is greater than the 0.15 regulatory limit. The 99Mo has not changed very much in the 6 hours since generator elution, but the 99mTc activity has halved, resulting in a large increase in the 99Mo/99mTc ratio.
The mathematics of this type of radionuclide generator are governed by the laws of radioactivity,2 relating radionuclides denoted as a parent-daughter-granddaughter (and so on) decay chain. The parent radionuclide 99Mo decays to the daughter 99mTc, which in turn decays to the granddaughter 99Tc, and the decay chain continues. Without dealing with the exponential algebra for this type of decay, it is possible to calculate the 99mTc radioactivity expected to be eluted from the generator by knowing three values:
1. The activity of 99Mo in the generator, which is given by the manufacturer’s calibration date and the decay law for 99Mo (t1/2 = 65.9 hours).
2. The time since the last elution of the generator, which is commonly 24 hours for daily elutions.
3. The ratio of 99mTc to 99Mo in the generator, which depends on the time since the last elution (99mTc to 99Mo ratios are shown in Table 1-5 as a function of the time since last elution).
For example, a generator is delivered on Saturday and calibrated by the manufacturer for the following Monday at 6 pm to contain 2 Ci of 99Mo. The generator is eluted daily (Monday through Friday) at 7 am. What activity of 99mTc is available in the generator at 7 am on Tuesday? The required data are as follows:
1. 99Mo activity Tuesday 7 am, 13 hours after calibration time
A = 2 Ci × e 0.693×(13 hr 65.9 hr)
A = 1744 mCi of 99Mo in the generator,Tuesday 7 AM
2. Time since last elution is 24 hours, since the generator is eluted daily
3. From Table 1-5, the 99mTc/99Mo ratio is 0.87 for 24 hours since last elution, so:
Activity 99mTc = 0.87 × Activity 99Mo
= 0.87 × 1744 mCi = 1518 mCi
Depending on the quality of the generator, only a percentage of this 1518 mCi of 99mTc will appear in the eluate. This
is known as the elution efficiency of the generator. If the elution efficiency is 95%, then the 99mTc found in the Tuesday morning eluate would be calculated as 0.95 × 1518 mCi = 1442 mCi and would be for patient studies.
Half-Life: Biological, Physical, and Effective
In most clinical applications, the nuclear medicine gamma camera measures the radioactive counts over an organ of interest in the patient’s body. Typically, the patient’s organ excretes the radiopharmaceutical with some biological halflife tB, while the radioactivity decays physically with a physical half-life that is denoted as tP. The biological half-life is an indicator of the physiological fate of the radiopharmaceutical, tB. The counts observed by the gamma camera follow an exponential decay law based on the effective half-life tE, where:
or, in a format that is much easier for calculation purposes: tE tP tB tP tB = × + ()
For example, if the liver excretes a 99mTc radiopharmaceutical with tB = 3 hours, then the gamma camera over the liver would observe an effective half-life of
tE = (6 hr + 3 hr)= 2 hr 6 hr × 3 hr
The effective half-life is always less than or equal to the smaller of tP or tB
Attenuation of Radiation
The calculation of the intensity (I) of x-ray or γ-ray photons transmitted through some thickness (x) of absorbing material follows exactly the same algebra as the equations for radioactive decay. Figure 1-1 shows a beam of monoenergetic x-ray or γ-ray photons striking a thickness of absorbing material. Monoenergetic means that the photons all have the same energy, such as a beam of photons from a 99mTc radionuclide source that emits photons with an energy of 140 keV. The electron volt (eV) is a common unit for specifying energy. In nuclear medicine, the energy of photons is commonly expressed in units of thousands of electron volts, abbreviated keV. The initial intensity (number of photons per second) entering the absorbing material is called I0. The material attenuates, or absorbs, some fraction of the photons, and the photon beam emerges with a transmitted (i.e., not absorbed) intensity I. The intensity of the transmitted radiation is given by:
I = I0e−µx
where μ is the linear attenuation coefficient, or the fraction of the beam absorbed in some (very small) thickness x. The linear attenuation coefficient μ is the analog of the decay constant λ in radioactive decay. The linear attenuation coefficient μ depends on the type of absorbing material and the energy of the photons. A large μ value means a strong absorbing material. For example, 99mTc γ-rays the μ value in lead is about 23 cm−1, whereas the μ value in water is only 0.15 cm−1. Since 23 cm−1 is greater than 0.15 cm−1, lead is much more absorbent than water.
