…final slides from lecture 6 last week: ROC Curves Larry MacDonald macdon@uw.edu May 14, 2013
Image quality assessment Question: which is a better image? Answer: what are you trying to do?
Quantifying Detection Performance Possible method of reader scoring: 1 = confident lesion absent ? 2 = probably lesion absent 3 = possibly lesion absent 4 = probably lesion present 5 = confident lesion present diagnostic threshold false true 0.5 lesion present lesion absent image (positive) image (negative) 0.4 0.3 Frequency of reader 0.2 scores 0.1 0 0 1 2 3 4 5 score
Class Separability (e.g. detectability) “ easy ” task “ difficult ” task 2.5 0.7 lesion lesion 0.6 2 absent present 0.5 (negative) (positive) Histogram Histogram 1.5 0.4 0.3 1 0.2 0.5 0.1 0 0 0 1 2 3 4 5 0 1 2 3 4 5 Reader score (1 = confident lesion absent, 5 = confident lesion present)
Quantifying Detection Performance Is the object present? (“truth” or gold standard ) Positive Negative False Positive True Positive Does the (FP) (TP) True observer say the object is False Negative True Negative False present? (FN) (TN)
Key concepts • Sensitivity: True positive fraction Is the object present? (TPF) = TP/(TP + FN) = TP/P Positive Negative True True Positive False Positive (TP) (FP) • Specificity: True negative fraction (TNF) = TN/(TN + FP) = TN/N False False Negative True Negative (FN) (TN) • Accuracy = (TP + TN) / (P + N)
Dependence of Sensitivity and Specificity on “threshold of abnormality”: 1.0 actually -ve cases actually +ve cases t 1 t S e n s i t i v i t y 2 Sensitivity Specificity ⇒ ( a t ) t 3 (at t 3 ) t 3 t 4 Confidence t t t t that case is + 0.0 1 2 3 4 0.0 1.0 Specificity Four possible “thresholds of abnormality”
Receiver Operating Characteristic (ROC) Curve 1.0 1.0 True Positive Fraction Sensitivity Sensitivity ⇐⇒ ⇐⇒ ROC curve 0.0 0.0 0.0 1.0 0.0 1.0 Specificity False Positive Fraction (false alarm rate) = 1.0 − Specificity
The ROC Curve Points A, B, & C correspond to different thresholds 1 Note, for example, it is always C possible to make sensitivity = 1 if the threshold is low TPF (Sensitivity) enough! B actually +ve cases actually -ve cases Decreasing A Threshold S e n s i t i v i t y ( T P F ) 1- Specificity (FPF) 0 Score C A B 0 1 Threshold for diagnosis FPF = 1 - Specificity
A dilemma: Which modality is better? 1.0 True Positive Fraction Modality B Sensitivity Modality A 0.0 0.0 1.0 False Positive Fraction = 1.0 − Specificity
The dilemma is resolved after ROCs are determined (one possible scenario): 1.0 Conclusion : True Positive Fraction Modality B is better, because it can Modality B achieve : • higher TPF at same FPF, or Modality A • lower FPF at same TPF 0.0 0.0 1.0 False Positive Fraction However: modality-A and modality-B curves may cross, each being more advantageous in different regions of the TPF-FPF space
The ROC Area Index (Az) perfect : A z = 1.0 random: A z = 0.5 1.0 where we want to go TPF = Sensitivity A z 0.0 0.0 1.0 False Positive Fraction = 1.0 − Specificity
Comparing Imaging Systems Ideal 1 Ideal Better 0.8 Typical Good d 0.6 s 1 s 2 No separability TPF or detectability d 0.4 SNR = ( ) 2 2 + s 2 s 1 2 (SNR for detection task) 0.2 0 0 0.2 0.4 0.6 0.8 1 Useless FPF
Introduction to Nuclear Physics and Nuclear Decay Larry MacDonald macdon@uw.edu May 14, 2013
Atoms •Nucleus: ~10 -14 m diameter ~10 17 kg/m 3 •Electron clouds: ~10 -10 m diameter (= size of atom) Nucleons (protons and neutrons) are ~10,000 times smaller than the atom, and ~1800 times more massive than electrons. (electron size < 10 -22 m (only an upper limit can be estimated)) Nuclear and atomic units of length 10 -15 = femtometer (fm) 10 -10 = angstrom (Å) Molecules water molecule: ~10 - 10 m diameter ~10 3 kg/m 3 mostly empty space ~ one trillionth of volume Hecht, Physics , 1994 Water occupied by mass (wikipedia)
Table of Elements Elements distinguished by their numbers of protons X = element symbol Z ( atomic number ) = number of protons in nucleus alternative denotations A X N A X A X N = number of neutrons in nucleus A ( atomic mass number ) = Z + N Z Z [ A is different than, but approximately equal to the atomic weight of an atom in amu] Examples; oxygen, lead 16 O 8 208 Pb 126 A X N Electrically neural atom, has Z electrons in its Z 8 82 atomic orbit. Otherwise it is ionized , and holds net electric charge. Z
Mass and Energy Units and Mass-Energy Equivalence Mass atomic mass unit, u (or dalton, Da): mass of 12 C ≡ 12.0000 u = 19.9265 x 10 -27 kg Energy Electron volt, eV ≡ kinetic energy attained by an electron accelerated through 1.0 volt 1 eV ≡ (1.6 x10 -19 Coulomb)*(1.0 volt) = 1.6 x10 -19 J 2 2 = m 0 c E = mc E = total energy (rest mass + kinetic) ( ) m 0 = rest mass 2 1 ! v c c = 3 x 10 8 m/s speed of light mass of proton, m p = 1.6724x10 -27 kg = 1.007276 u = 938.3 MeV/c 2 mass of neutron, m n = 1.6747x10 -27 kg = 939.6 MeV/c 2 = 1.008655 u mass of electron, m e = 9.108x10 -31 kg = 0.511 MeV/c 2 = 0.000548 u
Nuclide Groups/Families A nuclide is a nucleus with a specific Z and A ~1500 nuclides exist (Periodic Table typically lists distinct Z ) Nuclides with the same: Z (#protons) are Isotopes N (#neutrons) are Isotones A (#nucleons) are Isobars A, N , and Z are Isomers A nuclide with the same Z and A (& thus also N ) can also exist in different (excited & ground) energy states
Factors in Nuclear Stability • Nuclear stability represents a balance between: – Nuclear “strong force” (basically attractive) – Electrostatic interaction (Coulomb force) between protons (repulsive) – Pauli exclusion principle – Residual interactions (“pairing force”, etc.) – • Stability strongly favors N approximately equal to (but slightly larger than) Z. This results in the “band of stability” in the Chart of the Nuclides.
N vs. Z Chart of Nuclides N > Z for the majority ( N = Z for low Z elements) The line of stability (gold band) represents the stable nuclei. Distribution of stable nuclei: Z N #stable nuclei even even 165 even odd 57 odd even 53 odd odd 4 isobars 279 stable nuclei exist (all have Z < 84) isotones ~1200 unstable (radioactive) isotopes (65 natural, remaining are human- made) Hecht, Physics , 1994
Nuclear Shell Structure Schematic energy diagrams • Similar to atomic structure, the nucleus can be E=0: particle is unbound (free) modeled as having quantized allowed energy states E<0: particle is bound (e.g. in (shells) that the nucleons occupy. nucleus, in an atom) E>0: free & has excess energy (can • The lowest energy state is the ground state . be potential or kinetic) • Nuclei can exist in excited states with energy greater E than the ground state. • Excited nuclear states that exist for > 10 -12 sec. are metastable states ( isomeric ). • Nucleons held together by the ‘strong force’; short range, but strong. • This overcomes the repulsive electrostatic force of similar charged protons • Also similar to atomic theory: → Electrons swirl around in clouds about the nucleus; likewise, the nucleus is a dynamic swirl of nucleons. → Nucleons, like electrons, are paired in energy states - each with opposite spin. → Closed electron shells lead to chemically inert atoms. Magic numbers of nucleons (analogous to closed shells) form particularly stable nuclei. Hecht, Physics , 1994
Binding Energy The mass of a nuclide is less than the mass of the sum of the constituents. The difference in energy is the binding energy . The consequence is that energy is liberated when nucleons join to form a nuclide. The binding energy per nucleon dictates results when nuclides break apart (fission) or fuse together (fusion) (keep in mind that binding energies are thought of as negative, as in energy level diagrams on previous slide) Bushberg
Phenomenology of Stability • Stability strongly favors nuclides with even numbers of protons and/or neutrons – ~50% are Even-Even – ~25% are Odd-even – ~25% are Even-Odd – Only 4 out of 266 stable nuclides are Odd-Odd! The heaviest stable Odd-Odd nuclide is 14 N. • “Magic Numbers” -- analogous to closed atomic shells – Result in many stable isotopes or isotones – Magic nuclei are particularly stable and more “inert” – Magic #’s: 2, 8, 20, 28, 50, 82, 126
Nuclear Binding and Stability • Protons and neutrons are more stable in a nucleus than free. The binding energy is the amount by which the nucleus’ energy (i.e. mass) is reduced w.r.t. the combined energy (i.e. mass) of the nucleons. • Example: N-14 atom - Measured mass of N-14 = 14.00307 u mass of 7 protons = 7 * (1.00727 u) = 7.05089 u mass of 7 neutrons = 7 * (1.00866 u) = 7.06062 u mass of 7 electrons = 7 * (0.00055 u) = 0.00385 u mass of component particles of N-14 = 14.11536 u Binding energy is mass difference: E bind = 0.11229 u = 104.5 MeV
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