X-ray Physics

Ho Kyung Kimhokyung@pusan.ac.kr

Pusan National University

Lectures on Digital Radiography

Ch. 1 HMI

References

J. M. Boone, "X-ray Production, Interaction, and Detection in Diagnostic Imaging," in Handbook of Medical Imaging Perception and Techniques, J. Beutel, H. L. Kundel, and R. L. Van Metter, Eds., Bellingham, WA, USA: SPIE, 2000, ch. 1, pp. 1-77.

2

X-ray as a wave

𝐸 = ℎ𝜈

• ℎ = 4.135 × 10−15 eV s

• 𝜈 =𝑐

𝜆

• e.g., 1 eV = 1240 nm

X-ray tube

• Electron interaction with metal

• Bremsstrahlung

• Characteristic radiation

3

Bremsstrahlung

Electron interactions (e.g., deceleration) with nuclei

• Braking radiation

𝐼𝑏𝑟𝑒𝑚 ∝(𝑍𝑒)2(𝑧𝑒)4

𝑚2

• 𝑚−2 𝑒− as an incident ptl

• 𝑍2 W (𝑍 = 74) as a target mat'l

Ψ 𝐸 = 𝑘𝑍(𝐸𝑚𝑎𝑥 − 𝐸)

• Intensity spectrum

• Thick-target approximation

4

Characteristic x-ray

Electron interaction with (tightly) bound electrons

• Ejecting orbital electrons

• Cascaded electron transitions until valence shells are filled

• Each transition emits an energy (unique to each elemental atom)

– Characteristic radiation

– Auger electron

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6

X-ray tube

Notes:

• Mo anode stem (or shaft)

– Poor heat conductor to reduce heat transfer

– Not practical to pass a rotating shaft thru a high-vacuum seal

• Cathode cup

– Containing two different W wire filaments for two different sizes of focal spots

• Tube current << filament current

– Cathode to anode inside the vacuum tube

– 1 mA (fluoroscopy) to 1200 mA (cardiac catheterization)

• X-ray production efficiency ~0.5%

– W melting point = 3300C

– Major engineering problem of heat dissipation

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• Small anode angle (7° ≤ 𝜃 ≤ 15°)

• Line focal principle

– Large 𝑆𝑎𝑐𝑡𝑢𝑎𝑙 for heat dissipation

– Small 𝑆𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 for imaging

• 𝑆𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 = tan 𝜃 × 𝑆𝑎𝑐𝑡𝑢𝑎𝑙

• 0.12 ≤ tan 𝜃 ≤ 0.27

• Rotating focal track

– Allowing higher tube current (mA s)

• e.g., 3300 rpm 18 ms (per rotation)

– Increasing heat dissipation area from 𝑤(𝑟2 − 𝑟1)to 𝜋(𝑟2

2 − 𝑟12)

8

9

• Coverage is determined by the anode angle

• Non-uniform x-ray attenuation within angled anode results in non-uniform x-ray intensity distribution along the anode-to-cathode direction (Heel effect)

• 𝑀𝑜𝑏𝑗𝑒𝑐𝑡 =image size

object size=

𝐴+𝐵

𝐴

• Penumbra due to a finite-sized focal spot

– 𝑝 = 𝑠 ×𝐵

𝐴= 𝑠 × (𝑀𝑜𝑏𝑗𝑒𝑐𝑡 − 1)

10

Generator

Circuitries & control electronics for HV, filament current, anode rotation, and exposure timing

• Step-up (center-tap) transformer

• Bridge rectifier

11

As the ripple decreases, the spectrum becomes higher in average energy and the output of the x-ray tube (photon fluence or mR per mAs) increases

12

X-ray interactions

Photoelectric effect

Rayleigh scattering

Compton scattering

Pair production

Triplet production

13

Photoelectric effect

Interactions with tightly bound electrons

• 𝐸0 is transferred to a photoelectron

• Most probable when 𝐸0 − 𝐸𝐵𝐸• Followed by a cascade of electron transitions thru vacancies

– Characteristic (or fluorescence) x rays

• Reabsorption or escape

• K edges (or binding energies)

» 𝐸𝐾 = 0.5 keV for O (𝑍 = 8)

» 𝐸𝐾 = 4 keV for Ca (𝑍 = 20)

– Auger electrons

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𝑇 = 𝐸0 − 𝐸𝐵𝐸

Rayleigh scattering

Elastic (coherent) scattering: 𝐸′ = 𝐸0 More likely for low-E & high-Z

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Compton scattering

Interactions with loosely bound (unbound) electrons

Inelastic (incoherent) scattering: 𝐸′ ≠ 𝐸0

•𝐸′

𝐸0=

1

1+𝛼(1−cos 𝜃)

– 𝛼 =𝐸0

𝑚0𝑐2

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Pair production

Interactions with nuclear field (𝐸0 > 1.022 MeV)

• 𝐸0 = 2𝑚0𝑐2 + 𝑇+ + 𝑇−

Triplet production

• Interactions with electron field

• 𝐸0 > 2.044 MeV

Annihilation radiation

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Linear attenuation coefficient

Attenuation

• Removal of x-ray photons from the x-ray beam by either absorption or scattering as the x-ray beam passes thru matter

• Consider the attenuation of 𝑁 x-ray photons in a thin slab (d𝑥) with a probability of interaction 𝜇

Lambert-Beers law

• 𝑁 = 𝑁0𝑒−𝜇𝑡

Linear attenuation coefficient 𝜇

• Typically in units of cm−1

• The probability (per centimeter thickness of matter) that an x-ray photon will be attenuated (for a specific material & at a specific energy)

• 𝜇 = 𝜏 + 𝜎𝑟 + 𝜎 + 𝜋 + 𝛾

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Mass attenuation coefficient

𝜇 depends linearly on the 𝜌 of material (e.g., water: vapor, liquid, ice)

Mass attenuation coefficient 𝜇

𝜌

• To compensate the 𝜌-dependency

• In units of cm2 g−1

• 𝑁 = 𝑁0𝑒−

𝜇

𝜌𝜌𝑡

– 𝜌𝑡 = mass thickness in units of g cm−2

•𝜇

𝜌=

𝜏

𝜌+

𝜎𝑟

𝜌+

𝜎

𝜌+

𝜋

𝜌+

𝛾

𝜌

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Mass energy-transfer coefficient

Fraction of the mass attenuation coefficient which contributes to the production of kinetic energy in charged particles

• Do not consider radiative loss

– characteristic x rays, bremsstrahlung

• e.g., Initial PE: 𝑇

𝐸0=

𝐸0−𝐸𝐵𝐸

𝐸0

Photoelectric mass energy-transfer coefficient

•𝜏𝑡𝑟

𝜌=

𝜏

𝜌

𝐸0−𝑃𝐾𝑌𝐾 𝐸𝐾

𝐸0for 𝐸0 ≥ 𝐸𝐾

•𝜏𝑡𝑟

𝜌=

𝜏

𝜌

𝐸0−𝑃𝐿𝑌𝐿 𝐸𝐿

𝐸0for 𝐸𝐿 ≤ 𝐸0 < 𝐸𝐾

– 𝑃𝑗 = 𝑗-shell participation probability

– 𝑌𝑗 = 𝑗-shell fluorescence yield

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21

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Mass energy-absorption coefficient

𝜇𝑒𝑛

𝜌=

𝜇𝑡𝑟

𝜌

𝑇− 𝐸𝑟𝑎𝑑 𝑇

• 𝑇 = average energy for all charged particles from a single interaction

• 𝐸𝑟𝑎𝑑 = average energy of radiative loss (e.g., bremsstrahlung radiation)

• For low-Z, 𝜇𝑒𝑛

𝜌≈

𝜇𝑡𝑟

𝜌because 𝐸𝑟𝑎𝑑 ≈ 0

Attenuation coefficients for compounds

•𝜇

𝜌 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑= 𝑖=1

𝑁 𝑤𝑖𝜇

𝜌 𝑖

– 𝑤𝑖 = weight fraction of element 𝑖

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Density

• 𝜌 ∝ 𝑍3/4

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Energy

•𝜏

𝜌∝ 𝐸−3

•𝜎

𝜌∝ 𝐸0

25

Atomic number

•𝜏

𝜌∝ 𝑍4

•𝜎

𝜌∝ 𝑍0

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Attenuation geometry

Scattered photons not be included in the attenuation measurement

Measurement geometry (narrow-beam geometry)

• Pre-collimator to reduce the overall (production) number of scattered photons within a material

• Post-collimator to limit the number of scattered photons heading to x-ray meter

• Move the x-ray meter away from the scattering material

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Polyenergetic attenuation

𝜇 for a given material is energy-dependent

Monoenergetic beam follows the Lambert-Beers law

Attenuation of a polyenergetic spectrum

• 𝐴 𝑥 = 0𝐸𝑚𝑎𝑥 𝑎Φ(𝐸)𝜉−1(𝐸)𝑒

−𝜇𝜌 𝐴𝑙

𝜌𝑥d𝐸

0𝐸𝑚𝑎𝑥 𝑎Φ(𝐸)𝜉−1(𝐸) d𝐸

– 𝑎 in mm2

– Φ(𝐸) in photons mm−2 (per keV)

– 𝜉−1(𝐸) in mR per (photons mm−2) (per keV)

• Energy-dependent response of ion chamber

– 𝜉 𝐸 =5.43×105

𝐸𝜇𝑒𝑛(𝐸)

𝜌 𝑎𝑖𝑟

• Photon fluence per unit exposure

– 𝜉 𝐸 = 𝑎 + 𝑏 𝐸 ln𝐸 +𝑐

𝐸2

−1

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Half-value layer

A well-know parameter used to characterize beam quality (i.e., spectral distribution) in field measurements of attenuation

• Al thickness required to reduce the exposure of the x-ray beam by a factor of 2

• HVL =ln 2

𝜇

• HVL increases with increasing kVp

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Beam hardening

A process whereby the average energy of the x-ray beam increases as the beam passes thru increasing thicknesses of an absorber

Curvature in the polychromatic attenuation profile implies the beam hardening

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Filtration

Diagnostic x-ray spectra

• Tungsten (W) target at kVp ranging between 40 kVp & 150 kVp

Variations in the same kVp spectra from different x-ray systems due to:

• System calibration

• Generator waveform

• Anode angle

• Filtration both inside & outside the x-ray tube and its housing

– Inherent filtration

– Added filtration

When referring to beam quality, it is common to state both the kVp & HVL (in Al eq.)

• Homogeneity coefficient (HVL1/HVL2)

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X-ray spectral model

TASMIP (tungsten anode spectral model using interpolating polynomials)

• Empirical model

• Based on the 11 measured spectra (x-ray spectroscopy)

– Normalized to a tube current of 1.0 mAs

– No added filtration

• Polynomial interpolation from 0 keV to 140 keV in 1-keV step

– Φ 𝐸 = 𝑎0 𝐸 + 𝑎1 𝐸 kVp + 𝑎2 𝐸 kVp2 + 𝑎3 𝐸 kVp3

– Coefficient matrix 𝐴 = 141 𝐸′𝑠 × 4 𝑎𝑗′𝑠

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Effective vs. average energy

Estimates about the penetration capabilities or dose of an x-ray beam by assuming it is monoenergetic

• Average energy: 𝐸 = 0𝐸𝑚𝑎𝑥 𝐸Φ(𝐸) d𝐸

0𝐸𝑚𝑎𝑥 Φ(𝐸) d𝐸

– Require Φ(𝐸) as a known; unfortunately, only attenuation data are mostly available

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• Effective energy

– Measure 𝑁

𝑁0for a small thickness 𝑡 (e.g., 0.1 or 0.5

mm) of Al

– Calculate 𝜇

𝜌 𝐴𝑙from the (monoenergetic)

Lambert-Beers law

– Estimate 𝐸𝑒𝑓𝑓 from the tabulated 𝜇

𝜌 𝐴𝑙(𝐸) data

(e.g., NIST XCOM)

X-ray fluence

In general, the fluence is not directly measurable

Estimation of fluence using the spectral model:

• Φ = 0𝐸𝑚𝑎𝑥 Φ(𝐸) d𝐸

0𝐸𝑚𝑎𝑥 Φ(𝐸)𝜉−1(𝐸) d𝐸

– Φ in units of photons mm−2 per mR

– Φ(𝐸) from the spectral model (e.g., TASMIP)

– 𝑋 = the measured exposure in mR

• Φ = 𝑋 Φ

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Mammography x-ray spectra

To achieve reasonable subject contrast in mammography, the x-ray energies used are much lower than in general diagnostic radiology (why?)

Mo (𝐸𝐾 = 20 keV) & Rh (23.3 keV) are advantageous in mammography because the energies of their characteristic x-ray lines are near ideal for imaging the breast

K-edge filtration

• Rh/Rh spectrum is better for imaging thick, dense breasts, and reduce the glandular dose relative to a Mo/Mo spectrum

• Mo-Rh spectrum to compensate the low heat conductivity of Rh

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𝐾𝛼 = 17.4 keV

𝐾𝛽 = 19.6 keV

𝐾𝛼 = 20.2 keV

𝐾𝛽 = 22.7 keV

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Mammography spectral models

TASMIP extended to lower energies ranging from 18 kV to 42 kV

MASMIP (Mo)

RASMIP (Rh)

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Dosimetry

Exposure

𝑋 =d𝑄

d𝑚

• The absolute value of the total charge d𝑄 of the ions (of one sign) produced in air when all the electrons (negatrons & positrons) liberated by photons in air of mass d𝑚 are completely stopped in air

• 1 R = 2.58 × 10−4 C/kg

Absorbed dose

𝐷 =d𝜖

d𝑚

• The expectation value of the energy imparted to matter per unit mass at a point

• 1 Gy = 1 J/kg = 102 rad = 104 erg/g

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Considering the W-value of air = 33.97 eV = 33.97 J/C:

• 1 R ×𝑊 = 2.58 × 10−4C

kg× 33.97

J

kg= 0.00876

J

kg= 87.6

ergs

g

• 𝐷𝑎𝑖𝑟 = 0.876𝑋 (R-rad conversion)

– 1-R exposure 8.76 mGy

For an arbitrary medium:

• 𝐷𝑚𝑒𝑑 = 𝐷𝑎𝑖𝑟

𝜇𝑒𝑛𝜌 𝑚𝑒𝑑𝜇𝑒𝑛𝜌 𝑎𝑖𝑟

= 0.876

𝜇𝑒𝑛𝜌 𝑚𝑒𝑑𝜇𝑒𝑛𝜌 𝑎𝑖𝑟

𝑋 = 𝑓𝑋

– F-factor, 𝑓 = 0.876

𝜇𝑒𝑛𝜌 𝑚𝑒𝑑𝜇𝑒𝑛𝜌 𝑎𝑖𝑟

41

Kerma

Kinetic energy released in media

𝐾 ≡d𝜖𝑡𝑟d𝑚

• The expectation value of the energy transferred to charged particles per unit mass at a point of interest, including radiative-loss energy but excluding energy passed from one charged particle to another

• 𝐾 = 𝑘 0𝐸𝑚𝑎𝑥Φ 𝐸

𝜇𝑡𝑟(𝐸)

𝜌 𝑚𝑒𝑑𝐸 d𝐸

• For air, 𝐾𝑎𝑖𝑟 = 𝐷𝑎𝑖𝑟 because 𝜇𝑡𝑟

𝜌≈

𝜇

𝜌(negligible fluorescence)

42

Backscatter

43

𝑋 =𝐴

𝐴 + 𝐵

2

𝑋′

(𝐵 > 40 cm)

Equivalent & effective dose

Dose equivalent 𝐻𝐻 ≡ 𝐷𝑄

– Defined at a point (i.e., a point quantity)

– Sievert, 1 Sv = 1 J/kg

– 1 rem = 10-2 J/kg (equivalently to "rad")

– Not strictly a physical quantity

Equivalent dose 𝐻𝑇,𝑅

𝐻𝑇,𝑅 = 𝐷𝑇,𝑅𝑤𝑅

– Equivalent dose in an organ or in tissue 𝑇 due to radiation 𝑅

– Not a point quantity but an average over a tissue or organ

– 𝐻𝑇 = 𝑅𝐻𝑇,𝑅 = 𝑅𝐷𝑇,𝑅𝑤𝑅

– Not a measurable quantity

Effective dose

𝐸 =

𝑇

𝐻𝑇𝑤𝑇

– Designed to normalize the actual dose delivered to a small region to that of a whole-body exposure

– Not a measurable quantity44

Energy deposition in tissue

Mostly assessed by using Monte Carlo techniques because the Lambert-Beers law cannot address the secondary photons

45

Direct vs. indirect detection

Direct detectors

• Gas detectors

• Film (without intensifying screen)

• a-Se (photoconductor)-based detector

Indirect detectors

• Scintillator-based detectors

• Computed radiography using photostimulable phosphors (e.g., BrFBr:Eu)

46

Activators

Host-luminescent phosphors do not require activators (e.g., CdWO4)

Activators determines:

• Color of the luminescent emission

– Gd2O2S:Tb peak at 545 nm

– Gd2O2S:Eu peak at 626 nm

– Gd2O2S:Pr peak at 506 nm

• Scintillation efficiency (# light photons per absorbed energy)

• Decay time

• Note that the activators has negligible impact on the absorption efficiency of phosphors

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Absorption efficiency

The design goal of all x-ray detectors for medical imaging is to maximize the absorption efficiency of the detector, given the constraints of other performance parameters (e.g., spatial resolution)

Quantum detection efficiency

• Fraction of incident x-ray photons which are attenuated by the detector

• QDE = 0𝐸𝑚𝑎𝑥 Φ(𝐸)(1−𝑒−𝜇(𝐸)𝑡) d𝐸

0𝐸𝑚𝑎𝑥 Φ(𝐸) d𝐸

Energy absorption efficiency

• Fraction of incident x-ray photon energies which are attenuated by the detector

• QDE = 0𝐸𝑚𝑎𝑥 Φ 𝐸 𝐸

𝜇𝑒𝑛(𝐸)

𝜇(𝐸)(1−𝑒−𝜇(𝐸)𝑡) d𝐸

0𝐸𝑚𝑎𝑥 Φ 𝐸 𝐸 d𝐸

48

Absorption efficiency: energy dependency

49

Absorption efficiency: thickness dependency

50

Conversion efficiency

Intrinsic (conversion) efficiency

• Conversion of x-ray energy into visible light photons

– CdWO4 = ~5%

– Gd2O2S = ~15%

51

Conversion efficiency: thickness & reflector dependencies

Conversion efficiency will be lower as thickness increases (due to self-absorption)

52

Geometry

Dual screen/dual film

• Most common in general-purpose screen-film radiography

• Good compromise in terms of absorption efficiency (thick screen) & spatial resolution

Single screen/single film

• Chief mammography

• Radiography for extremities

Digital detectors

• Photodetector is not transparent to x rays

Optics-coupled systems

• Lens

• Fiber-optic faceplate

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X-ray fluorescence in detectors

54

Backscatterd 𝐸𝐿

Note, here, that scatter means the combination of Compton, Rayleigh, and characteristic x-ray re-emission