Appendix


 

Basic science

Equations

1 atomic mass unit (amu) = 1/12 the mass of a carbon-12 atom

Maximum number of electrons in a shell = 2n2

Kinetic energy = ½mv2 (m = mass; v = velocity)

Frequency = 1 / T (T = time between successive peaks in seconds)

Velocity (c, ms-1) = frequency (f) x wavelength (λ)

Intensity (E) = hf (h = Planck's constant; f = frequency)

Intensity ∝ 1 / d2 (d = distance)

Miscellaneous

  Relative mass Charge Symbol
Neutron 1 0 n
Proton 1 +1 p
Electron 0 (1/2000) -1 e-

1 Becquerel (Bq) = 1 transformation per second


X-ray imaging

Equations

Heat (J) = kVe x mAs = w x kVp x mAs (kVe = effective kV, W = waveform, kVp = peak kV, mAs = current per second)

Temperature = heat applied / heat capacity

Factor of reduction = 2HVL (HVL = half value layer)

Linear attenuation coefficient (LAC, m-1) = 0.693 / HVL

Mass attenuation coefficient (MAC, m2k-1) = LAC / density

Compton scatter ∝ density / energy

Photoelectric LAC ∝ ρZ3 / E3 (ρ = density, Z = atomic number, E = energy)

Screen film radiography

Optical density (D) = log10 (I0 / In)2 ( I0 = intensity of incident beam, In = intensity of transmitted light beam)

Film speed = 1000 / (air kerma required for density of base-plus-fog + 1)

Intensification factor of screen = exposure required to produce given optical density without screen ÷ exposure needed with screen

Anti-scatter grid

Grid ratio = h / d (h = height; d = distance between detectors)

Grid factor = exposure necessary with a grid ÷ exposure without a grid

Proportion of direct radiation stopped = t / (t + d)

Computed radiography

Image plate (IP) = barium fluorohalide activated with divalent europium ions. 0.3 mm

Red laser beam for read out

Light released in blue part of spectrum

Speed = 2000 / X (X = dose incident on IP)

Detective quantum efficiency (DQE) = SNR2out / SNR2in (SNR = signal to noise ratio)

  • 0.25 for standard IP
  • 0.12 for high resolution IP

Digital radiography

Indirect DR

Scintillator layer = 500 μm layer of caesium iodide with thallium (CsI:Tl)

X-ray photon → ~3000 light photons in green spectrum

Matrix = amorphous silicon layer doped with hydrogen (a-Si:H)

Fill factor = sensitive area / overall area

Direct DR

Photoconductor = amorphous selenium (a-Se)

Mammography

Maximum compression = 200 N (normal = 100-150 N)

Target/filter:

  • General use: MoMo
  • Dense breasts: MoRh or RhRh

Broad focal spot = 0.3 mm

Fine focal spot = 0.1 - 0.15 mm

Focus-to-film distance = 65-66 cm

Fluoroscopy

Equipment

II window made of aluminium of titanium foil

Input phoshor:

  • Sodium activated caesium iodide (CsI:Na)
  • 400-500 μm thick,
  • Produces light photons in blue spectrum

Photocathode:

  • Antimony caesium (SbCs3)

Output screen:

  • Silver-activated zinc cadmium sulphide (ZnCdS:Ag)
  • 25-35 in diameter, few micrometres thick
Equations:

Gbrightness = Gminification x Gflux (Gminification = minification gain; Gflux = flux gain)

Gminification = (Dinput / Doutput)2 (Dinput = diameter of input screen; Doutput = diameter of output screen)

Gx = L / X' (Gx = image intensifier conversion factor; L = luminance of the output; X' = entrance dose rate)

Gminification ~ 100

Gflux ~ 100

Gx ~ 10-30

Elements

Tungsten

Characteristic radiation:

  • Kα = 59.3 keV
  • Kβ = 67.6 keV

Mass number (A) = 184

Atomic number (Z) = 74

Molybdenum

Characteristic radiation:

  • Kα = 17.5 keV
  • Kβ = 19.6 keV

K-edge = 20 keV

Rhodium

Characteristic radiation:

  • Kα = 20.2 keV
  • Kβ = 22.7 keV

K-edge = 23.3 keV

Image quality

Subject contrast (c) ∝ (μ1 - μ2) x t

Geometric unsharpness (Ug) = f x b / a (f = x-ray focal size; a = distance from x-ray source to front surface of object; b = distance from object to detector)

Magnification (M) = image size / object size = d2 / d1 (d2 = focal spot to detector; d1 = focal spot to object)

Sampling frequency = 2 x Nyquist frequency 


CT imaging

Detector array = 8 - 64 rows; 700 - 900 detectors per row

Single slice pitch = detector pitch = couch travel per rotation / detector width

Multislice pitch = beam pitch = couch travel per rotation / total width of simultaneously acquired slices

Hounsfield unit (HU) = CT number = 100 x (ut - uw) / uw (ut = attenuation coefficient of tissue; uw = attenuation coefficient of water)

Focal spot: fine = 0.7 mm, broad = 1.0 mm)

Pixel size (d) = FOV / n (FOV = field of view; n = image matrix size)

Highest spatial frequency (fmax) = 1 / 2d

CT number values
Tissue CT number
Bone
Liver
White matter
Grey matter
Intravascular blood
Fresh clotted blood
Muscle
Kidney
CSF
Water
Fat
Air
+1000
40 to 60
20 to 30
37 to 45
40 to 45
70 to 80
10 to 40
30
15
0
-50 to -100
-1000

Dose

Dose = mAs / pitch

Measurement Definition Equation Unit
CT dose index
(CTDI)
Dose to the detector from single gantry rotation   mGy
Weighted CTDI
(CTDIw)
Adjusted for spatial variation of dose ⅓ CTDIcentre + ⅔ CTDIperiphery mGy
Volume CTDI
(CTDIvol)
Accounts for pitch CTDIw / pitch mGy
Dose length product
(DLP)
Total dose along distance scanned CTDIvol x distance scanned mGy cm
Effective dose
(E)
Physical effect of total radiation dose 1) Σ (HT x WT)

2) EDLP x DLP

HT = individual organ dose
WT = tissue weighting factor
EDLP = agreed value of EDLP for whole region
 

Ultrasound imaging

Equations

Audible range of soundwaves = 20 to 20,000 Hz

Medical ultrasound = 2 to 18 MHz

Velocity (c) = √ 1 / ƙρ (ƙ = compressibility;ρ = density)

c = f λ (f = frequency; λ = wavelength)

Speed of sound through tissue = 1540 ms

Intensity (dB ratio) = 10 log10 (I1 / I2) (I1 = intensity 1; I2 = intensity 2)

Acoustic impedence (Z, kg m-2 s-1) = density x speed of sound in that material

Reflection coefficient (R) = Z2 - Z1)2 / Z2 + Z1)2

Beam weight = focal length x λ / D (λ = wavelength; D = diameter of PZT crystals)

Doppler

Resistive index (RI) = (peak systolic frequency - end diastolic frequency) ÷ peak systolic frequency

Pulsatility index (PI) = (peak systolic frequency - minimum frequency) ÷ time averaged maximum frequency

In low resistance artery: normal RI = 0.6 - 0.7; abnormal RI = 0.8 - 1.0

Nyquist limit = PRF / 2

Equipment

Piezoelectric material = ½ wavelength thick; 256 crystals

Matching layer = ¼ wavelength thick

Near field distance = D2 / 4λ (D = diameter of transducer; λ = wavelength)

Pulse repetition frequency (PRF) = frame rate x lines per frame

Distance of wave = time x velocity x 0.5

Depth of view = 0.5 x sound velocity / PRF

Safety

Thermal index (TI) = power emitted / that required to increase temperature by 1°c. Keep < 0.5

Mechanical index (MI) = peak negative pression / √ultrasound frequency. Keep < 0.7. In fetal scanning <0.3

Time averaged intensity < 100 mWcm-2

Total sound energy < 50 Jcm-2


MR imaging

Equations

Larmor equation (F) = precessional frequency = K x B0 (K = gyromagnetic ration; B0 = strength of static magnetic field)

Larmor frequency of hydrogen at 1 Tesla = 42 MHz

Larmor frequency of hydrogen at 1.5 Tesla = 63 MHz

T1 = time for Mz (longitudinal magnetisation) to recover to 63%

T2 = time for Mxy (transverse magnetisation) to decay to 37%

Relaxation times at 1 Tesla
  T1 (ms) T2 (ms)
Fat
Kidney
White matter
Grey matter
CSF
Water
Bone, teeth
250
550
650
800
2000
3000
Very long
80
60
90
100
150
3000
Very long

Sequence

Spin echo
  • 90º RF → 180º RF rephasing pulse at TE/2 → Echo signal at time TE → repeat at TR
  • Scan time = TR x no. GPE x NEX (GPE = phase encoding steps; NEX = number of signal averages or slices)
  • Turbo spin echo = TR x no. GPE x NEX / ETL (ETL = echo train length)
  • T1 weighted: TR determines T1 signal. Short TR
  • T2 weighted: TE determines T2 signal. Long TE
  • Proton density: minimise T1 with long TR and minimise T2 with short TE
Inversion recovery
  • STIR: short TI of 130 ms (TI = time to application of 180º inversion pulse)
  • FLAIR: long TI of 2500 ms
Gradient recalled echo
  • RF pulse of certain flip angle → gradient applied to rephase spins → echo signal at time TE → repeat at TR
  • T1 weighted: large flip angle, short TE and short TR
  • T2* weighted: small flip angle, long TE and short TR
  • T2 weighted: can't achieve
  • Proton density = small flip angle, short TE and short TR

Localisation

  1. Slice select along Z-axis with gradient
  2. Segment along X-axis selected by frequency encoding
  3. Segment along Y-axis selected by phase encoding
  4. For 3D, segment along Z-axis selected by phase encoding
  5. Wave decoded with Fourier transformation

K-space: periphery for fine detail, centre for contrast information

Angiography

Time of flight (TOF): non-contrast bright blood technique. Uses flow-related enhancement artefact

Phase contrast: non-contrast bright blood technique. Uses spin phase artefact.

Contrast enhanced: IV contrast bright blood technique

Contrast agents:

  • T1 paramagnetic = shorten T1 = high T1 signal e.g. gadolinium, hepatobiliary agents that contain manganese
  • T2 superparamagnetic = speeds up T2 decay = low T2 signal e.g. iron oxide based SPIOs and USPIOs

Artefacts

Local field inhomogeneity artefacts occur in frequency-encoding direction

External RF signal artefacts occur in phase-encoding direction


Nuclear imaging

Equipment

Cyclotron: Fluorine-18, Gallium-67, Thallium-201, Krypton-81m

Nuclear reactor: Molybdenum-99 (for Tc99m production in generator), Technetium-99m, Iodine-131, Xenon-133

Radionuclide generator: Technetium-99m (most common source), Krypton-81

Collimator:

  • Low energy = 150 keV = 0.3 mm = 99mTc
  • Medium energy = 300 keV = 1 mm = Indium-111
  • High energy = 400 keV = 2 mm = 131I

Scintillation crystal: sodium iodide with thallium (NaI(Tl)); 6-13 mm thick

PET imaging

Positron decay → annihilation with electron → two 511 keV photons

Scintillation crystal: bismuth germanate (BSO), lutetium oxyorthosilicate (LSO and gadolinium oxyorthosillicate (GSO)

Image quality

Subject contrast (Cs) = (AL - AT) / AT (AL = activity per unit of lesion; AT = activity per unit mass of healthy tissue)

Image contrast (CI) = (SL - ST) / ST (SL = counts per unit area of lesion; ST = counts per unit area of healthy tissue)

Noise contrast (CN) = 1 / √(AS) (A = area; S = count density)

Collimator spatial resolution (RC) ≈ d (1 + b/h) (d = hole diameter; b = distance from radiation source to collimator; h = hole length)

System spatial resolution (RS) = √(RI2 + RC2) (RI = intrinsic spatial resolution; RC = collimator spatial resolution)

Energy resolution = FWHM (keV) / photopeak energy (keV) x 100 (FWHM = full width half maximum)

Scatter rejection = 20% acceptance window


Radiation dosimetry, protection and legislation

Dose

Absorbed dose = energy deposited per unit mass of tissue

Equivalent dose = entrance surface dose x radiation weighting factor

Effective dose = ∑(equivalent dose x tissue weighting factor)

Radiation Radiation weighting factor
X-ray and gamma ray
Beta particles and positrons
Neutrons < 10 keV
Neutrons 100 keV to 2 MeV
Alpha particles
1
1
5
20
20

External radiation: gamma and x-rays > beta > alpha

Internal radiation: alpha > beta > gamma and x-rays

For other dose effects see: Dose effects

Organ Tissue weighting factor
Skin, bone, brain, salivary glands
Bladder, oesophagus, liver, thyroid
Gonads
Red bone marrow, colon, lung, stomach, breast, remainder of tissues
0.01
0.04
0.08
0.12

Protection

Protection Lead equivalence
Lead aprons

0.25 mm for 100 kV

0.35 mm for 150 kV

Thyroid shields 0.5 mm
Lead glasses 0.25 x 1.0 mm
Lead gloves 0.25 mm
Modern gloves have 0.5 or 1.0 mm

Legislation

IRMER 2000: exposure to patients. ALARP

IRR99: exposure to employees and members of the public

Effective dose limits per year:
Radiation workers > 18 yo 20 mSv
Members of public 1 mSv
Radiation workers < 18 yo 6 mSv i.e. 3/10 of adult dose
Dose limit to abdomen of person of reproductive capacity 13 mSv in any consecutive 3 months
Comforters and carers 5 mSv
Any other person / member of public (fetus counts as members of the public) 1 mSv
Pregnancy employees dose to fetus 1 mSv for remainder of pregnancy
Equivalent dose limits per year:
Lens of the eye 150 mSv
Hands, forearms, feet and ankle 500 mSv
Skin 500 mSv
Classified workers

Anyone who is likely to receive:

  • Effective dose of 6 mSv in a year (3/10 of dose limit)
  • Equivalent dose of > 3/10 of any dose limit
Reporting overexposure:
Investigation Intended dose multiplying factor
Fluoroscopic procedure with contrast agents
CT scans
Interventional procedures
1.5 x
AXR
CXR
Mammography
10 x
Skull, dental and extremity x-rays 20 x

Nuclear medicine

MARS78: governs administration of radioactive substance

RSA93: governs storage and safe disposal of radioactive materials

Radioactive Material (Road Transport) (Great Britain) Regulation 2001: governs transport of radioactive substances by road

MRI safety

MHRA guideline for whole body exposure of patients

  • Normal and pregnant < 2.5 Tesla
  • Controlled 2.5 to 4.0 Tesla
  • Research > 4.0 Tesla

MHRA guideline for exposure of staff

  • < 2 T for whole body
  • < 5 T for limbs
  • < 0.2 T over 24 hours

Controlled area = 5 Gauss, 0.5 mT boundary

1 SAR = 1 W/kg = whole body temperature rise of 0.5ºc

Patient temp not to exceed 1ºc

Hearing protection needed at 90 dB


Miscellaneous

Resolution

Imaging modality Resolution
Film screen radiograph 6 lp/mm
Digital radiograph 3 lp/mm
Film screen mammography 15 lp/mm
Digital mammography 5 to 10 lp/mm
Fluoroscopy 1 lp/mm
Direct subtraction angiography 2 lp/mm
Fluoroscopy - flat panel detector 3 lp/mm
CT: transaxial 2 lp/mm
CT: Z-sensitivity 2 to 0.4 lp/mm

 

Effective radiation dose

Procedure Dose Procedure Dose (mSv)
Abdominal
CT Abdo/pelvis 10 XR lower GIT 3
CT colonography 6 XR upper GIT 6
Barium enema 7 XR pelvis 0.5
CNS
CT head 2 XR cervical spine 0.2
CT spine 6 XR thoracic spine 1.0
CT neck 3 XR lumbar spine 1.5
    XR skull < 0.1
Chest
CT chest 7 XR chest 0.1
Cardiac CT 3 Mammography 0.4
Cardiac CTA 12 - 20    
Extremities
    XR hand / foot 0.005
Nuclear imaging
Brain PET (18F FDG) 14 DEXA 0.001
Brain perfusion (99mTc HMPAO) 9 to 10 Renal MAG3 2
Brain SPECT (99mTc sestamibi) 10 Renal DTPA 2
Bone scan 6 DaTscan brain 6 to 10
Heart stress (99mTc sestamibi) 9.4 V/Q 2.5
Thyroid scan 5 Gastric emptying 1
Whole PET/CT 24    

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