An Incomplete Medical Physics Review

X-ray principles

X-ray tube has most component in a vacuum (don't want electrons interacting with air)

Electrons generated in cathode with low voltage boiling electrons from filament. High voltage electric potential accelerate electrons to anode. Negatively charged focusing cup used to focus electrons. Filament size determines spot size (usually have two sizes). Anode spins to spread out heat. Cathode to anode in vacuum and cooled with oil.

Energy deposited in anode: $E(J) = mAs \times kVp$

Anode angle alters true focal spot to an effective focal spot: $l_{eff} = l_t\sin(\theta)$
Angle for imaging is usually 6-17 degrees, and for therapy is ~30 degrees

Focal spot blur: occurs because spot is not a point.
$\text{blur} = F \times OID / SOD $ where F is size of spot, OID is object image distance, SOD is source object distance

Heel effect - target self-absorption. Intensity reduced from cathode to anode.

AC voltage need rectification to produce current consistently (otherwise anode positive only half the time).
Voltage ripple: $(V_{min} - V_{max}) / V_{max} * 100$
Three phase generators: size pulses per cycle, 13-25% ripple
Constant potential generators: uses rectifiers, capacitors, and triode valve, <2% ripple
High-frequency generator rectified and smoothed (via capacitors) and fed to a chopper and inverter circuits, <2% ripple

Fluorescent/characteristic and bremsstrahlung x-rays produced. See atomic/nuclear physics page for details

Fluorescent/characteristic x-rays: photoelectric effect, electron capture/internal conversion, heavy ions
Energy of fluorescence depend on binding energy: $E = h\nu - BE$
Energy proportional to Z, and Z linear with sqrt of frequency for K and L series.
Fluorescence yield is probability an x-ray will escape an atom. About 0 for Z < 10, ~ 0.95 for Z = 74 (W)
Isotropically emitted
Characteristic x-rays exploited mostly in mammography

Bremsstrahlung: photons emitted from accelerating charged particles (deflected by nucleus of atom)
Any amount may be emitted up to the energy of the charged particle → continuous energy spectrum
Probability is proportional to Z² and to 1/m_e (i.e. very limited for particles heavier than electrons).
Efficiency is energy emitted by photons / energy of deposited by electrons: $\epsilon = 9 \times 10^{-10} ZV$ where $Z$ is atomic number and $V$ is tube voltage. For typical imaging devices, $\epsilon < 1%$
As electron energy increases, photons are emitted in more forward direction

Quality: characterizes beam spectrum, i.e. energy, penetrating power, hardness. Often expressed in HVL

Typically use aluminum (0.5 - 1 mm) to filter out low energy bremsstrahlung x-rays (that won't pass through object to be included in image).
Hardens beam, but need to balance hardness and intensity (more filtering lowers intensity).
Use higher Z material (e.g., copper) for pediatric or thin patients.

X-ray output exposure is proportional to current, time, and peak voltage squared: mAs and kVp².

Radiography

Magnification: M = source to image / source to object

Diagnostic x-ray exposure limit at 1 m: 100 mR/hr

Optical density: darkness of film after radiation exposure. Proportional to the log of the ratio of intensity of transmitted light relative to intensity of unexposed film.
$\text{OD} = -\log(I/I_0) = \log(I_0/I)$

Fluoroscopy

Barium swallow - to check GI tract (dysphagia, hiatal hernia, GERD, ulcers, tumors, etc.)

Mammography

Uses Rh and Mo targets and filters to take advantage of characteristic x-rays in 20-40 keV range (good image contrast and penetrating enough for breast).

Mo - 17.5, 19.6 keV
Rh - 20.2 keV, 22.7 keV

Mo target, Mo filter - good for fatty breast, 24-26 kVp
Mo target, Rh filter - good for glandular breast, 27-31 kVp
Rh target, Rh filter - good for thicker breasts (> 7 cm)
Rh target, Mo filter - no, stop, this will do no good. Mo will filter out the Rh characteristic x-rays.

More coming soon

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Hardware & scanning

Uses same sort of CT tube as for planar x-ray.  Tube is opposite an arc scintillator detector (rare earth oxides) to convert x-rays to light photons, and photodiode to  convert light to electrons. Electrons are integrated and digitized by ADCs.
Multi-row detectors typically (~96-320 rows in Z x 600-1000 channels in X). Collimation can change number of rows used. 

Bowtie filter: compensates for different x-ray path length (reduce dose at periphery).  Al + graphite.

Scanning parameters include current (mA), time (s), and voltage (kVp).
Current and time are typically given together as mAs.  

Axial scanning: step and shoot. Table moves to position, stops, data acquired (tube/detector rotate around)
Helical scanning: continuous data acquisition as table moves.Pitch, table increment per rotation / beam width, can be adjusted. 
Pitch < 1, overlap, pitch = 1, contiguous, pitch > 1, gaps. 

Projection data reconstructed using FBP or iterative reconstruction for 3D volumes (similar techniques are also used for PET and SPECT).

Hounsfield unit: basic scale for CT images ("CT number"), based on attenuation coefficients, set such that water is 0 and air is -1000
$\text{HU} = 1000 \cdot \frac{\mu - \mu_w}{\mu_w - \mu_{air}}$

Image quality metrics

Spatial resolution: ability to resolve small objects. Tested with in-plane bar patterns, cross-plane by slice sensitivity profile or slice thickness.  Depends on focal spot size, detector size, post-patient collimation, reconstruction kernel, reconstruction FOV (pixel size), and slice thickness.
Changing kernel will affect the MTF, and can be selected to highlight lung, bone, soft, details, etc.
Noise will increase with better resolution.

Low contrast detectability: depends on mAs, slice thickness, reconstruction kernel/algo.

Image noise: depends on mAs (linearly), kVp (not linearly), and reconstruction kernel/algo. Noise power spectrum (vs spatial frequency) provides magnitude and spatial correlation (texture) of the noise.

Artifacts
Cupping: reduction in HU at center due to beam hardening. Apply BH correction
Bone and metal can also cause streaks and shadows (reduced HU) from BH and photon starvation. Apply bone BH and/or metal artifact reduction correction, if available.
Aliasing: caused by sharp edges (metal, bone). Increase sampling rate or use smoothing kernel to reduce
Windmill: under-sampling in z-direction, increase sampling rate or use z-flying focal spot
Ring/bands: defects in detector cells, recalibrate or replace bad detector
Smudge: in center, from tube focal spot thermal drift or detector gain drift. Warm up or recalibrate
Tube Arching: debris inside tube insert, replace or season, can software correct also.
Motion: streaking and blurring, inadequate immobilization. Can use high pitch mode to reduce time.
Truncation: patient partly outsize FOV. Center as possible. Use largest FOV. 

Dosimetry

CT dose depends on mAs, kVp, pitch.
mAs: dose increases with mAs linearly. Usually use 50 - 300 mAs. Can also be modulated depending on object thickness.
kVp: needs to be high enough to get through patient.  Typically 100, 120, 140 kVp. Dose ∝ kVp^2.6 depending on tube and filtering.  Higher kVp allows lower absorbed dose for same detector exposure but also lower contrast (less photoelectric effect). 
pitch: larger pitch, less dose (linearly)

CT Dose Index (CTDI): standard method of comparing doses for various scans. Given by CTDI_w = (1/3) * radiation_center + (2/3) * radiation_periphery for one slice in a phantom of length 100 mm (measured with 100 mm pencil ionization probe).
Use 16 cm head phantom and 32 cm body phantom.
CTDI_vol = CTDI_w / pitch for helical scans

Note: CTDI ≠ dose to patient! Corrections must be made based on patient size and what size phantom (32 cm or 16 cm) was used for the initial measurement.
See AAPM Report 204: Size-Specific Dose Estimates (SSDE) in Pediatric and Adult Body CT Examinations

Same CTDI will have higher SSDE for smaller patients.

Dose product length (mGy · cm): DPL = CTDI_vol × scan (exposure) length 

Effective dose: include tissue weighting factors: $D_E = \sum{W_T D_T}$ (mSv),
$D_E = \text{DLP} · k$ where $k$ is a factor depending on age and region scanned.
$k$ values in mSv / mGy / cm (AAPM Rpt No 96)

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Transducer generates sound wave and also receives it. Resonant frequency of transducer is determined by thickness of piezoelectric crystal
$d = \lambda/2$

Acoustics

Diagnostic frequency in the 0.8-15 MHz range.

Speed of sound: $C = \lambda f$
$C = (K/\rho)^{1/2}$ where $\rho$ = density and $K$ = adiabatic bulk elastic modulus (resistance to deformation)
Impedance: $Z = \rho C = (K\rho)^{1/2}$ (kg/m²s = rayl)

Speed of sound in various tissue types

Intensity: power/unit area = A²/area (W/m²)
Relative intensity: $\text{RI} = 10 \log(I_2/I_1) = 20 \log(A_2/A_1)$ (dB)
Half value layer (HVL) is $I_2/I_1 = 0.5 \rightarrow 10 \log(0.5) = -3$ dB

Reflection fraction: $(Z_1 - Z_2)^2/(Z_1 + Z_2)^2$
Transmission = 1 - reflection
Obeys Snell's Law: $\sin(\theta_1)/\sin(\theta_2) = v_1/v_2$

Attenuation: $I = I_0 \exp(-\mu x)$
In dB: $\mu = \alpha f$ where $\alpha$ is material coefficient in dB/cm/MHz and $f$ is frequency in MHz.
$\alpha \approx 1$ db/cm/MHz for soft tissue.
Higher frequency → more attenuation.
Half-value-layer (HVL) = -3 dB → 10*log(0.5)
Tenth-value-layer (TVL) = -10 dB → 10*log(0.1)

Spatial Pulse Length (SPL) = $n\lambda$ = number of cycles * wavelength
Axial resolution = $n \lambda/2$ = SPL/2. $n$ is typically 3.
Axial resolution is better than lateral. High frequency → shorter spatial pulse width → better axial resolution but increased attenuation → less depth

Fresnel Zone (near field): Relatively constant pulse diameter (determined by transducer diameter), useful for imaging
Length of near field: $\text{L} = \frac{D^2}{4λ} = \frac{r^2}{\lambda}$

Fraunhofer zone (far field): Field diverges as $\theta = 70 λ / D \,\text{degrees}= 1.22 \lambda / d \,\text{radians}$
Higher frequency means less divergence, less blur.

Transducers

Piezoelectric crystal (voltage distorts crystal and creates pressure waves).
Thickness is half of wavelength: $d = \lambda/2$
Matching layer in front at $d = \lambda/4$ to prevent large reflections from acoustic impedance mismatches: $Z_t < Z_m < Z_p$
Damping block behind crystal to damp vibrations so it can receive signal

Listen to 20 cm depth is about t = d/v = 40 cm /1540 m/s = 260 μs
Can image, e.g., 225 lines at 17 frames/sec.
Pulse repetition frequency (PRF) = 1/travel time = 1/(d/v) = v/(2*depth). Typically 2-4 kHz.
Increase frame rate by decreasing depth, or reducing FOV

Doppler

Longer pulses, high quality factor: $Q = f/BW$
$\Delta f \propto \text{reflector speed}$
$f_{shift} = 2 \cdot (v/c) \cdot f_0 \cdot \cos(\theta)$ where $v$ is velocity of reflector, $c$ is speed of sound, $f_0$ is transducer frequency, and $\theta$ is angle of velocity wrt sound wave.

US Artifacts

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Units: Tesla (SI), 1 T = 10,000 gauss (historical)

Magnetic fields can be induced by moving charges (Maxwell's equations). Direction of induced magnetic fields are determined by the right hand rule: fingers along direction of moving positive charge, thumb then points in direction of the magnetic field.

Nuclear Magnetic characteristics of elements

Elements must have spin and charge --> Odd # of protons or neutrons
If Nn+Np is odd, nucleus will have 1/2 integer spin
If Nn and Np are odd, nucleus will have integer spin
If Nn and Np are even, nucleus will have zero spin
Common MR active nuclei:
¹H, ¹³C, ¹⁹F, ²³Na, ³¹P

Most signal comes from ¹H: largest magnetic moment along with highest abundance (of atoms with MR signal).

Magnetic moment: $\mu_z = \pm \frac{\gamma h}{4 \pi} $

Magnetism in materials:
Susceptibility: how magnetized a material becomes in a magnetic field.
Diamagnetic: slight negative susceptibility, oppose the applied field because of paired electrons in orbitals. Calcium, water, organics (carbon and hydrogens)
Paramagnetic: slight positive susceptibility, enhance applied field because of unpaired electrons. O₂, some blood parts, gadolinium contrast agents.
Ferromagnetic: large positive susceptibility, substantially augment applied field, have self-magnetism. Iron, cobalt, nickel. Will distort signals.

MR fields

The performance of magnets in medical imaging depend on field strength, temporal stability, and field homogeneity. Electromagnetic core wires must be superconductive to achieve the >= 1 T used in clinical imaging. Superconductivity, no resistance to electric current, is achieved through the use of special metals held at extremely low temperatures (a few Kelvin). Cooling is typically done with continually replenished liquid helium (~4K).

In addition to the main magnetic field of an MR machine (1.5 - 3 T clinically, 4 -7 T research), B₀, there are also shim coils to aid in generating a homogeneous field, gradient coils to produce linear variations in field strength (for localization), and RF coils for sending an receiving signals that generate contrast in the tissues to form images.
Fringe field limit: 0.5 mT = 5 G.

The gradient field is generated from two or more coils each creating magnetic field variations that combined induce a slight linear change in the main B₀ on the order of 0.005 T/m. [more on localization later]

Excess spins based on Boltzmann statistics - difference in energy between states
$N_{anti}/N_{parallel} = \text{exp}(\frac{\Delta E}{kT}) = \text{exp}(\frac{\gamma h B_0}{2 \pi k T}) \approx 1 + \frac{\gamma h B_0}{2 \pi k T}$

Each tissue voxel contains ~10²¹ protons → ~3x10¹⁵ excess spins in low energy state per voxel.

Larmor equation: $$\omega = \gamma B_0$$
Angular momentum of proton spin is determined by the nucleus-specific gyromagnetic ratio $\gamma$ and the main magnetic field $B_0$
For hydrogen (¹H), $\gamma$ = 42.57 MHz/Tesla

Bloch equation: $\frac{dM}{dt} = \gamma M x B$

$M_z$ is magnetization along $B_0$ field. $M_{xy}$ is magnetization in the perpendicular plane. RF pulses can be used to flip spins to different combinations.
The flip angle depends directly on the power of the RF field and the length of the pulse.
For the same power, a pulse that lasts twice as long will flip 2x the angle.

Apply 90 degree RF flip → $\theta = \gamma B_1 t = \pi / 2 $ → need field of ~ 0.1 G for 1 ms
Maximizes $M_{xy}$ and minimizes $M_z$
Precession in transverse plane (only) generates the MR signal (induces current in receiving coil).

$M_z$ exponentially relaxes over time to return to net magnetization along $B_0$ field.
$$M_z=M_0 (1-\text{exp}(-t/T1))$$
Different tissues relax at different times (T1s) and that is where contrast comes from. (Microscopic energy interactions between molecules.)
Spin-lattice relaxation, T1, is time to return to 63% of $M_0$. Fully recovered in 5*T1. T1 depends on field strength. Fat relaxes faster than water (smaller T1). Typically 0.1 to 1 second.
Long T1 is dark, short T1 is bright. Useful for anatomy.
air, calcium, cortical bone, fast blood : fluid, muscle, organs, proteinous tissue : blood, fat, bone marrow, contrast agents
(similar ordering as O/C ratio is tissues (light to dark): Fat < liver < organs < WM < muscle < GM < CSF)

Spin-spin relaxation
After 90 degree RF flip, $M_z$ is zero, $M_{xy}$ is maximized and rotates at Larmor frequency. Coherence in xy plane is gradually lost. Decay of signal is detected by receive coil.
Loss of MR signals is called Free Induction Decay (FID).

Demo of xy plan coherence loss (WORK IN PROGRESS)

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T2 is the decay time (half-life) of the transverse magnetization. It occurs because of magnetic field inhomogeneities (each spin sees slightly different $B_0$), from external field and structure of material. It is independing of the main field strength. T2 fat < T2 water
$M_{xy} = M_0 \text{exp}(-t/T2)$

short T2 is dark, long T2 is bright. Useful for pathology (edema).
air, calcium, cortical bone, fast blood : fat, ligaments, cartilage, mucles, bound water : fluid, CSF, pathology

T2 decay from tissue effects, T2* includes external effects, and is always faster.

T1 > T2 > T2*

High tumbling rate (non-viscous), high T1 and high T2 Low tumbling rate (solids), high T1 and low T2

Measure T1 using two sets of 90° pulses. Estimate from recovery curve.

T1-weighted images use a short TR and short TE (emphasize differences in T1 and deemphasize T2 differences).
T2-weighted images use a long TR and a long TE

Slice selection and localization

Sequences in MR

Artifacts

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Detective Quantum Efficiency (DQE): Characterizes the frequency dependent SNR performance of an imaging system.
$\text{DQE} = (\text{SNR}_{out}/\text{SNR}_{in})^2$

Signal-to-Noise Ratio (SNR): signal relative to the noise in an image:

$$\text{SNR} = \frac{\mu_{sig}}{\sigma_{bkgd}}  = \frac{\mu_{sig}}{\sqrt{\sigma_{bkgd}^2 + \sigma_{sig}^2}}  = \frac{\sum_i{(x_i - \bar{x}_{bg})}}{\sigma_{bkgd}}$$’

Contrast-to-Noise Ratio (CNR): signal-background relative to the noise in an image: $\text{CNR} = \frac{(\mu_{sig} - \mu_{bkgd})}{\sigma_{bkgd}}$ or $\text{CNR} = \frac{(\mu_{sig} - \mu_{bkgd}) \,}{\sqrt{\sigma_{bkgd}^2 + \sigma_{sig}^2}}$ 

Rose criterion: SNR limit for likely detection of an object. Usually around SNR > 3-5

Noise power spectrum: visualization of the frequency dependence of noise (how much structure is has).
White noise has no frequency dependence

$\int{\text{NPS}} = \sigma^2$ (variance)

For MRI, SNR proportional to voxel volume, and square root of number of averages and phase steps. Can also increase signal by decreasing TE and increasing TR
Higher SNR takes more time.

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Typical radiation doses:

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