Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Medical Imaging
By Satyam Ladva
March 22, 2012

1

X-rays and X-ray CT

1
...

1
...
1

The Crookes tube

• Electrons emitted from cathode by thermionic emission
• Passes through an evacuated tube and into an anode (both cathode and anode are connected to a high voltage
source)
• Depending on the voltage, the high energy electrons collide with the phospor screen and create x-rays
...
1
...
1
...
An Electric field accelerates the electrons into collisions with the anode
resulting in electron-atomic x-ray emission
...

1
...
4

X-ray tube current

• Feedback circuits have taken over from resistors in series, in order to control modern models
...

Characteristic radiation (Bohr model)
• Absorption of energy will excite an electron to a higher state
...

• Energy released is known as characteristic x-ray
...

• An L electron decays to K shell releasing a photon, of energy equal to the energy level gap, in an isotropic manner
...
6 Z2
...

• Inner electron shielding and fine structure (spin -orbit coupling) effects present in real atoms
...
Electron hits the nucleus - X-ray emitted has maxmum possible energy
2
...
Electron approaches nucleus - Scattered at a smaller angle and x-ray has low energy
...
of photons n ∝ Z(Emax − E) where Emax is the x-axis cut-off energy of the photon (electron acceleration
energy)
...

• Bremmstrahlung generation efficiency η =

Px−ray
Pe−beam

∝ ZEmax ∝ ZV

Absorption
• Mass attenuation coefficient - Lower energy x-rays experience higher attenuation, which leads to “beam hardening”
• Adding more layers of the material removes lower energy x-rays
1
...
5

X-ray tube dials

• Tube has 3 controls - Current, voltage and exposure time
...

1
...
6

Scattering in tissue

• Absorption of x-rays produces image contrast but increases the dose to the patient
...

Rayleigh Scattering
• Incident photon interacts with all of the electrons, causing them to oscillate in-phase
...

• Accounts for ~12% interactions, E=30keV and less than 5% for above 70keV
...
Photon has energy ~100keV
• PP E ∝

2
Zef f
E3

• Characteristic radiation emitted - M shell electron decays to K-shell
...
KE of auger electron = transferred energy - binding energy
...

Auger electrons dominate for low Z elements and outer shells of high Z
...


2

Compton scattering
• Inelastic scattering, dominant in soft tissue above 26keV
...

• Atom is ionised and as energy of incident photon increases, scattering angle reduces
...

• Pcomp ∝ constant
• Ei = Esc + Ee− where Esc =
1
...
7

Ei
(1−cosθ)

Ei
e c2

1+ m

Linear attenuation coefficient

• I(x) = I(0)exp(−µx) where x is the cm of absorber the photon has passed through, where I is photon intensity
...

• Half value layer - tissue thickness that attenuates x-ray by 1/2
...


1
...
E
...

1
...
1

Focal spot size and coverage

• Effective focal spot width: f = F sinθ where F is the focal spot width on the anode from the cathode and θ is the
angle which the heat spreads from the anode - See diagram
...

• Coverage: c ∼ 2dsinθ where d is the distance from the anode to the patient - See diagrams
...
Increasing the potential can result in pulses and, if high enough, be switched off
...
2
...

• Intensity is not uniform across the x-ray beam (X-rays are created inside the target) and the x-rays near the anode
end are more likely to be scattered or absorbed resulting in an intensity drop from cathode to anode, as a function
of distance between them
...

1
...
3

Scattered x-rays

• Image contrast: Fractional difference in signal intensity, S, from an area of interest compared to surrounding normal
−S
tissue: C = ∆S = (SlSn n )
S
• Can reduce contrast of the image by a reduction factor:
field of views FOV
...


This increases for larger

• Can place an anti-scatter grid after detector with dimensions: thickness t, separation of grids d, and height of grids
h
...
Crossed grids are better but cannot be moved
...
2
...


• Since the focal spot in x-ray generators is of a finite size, geometric unsharpness will form - penumbra (often at the
edges)
...
To avoid
this change the tube geometry
...
If all are gaussian, can add components in quadrature
...
2
...

• Optical density - Measure of film blackening: OD = log10 (Ii /It ) depends on exposure time and x-ray rate on screen
...
2
...

• Determined by efficiency by which x-ray converted into light
...
2
...

• Computed - Phospor imaging plate absorbs x-ray resulting in excitation of phosphor molecules (lasts for a long
time)
...
Bright
lights will erase the plate data and be re-used
...
Amorphous Si transistor array detects light
...
The scintilator acts as a light pipe in order to reduce blurring
...
2
...
The latter
dominates
...
Noise = standard deviation
√
on the signal and SNR ~ N/ N
√
• SN R ∝ exposure time × current
• Can increase kV on x-ray tube (less attenuation and contrast), reduce filtration etc
...


• Spatial resolution: Ability to resolve structures
...

• Image contrast: Depends on x-ray energy, detector absorption and scattering quantity
...
The CNR for lesion is: C|CL | = |CL |dL RN t
noise
• Contrast agent can be used to increase contrast which have appropriate K-edge energy
...
2
...
Consists of
electron beam travelling through vacuum tube with an increasing negative potential until it reaches a highly positive
anode
...
3

X-ray and CT image formation

• Imaging system takes 1D view through thin slice of patient for many different angles
...
µi is the linear attenuation coefficient of different sample layers
(of thickness ∆x)
´
• Instead of sections, one large sample is used: p = µ(x)dx
1
...
1

Radon transform

• Acquiring a projected image through a sample
...
e
...

• As opposed to cartesian coordinates, use polar coordinates to keep the system stationary WRT the detector
...

1
...
2

Simple Backprojection

• A rotating object produces a sinogram
• Use backprojection to acquire original image at various angles
...
e
...

• Mathematically: Define co-ordinate system transformation from r,s system to cartesian and rotation coordinates
...
The reconstructed image f (x, y) = N i=0 p(xcosφ1 + ysinφ1 , φ1 )
• Image is blurred since counts projected outside of object and so only useful for high contrast objects
...


1
r

where the latter is the blurring

1
...
3

Fourier transform and Central slice theorem
´∞ ´∞
• F (u, v) = −∞ −∞ f (x, y)e−i2π(xu+yv) dxdy - 2D transform
...


Direct FT reconstruction:
• Acquire projection profiles at various angles and take the 1D FT of each projection
...

• Then take the inverse transform to compute image of object
• Noise free data but time consuming
...

5

Filtered backprojection:
• Acquire various projections at N angles
...

• Multiply projection by FT of ramp filter (in freq space) - The equivalent of deconvolving the blurring function in
image space
...

• Perform conventional backprojection with the filtered profiles
...
At high freq, not much signal but white
noise present
...
linear and fast
...

1
...
4

Sampling

• Linear sampling distance ∆r is the distance between the sample points in the projection path
...

• The minimum sampling required: ∆rmin <

1
2ξmax

where ξmax is the maximum frequency (Nyquist freq) of the signal
...

• ∆r < RF W HM /3 where RF W HM is the spatial resolution
...

• Required number of angular views: = πD/(2∆r)
...

• Beam hardening - As x-ray passes through tissue, lower energy beams are filtered out leading to µ decreasing for
increasing distance
...
3
...
Overlap
this wave with the wave obtained from the original object and reproduce the final image
...
4
1
...
1

X-ray CT hardware
1st generation

• X-ray tube and single detector translate across FOV, producing series of parallel rays and this occurs for many
angles within 180 degrees
...
4
...
Finer
sampling and faster
...
4
...
Tube and detector array linked together
and rotate around patient
...

• Slip ring gantry - Allow for continuous rotation in one direction
...

• Ring artefact problem - Each detector forms part of the whole image so, if one breaks, part of the image is lost
...


6

1
...
4

4th generation

• X-ray tube rotates around detector ring and each of its ~4800 detectors act independently
...
Improve accuracy
by combining parallel and anti-parallel views
...
4
...

1
...
6

6th generation

• Table moves in and out of helical tube, which itself rotates around the table (with patient on it)
...

• Spiral pitch - Depends on the table feed d per revolution of the x-ray tube and collimated slice thickness S: p = d/S

...
p>1, then partial scanning
...
5, since this accompasses
the 180 degree requirement
...
4
...

• ’Binning’ and collimation changes the slice thickness and number of slices
...
4
...

• Ceramic scintillator coupled to photodiode which converts x-ray into visible light, which is converted into a current
...
4
...
A filter causes the flux to be uniform and, for
CT, ’bow-tie’ filter used
...
4
...

µwater
• CT carried out with highly filtered, high energy x-ray beam where Compton scattering dominates
...
4
...

• Window width - The CT number range from black to white
...
4
...

• Effective dose - E =

T

wT HT where wT is based on the radiation susceptibility of the tissue
...

• Radiotracer - Compound tagged with radionuclide in very tiny amounts and emit gamma rays - usually
...


2
...

• For low Z, N=Z and for high Z, N=1
...

2
...
1

Decay

• Radioactive samples decay with time:
factor
...

• Activity - Q = | dN | = λN
...

dt
• 1 Curie (Ci) =

λN
3
...


• Decay factor - Substitution of activity, leads to Q decreasing exponentially with time
...
τ 2 =

2
...
2

ln(2)
λ

Decay pathways

• Alpha decay not used since poor penetration (µm)
• Beta - decay: Not used alone, since <2mm penetration
...
E
...
A second method is of the daughter nuclei being in an excited state and
releasing a gamma photon to de-excite
...

• Electron capture: Inverse Beta -, not used alone, but can be followed by x-ray emission after
...
orbital vacancy results in x-ray emission but daughter nuclei may become
excited, resulting in gamma emission
...

• Isometric transitions: long-lived metastable nuclei (subscript m), emitting continuous gamma rays
...

• Internal conversion: energy from decaying metastable nuclei is transferred to an orbitting electron, and ejected
(conversion electron)
...


2
...

• Nuclear reactor - Fuel cells contain fissionable materials
...
Control rods
shield or expose fuel cells and position determines the chain reaction
...

• Cyclotron - Circular motion, Lorentz force law etc
...


8

2
...

• Multiple detections from the PMT arrays are combined in a summing circuit, in order to provide the maximum
gamma ray intensity height
...

• Image formation: absorptive collimation (collimator must be thin) in order to allow maximum gamma rays to be
absorbed however, only those in a specific direction are detected
...
3
...

• Image non-inverted and magnification M=1
...

• Geometric separation, probability of gamma rays reaching detector from point source: G =
collimator thickness
...
3
...

• Minimised, non-inverted image
...
3
...


Converging collimator

• Holes converge to a point in front of collimator
...


• Image is inverted and magnified
...
3
...

• Pinhole size gives good resolution at small z but poor efficiency at large z
...
3
...
The
D’s represent the distance of the centre of the detector to the gamma ray peak and D = D1 + D2
...
3
...

• Some iodine excited from detector K-shell causing energy to decrease
...
E
...


9

2
...
7

Gamma ray detection efficiency

• Es = G f where G is the geometric collimator efficiency, f is the fraction of photons accepted by the energy
discriminating circuit and
•

N o
...
of γ−rays hitting scintillator

=

2
...
8

= 1 − exp(−µx) where x is the detector thickness
...

• τ=
2
...
9

N −n
Nn

where N is the true event rate and the measured count rate n = N (1 − nτ ) where nτ is the total deadtime
...
SNR∝
radiopharmaceutical, imaging time and sensitivity etc
...

2
...
10

SPECT

• Similar to x-ray CT
...

• Geometric response: Attenuation effects (corrected by using mean shape and attenuation of an average patient, use
Chang’s method)
...
In addition, take a reference blank scan
...

• Use dual energy windows to correct the scattering: Multiply data in scatter window by a weighting factor and
subtract from photopeak window
...

• Coincidence detection method
• Intrinsic spatial resolution: δrint ∼

d
2

where d is the scintillator width
...

• System spatial resolution in PET: δrsys ≈

3
...
10
...
2

Positron range

• Depends on maximum kinetic energy of emitted positrons (exp
...

• Inversely proportional to absorber density
...
3

Non-colinearity

• The photons are not released opposite to each other due to residue momentum of electron and positron at point of
annihalation
...
4

D
2

×

0
...
5 and 0
...


Sensitivity

• Determined primarily by detector efficiency and solid angle coverage
...


3
...

• Scatter coincidences - One or both gamma rays scatter in tissue
...
subtract Rrandom (i, j) =
Rs(i) × Rs(j) × 2τ from true coincidence rate
...

• 2D and 3D acquistion - 3D sensitivity increases and rate of scatter events increases
...


3
...

• Attenuation independant of source position along a line of response
...
7

Image reconstruction

• Similar to Spect
• PET data provides integral of activity along line between each source and detector pair
...

• Use filtered backprojection algorithms etc
...
1

Absorption

• Defined by the Beer-Lambert law IT = I0 exp(−µabs L) (Intensity of the transmitted light as a function of absorber
thickness)
...


11

4
...

• Collimating grid: Suitable for weak scattering, due to presence of in-line scattering, but reduces the overall scattered
light detected
...

• Temporal gating (see slides)
• Optical coherance tomography (see slides)

4
...
In-line scattered light limits
OCT imaging depth
...


5

Ultrasound - The physics
• A higher frequency provides better resolution although a lower penetration
...
1

Basics

• Definition: Sound above the audible frequency: 20kHz where higher frequency used for research applications
...
Density and elasticity of the particles
determine this ability
...

• Pulse waves: Required for range finding method
...


12

5
...
As ∆P → 0, β = −V0 ( dV )0 - Conservation
V0
of mass
...

ρ0
(
...
mass, N2 and Equation of state
...


• Three distinct classes of tissues: Bone (mainly solid), Lung (mainly gases) and soft tissues/liquids
...
3

Sound interactions - Derived parameters: Energy, Intensity, Acoustic impedance
2

2

v
p
• Wave energy = Kinetic energy ( ρ02 ) + Potential energy ( 2ρ0 c2 )
...


dξ
• Can determine time averaged intensity and energy density by using: v = dξ and p = −ρ0 c2 dx
...
dS and, this equals I
...


• Acoustic impedance - Ratio of the push variable to flow variable: Zsp =

p
v

• Issues with model: Not an infinite plane wave or a lossless medium
...


5
...

• Power conserved across the boundary

5
...
Remember derivation consists of 2 B
...
’s
1
...
v + + v − = v tr where changing into pressure form and dividing by p+ gives 1 − R =

Z1
Z2 T

• Derivation of Power reflection and transmission coefficients leads to r + τ = 1
• Reflection and transmission at oblique boundaries

5
...
scattering
• Rayleigh scattering - scattering cross sections
• Collection of incoherent scattering - Speckled/Interfering waves
...
7

Other ways to lose energy

1
...
Finite time for energy redistribution and
density fluctuations out of phase with pressure
...
Inhomogeneous media: viscous damping, density variations in surrounding media, leads to rotational motion and
shear waves
...

3
...
Absorption is more dominant in biological tissues
...
8

Some coefficients

W
• In a thin tissue, of thickness ∆x, total power ∆W is lost: ∆W = −µW ∆x where µ = (µs + µa ) = − 10 log10 ( W0 )
x

• Attenuation = µ × l × f

5
...

• Image artefacts: Artifical bright and dark regions
...


6

Ultrasound - Engineering and system design

6
...

• Electric field is proportional to strain: ∆T ∝ V

6
...

• Fundamental resonance - ZB < Z0 > ZL leads to 2nπ = T ×
the active element thickness
...
T =

2π
λ

−π+T ×

2π
λ

− π
...

• Backing layer - Acts as a mechanical support and attenuates sound (Stops reflections from back)
...


6
...

• Can increase sensitivity of detector but reflection amount increases, despite noise being reduced
...
Spatial impulse
response used and delta function excitation of elemental unit of transducer surface
...
4
6
...
1

Acoustic field
From a circular vibrating plate

• Beam formed by interference between spherical waves from each point on transducer’s surface
...


2λ
D

and distance from transducer to far field is:

• In far field: Beam profile (Airy discs) is FT of the aperture and Rayleigh’s criterion: sinθ =
• Apodization: Changing intensity profile of aperture to change far field pattern
...
22λ
D
...
4
...


• Near field: Rect apertures have less extreme fluctuations
...
5

Impulse response function

• Axial resolution: ∆r =

cTp
2

• Lateral resolution: DF 3dB =

λ
DF

• Multiple isolated transducers allow greater control of the beam pattern
...
6

Beamforming

• Synchronus, focused, steered and (steered and focused) beam type
...

• After beamforming, B-mode image (instantaneous amplitude of RF signal created) but in order to filter, requires
knowledge for filter design
...
7

The Hilbert transform

• Detects the envelope of the time-varying signals: FHI =

−1
πx

∗ f (x)

• Wave forms differentiated - signals shifted by π/2 and two hilbert transforms result in the original image being
inverted
...
2d where c =
c
ρ0
...


7

Ultrasound imaging of moving targets

7
...

• Only the component towards and away from the measurement region is used: dν = − 2vcos(θ)ν and no shift perpenc
dicular
...
2

Detecting the Doppler signal

• Coherent demodulation cannot determine direction of target, only velocity
...

• Multiply the signals and then perform a low pass filter
...


15

π
2

out of phase
...
3

Continuous vs
...

• Pulsed waves provide range information via time delay information albeit multiple pulses required in order to
determine phase change
...
4

Colour Doppler imaging

• A target moving towards or away from a transducer results in gradient of phase
...
time graph reversing
...

• See simulations

7
...


• More pulse transmissions - More noise and greater precision
• Less pulse transmissions - Aliasing and for larger regions of interest
...
1

Original motivation

• Contrast agents boost the weak echoes associated by sound by strengthening the pulse signal greatly above the noise
threshold
...


8
...

• The bubbles must be the same size as red blood cells - so they can travel around the body unimpeded
...

• Heavier, less soluble gases last longer
...
e
...

• The system mass is that defined of the surrounding liquid and the restoring force is the elasticity of the gas
...


8
...
11-12
...

• Pressure dependent response: Linear → non − linear → Destruction (Path of bubble)
...

• For slow flows, the blood signal is very close to tissue signals and so 3D image required with 3rd axis being nonlinearity information
...
4

Non-Linear imaging

• Destruction imaging - High amplitude pulses destroy the bubbles, produces good sensitivity and resolution imaging
although can only be done once and not in real time
...

• Harmonic imaging - Filtering on the 2nd harmonic response but reduces the spatial resolution
...

• Low amplitude multipulse imaging - nonlinear detection of harmonics results in no resolution loss but a lower frame
rate than harmonic and lower sensitivity than destruction imaging
...


9

Magnetic Resonance Imaging
• NMR – Nuclear: Spectroscopic study of the magnetic properties of the nucleus of the atom
...

– Resonance: An energy coupling when nuclei in a strong external field which causes absorption and release of
energy
...

¯
¯
ˆ
• Due to external B-field B0 , moment will precess
...


9
...

• Zeeman energy levels due to external magnetic field
...
Em = −ml γ B0
• From Planck energy and Zeeman shift energy: ∆E = ω = γ B0 where cancellations gives the Larmor equation
...

• The +/- energy level difference is small compared to thermal energy level difference which results in average population density mismatch
...
2

Classical derivation of the Larmor equation

¯
¯ ¯
¯
¯
• M = γ S, N = M × B and

¯
dS
dt

¯
= N are combined to give Bloch equation:

¯
dM
dt

¯
¯
= γ(M × B)

• The equation describes the Bulk motion of the magnetic moment in the external field
...


• Thermal energy perturbs the hydrogen energy level
...


17

9
...

• At equilibrium net magnetisation vector orientated along the direction of the main magnetic field
...


• Time varying magnetic moment in the transverse plane can be detected by an RF freq
...
When switch off
pulse, they start in phase but some end up slower (or faster)
...
4

Relaxation processes

• Destruction of transverse magnetism - Spin/Spin relaxation has decay time T2
• Growth of longitudinal magnetization - Spin/Lattice relaxation has decay time T1
• These drive the magnetization back into equilibrium
...

• From Spin-Spin: The magnetic moment in the x-y plane will smear out resulting in rapid dephasing
...

Signals “fan” out
...

inhomogeneities
...
5

1
T2 ∗

=

1
T2

+

1
T2

where T2 is the field

Spin Echo pulse sequence

• Phase change results in slower waves lagging behind whereas faster ones spread out further
...

• The Echo-time: TE = 2τ
...


9
...

• The energy is given up to the structure of the solid in order to restore thermal equilibrium via a stimulated emission
...
7

Inversion and Saturation recovery

• Inversion recovery
– Flip by 180 and then by 90
...
e
...
Think of a pendulum
...

– The signal will reflect T1 differently in tissues because of different amounts of longitudinal recovery during the
TR period
...
8

Relaxation

• T1 (longitudinal relaxation) and T2 (transverse) depend on Temperature, spin mobility and paramagnetic ions/molecules
(and their sizes)
...

• Molecular correlation time τc is the time between collisions and rotational changes
...


• T1 (in s), T2 (in ms) and T2 ∗ (few ms)
...
9

Magnetic gradients

• Larmor frequency is dependent on B0 and, by adding a small gradient xGx , B becomes spatial dependent resulting
in a varying precession ν(x)

9
...
ν(z) =
• There is a selective spread in Larmor frequency: ∆ν(z) =

γ
2π B(z)

γ
2π zGz

• If excited with a single frequency, a very long pulse required in order to increase exposure time
...

• Cross-talks between slices eliminated via refocussing process and de-phasing
...
11

Spatial encoding - Frequency encoding

• FT from time to frequency domain and vice versa
...

• Can use a negative pre-gradient in order to pre-wind the phases in the frequency encoded direction however, inhomogeneities still present
...

• BW =

9
...


•
...
The signal is then “read” on
...
the time domain, which can be determined via FT of freq
...
∆t ≤
• Map onto K-space, “spatial frequency” domain by using: k =
•
...


1
F OV

• Frequency encoding with a 2D or higher environment is impossible because frequency is a scalar parameter and, if
done so, the parts superimpose creating non-unique frequency terms
...
13

Spatial encoding - The 3rd dimension: Phase encoding

• Slice selection, frequency encoding and phase encoding all do the same things to spin but this must be done during
excitation and during readout and passively during free induction decay FID
...
y
...
5)
...

• The maximum resolution adjacent pixels vary by half a cycle (π)
...


• MRI is pixel limited and ∆y = F OV /Np where there is a trade off between the resolution, the SNR and acquisition
time (Time required for system to relax before taking another set of readings)
...
14

Consequence of acquiring data in k-space

• The sampling theorem: ∆kx ≤
encoded
...

• The signal is sampled discretely at every ∆t where the sampling is perfect if ∆t ≤
– νapparent =

ν − nνsampling
ν + nνsampling

1
2νmax

where:

ν>0
ν<0

• The patient moving or organ moving results in signal modulation
...


9
...

• Air cored resistive, iron cored electromagnetic, permanent and superconducting magnets are the 4 main types
...
This results in
a lower sensitivity which effects scan time and resolution
...

• Strong gradients with high slew rates get hot
• Large homogeneous fields lead to image distortion, rapid switching of the gradient fields increases acquisition time
rate, low power dissipation reduces heating output and eddy currents not generated result in artifacts
...
B0
...

• Emitted RF signal is bandlimited at the Larmor frequency

20

9
...
(voxel size)
...
Ny ∆z

– measurements: Nx Ny Nrpts
– Wide band has more noise and

Nx
Ts

where the t is the sampling time
...


9
...

• If TR ~ T2 Mxy remaining after each acquisition which can be removed with additional post readout gradients
...
Same as above limits
...


9
...

• FLASH - Fast low angle shot: Gradient Echo
– Shortern TR - Small flip angle image, takes a while
– See Ernst angle derivation
• TSE - Turbo spin echo: Fast spin echo
– Collect multiple spin echoes for each excitation
– See pulse sequence diagrams
...

– If T2’s are short, resolution loss and SNR loss
...

• GRASE = EPI+TSE
– Low exposure vs

---