Ilkka Nissil 3.10.2019 Aalto University School of Science Department - - PowerPoint PPT Presentation

Ilkka Nissilä 3.10.2019 Aalto University School of Science Department of Neuroscience and Biomedical Engineering (NBE)

Practicalities

 Lectures: 3.10. and 10.10.  Exercise sessions: 4.10., 9.10., 11.10. and 16.10.  Teachers: Ilkka Nissilä and Tuomas Mutanen  The assignment involves familiarization with X‐ray

propagation and computed tomography

 Return the exercise by 17.10. and the learning journal

by 21.10.

X‐ray imaging

Computed tomography – three‐dimensional imaging using x‐rays 1979 Nobel Prize in Medicine: Hounsfield and Cormack

Applications of CT

This week’s lecture

 Physics of X‐ray imaging

 Generation of X‐rays  Interaction between X‐rays and tissue  Detection of X‐rays

 Imaging geometry and forward model

 Planar imaging (2D)  Computed Tomography (3D imaging)

Principle of 2D X‐ray imaging (planar imaging)

Scintillator Photodiode array TFT matrix

• Photoelectric effect (absorption)

creates contrast between tissues

• Scattered x‐ray photons reduce contrast

Planar X‐ray imaging creates a 2D projection of the tissue

What are x‐rays?

• X‐rays are electromagnetic

radiation in the gamma range

• Photon energy E = hν = hc/λ ~

6e‐34*3e8/1e‐10J ~ 2 fJ ~ 10keV

• X‐rays penetrate tissue quite

well but are attenuated due to photoelectric effect and Compton scattering

• High‐energy gamma rays have

higher probability of Compton scattering than x‐rays

Radiation dose in clinical use

• Effective dose equivalents HE = Biological effect of radiation

– Dental

0.01 mSv

– Breast

0.05 mSv

– Chest

0.02‐0.2 mSv

– Skull

0.15 mSv

– Abdominal

1.0 mSv

– Barium fluoroscopy

5 mSv

– Head CT

3 mSv

– Body CT

10 mSv

• Natural background radiation 0.3‐3 mSv/year in Finland
• Diagnostic x‐ray amounts to 14% increase in total radiation

worldwide

1 Gy = 1 J/kg absorbed dose 1 Sv = 1 J/kg ”equivalent” in terms of biological effect For gamma and x‐ray, 1 Gy => 1 Sv For alpha, 1 Gy => 20 Sv

X‐ray source: the x‐ray tube

• 10‐7 atm pressure
• 15 to 150 kV rectified alternating voltage between cathode and

anode

• Cathode heated (~2200 deg C) tungsten

filament wire

• Thermionic emission
• Anode: rotating disc, covered by a layer of tungsten, tungsten‐

rhenium or molybdenum, liquid cooling

Number of x‐ray photons in the beam ∝

) (mA)

kVp = accelerating (peak) voltage mA = filament current

X‐ray tube output energy spectra

Different filters affect x‐ray energy content Aluminum filter = standard beam energy content (e.g. for imaging the torso) Molybdenum filter = low energy content (used e.g. in mammography) X‐ray tube housing absorbs low‐energy X‐rays Brehmsstrahlung continuous spectrum (deflection of incoming electron) Peaks correspond to characteristic X‐rays (anode material property) Electrons hop from outer to inner shell => X‐ray

X‐ray interaction with tissue

• Interaction of x‐rays with

tissue includes absorption (photoelectric effect); Compton scattering and Rayleigh scattering

• Photoelectric effect is the

most frequent event in tissue‐x‐ray interaction and it produces useful diagnostic contrast

• Probability of each event type

depends on the energy of the radiation and the material properties

Mass attenuation coefficient is absorption coefficient [1/cm] divided by density [g/cm^3]

Photoelectric interaction

 In the photoelectric interaction between X‐ray and tissue,

an inner electron is ejected by the X‐ray

 An outer electron takes up the vacancy and emits a low‐

energy characteristic X‐ray which is absorbed quickly.

∝

Photoelectric absorption in tissue

K edge:

• Probability of PE event is

more likely when the energy of incoming X‐ray is just above binding energy of K electron

• Contrast between bone

and soft tissue increased

• Dual‐energy imaging can

highlight the contrast

Compton scattering

 In Compton scattering, an outer electron is ejected

from a molecule

 The original X‐ray is deflected by an angle θ

Rayleigh scattering

 Rayleigh scattering is elastic i.e. the emitted X‐ray has

the same wavelength as the incoming X‐ray

 The angle of deflection is small.

Half‐Value Layer (HVL)

How thick a slab of given tissue reduces the X‐ray beam intensity by 50%? Higher energy X‐rays are needed to get a useful image of the torso In mammography, the breast is compressed to a thickness of ~ 4 cm

Dual‐energy imaging

 By starting from images

• btained using X‐rays

generated with two different tube voltages, it is possible to produce different weightings of bone and soft tissue, enhancing contrast

 Can also suppress artifacts

due to metal objects

 Measurement of electron

density

Instrumentation for planar x‐ray imaging

Digital Radiography TFT Array Detectors

 TFT array detectors can be large  Indirect method: use scintillator and optical coupling to

TFT matrix

 Direct method: X‐rays release ion pairs; electrical

coupling to TFT matrix

Anti‐scatter grids

Lead strips Aluminium Length = h Thickness = t Separation = d If the X‐ray source is close the beam divergence should be considered Grid ratio = h/d Grid frequency = 1/(d+t)

Noise in x‐ray imaging: photon shot noise

 Photon shot noise is a key image quality parameter  η is the quantum efficiency, N number of photons hitting

the detector during the exposure

 N follows Poissonian statistics

N n N SNR N n shot      

Additional sources of noise in x‐ray detection

 In addition to photon shot noise, detectors and

electronics introduce additional noise sources

 Dark current is the current that the detector generates

when not exposed to X‐rays

 Thermal electrons are separated from the photodetector

material and amplified

 The amplifiers add some noise of their own to the signal

 In this course we can model these additional noise

sources optionally with a Gaussian white noise term added to the photon count or intensity

Instrumentation for CT



First and second generation devices used synchronized translation of both the X-ray source and detector on opposite sides of patient



First generation used a pencil beam



Second generation used a narrow fan beam

Instrumentation for CT



Third generation devices use a rotating assembly with an arc of detectors and X-ray source on opposite side



Fourth generation systems use a rotating X-ray source and a full ring of detectors

Detectors in CT systems



Scintillator crystals convert X-ray into light



Optical filler material between each crystal to prevent cross talk



Photodiodes convert the light into electrical signals

Mathematical principles of CT‐ measured data

 Considering only the PE effect for simplicity, the reduction

• f the intensity of the x‐ray beam along the projection line

is proportional to the absorption

 The measured intensity at a single point at the detector is  I0 = source intensity  I = intensity at detector; n = noise

Projection

• log 0

log

• The projection p is an integral of the absorption

coefficient along the line of propagation of the x‐rays

Calculating the projections

1 3 2 3 2 5 1 2 Measurements are electrical signals which are proportional to intensities

• =

p log Δ

• X‐ray tube positions for

the second orientation X‐ray tube positions for the second orientation

Detectors Anti‐scatter grid

10 6 3 4 6 9 1 2 3 4 5 6

Scanning

• rder

Rotate tube and detector array

Model of an axial slice with 3 x 3 resolution

Ray‐by‐ray image reconstruction

1.33 2 3 1.33 2 3 1.33 2 3 1 3 2 3 2 5 1 2 1.22 1.89 2.89 2.56 3.22 4.22 0.22 0.89 1.89 Original Initial guess 4 6 9 6 10 3 , , 4 6 9 3 10 6 4 6 9 6.33 6.33 6.33 1st iteration 2nd iteration 4 6 9 6.33 6.33 6.33 4 6 9 6 10 3 4 6 9 3 10 6 Correct by

,,

• Correct by

,,

• x

y

Projection with multiple directions

 For the first projection angle  For a general projection direction α

, ∆

• , ∆
• ∆

First projection along y axis; translation along x axis. r and s replace x and y in a rotated coordinate system x y r s

Sinogram

The sinogram contains the measured (or simulated) projections; each row corresponds to a different projection angle and each column to a different translational position

Original image: attenuation varies between 0 and 1.5

First projection incidated by arrow

Mathematical principles of CT – the forward problem

 The forward model can be expressed using a matrix

equation:

 μ = image of tissue absorption as a vector (”source”)  p = attenuation along a projection line L

(”measurement”)

 A is the system matrix which describes the projection

geometries for all source and detector positions

n Aμ p  

)dl x ( μ I I log I dI p

 

       

L L

= =

Constructing the linear system

 The projection vector p is a column vector consisting

• f subvectors for each projection angle; if Ntrans is the

number of translation positions and Nrot is the number

• f rotation angles, the vector has dimensions

(NtransNrot) × 1.

p = , , … , ∆, …

• ,

Constructing the linear system

 The μ vector is a

NxNy × 1 column vector representation of the two‐dimensional axial slice

 Each element of the

A matrix gives the pathlength of the x‐ray beam (x’,φ) in a given voxel (x,y)

X‐ray tube X‐ray detector Translation x’ Rotation φ xlab ylab xlab ylab x’ y’ φ (x,y) l l = pathlength in voxel (x,y)

Constructing the linear system

 To repeat, the A matrix describes how sensitive each

measurement (x’,φ) is to attenuation in each voxel of the image at position (x,y)

 This sensitivity is equal to the length of the beam’s

travel in the given voxel (= l in the previous slide)

(NtransNrot) × 1 NxNy × 1 (NtransNrot) × 1 (NtransNrot) × (NxNy)

Projection p(r,φ) is expressed as a Radon transform of the absorption coefficient :

=

: ,

• To estimate the attenuation μ(x,y), we may apply the

inverse Radon transform:

Forward model: Radon transform

• Normalization?

CT number in Hounsfield Units vs. linear attenuation



• 2

2

where

0 is the linear attenuation coefficient of tissue in

each voxel of the image



2 is the linear attenuation coefficient of water

 Mass attenuation of water:

 0.3756 cm2/g at 30keV; density 1 g/cm3 ; 2, 30 = 0.3756 1/cm  0.1707 cm2/g at 100keV; 2, 100 = 0.1707 1/cm

Lecture 1 ends here