Basic concepts

Exponential Behavior

Exponential Decay/Growth:

\frac{δ N}{δ t} = \pm λ

$\frac{\delta N}{\delta t} = \pm\lambda$

X-Ray Production

Coolidge Tube

High-voltage Generator for heating ( $U_h$ ) and cathode/anode ( $U_a$ )
Filaments is heated and gives off electrons
Electrons are accelerated from cathode to anode
Electrons collide with the anode material and accelerate other electrons
About $1\%$ of the energy generated is emitted as X-Rays

Tourner l’anode genere de la chaleur

Interaction with Anode Material

Bremsstrahlung

Complex model depending on:

Path of electron in the target
Change in direction at each interaction
Change of ionization and radiation loss
Direction of emission of the bremsstrahlung
Attenuation and scattering inside the target

Thin to thick target emission model:

I (E) = \underset{constant}{\underset{⏟}{C}} \cdot \underset{atomic number}{\underset{⏟}{Z}} \cdot (\underset{energy of the bombarding electron}{\underset{⏟}{E_{m a x}}} - E)

$I(E)=\underbrace{C}_{\text{constant}}\cdot \underbrace{Z}_{\text{atomic number}}\cdot (\underbrace{E_{max}}_{\text{energy of the} \\ \text{bombarding} \\ \text{electron}}-E)$

Emitted Spectra

Pour un tungstene:

X-Ray Interaction with Matter

X-Ray photon life span:

Photon is transmitted through the matter
Photon is absorbed (end of life)
Photon is scattered ( $E_{new}\le E$ )

If $E_{new}\gt0$ , then more 1, 2 or 3

Photoelectric Absorption

Interaction with an electron of the K, L, M, … atomic shell

All energy is absorbed
Ejects an photoelectron » ionizing radiation
Vacancy is filled from a electron of a higher shell
Produces either characteristic radiation (fluroescence) or an “Auger electron”

Probability of occurrence (or cross-section):

σ_{p h o t o n} \propto \frac{Z^{3}}{E^3}

$\sigma_{photon}\propto \frac{Z^3}{E\text{^}3}$

Compton (Incoherent Scatter)

Interaction with free electrons (outer shell):

Part of photon energy is transferred to the electron (ionization)
Photon is deflected with a certain angle and new energy $E_{new}\lt E$
Energy loss depends on the scattering angle (energy conservation law)
Scatter angle ( $\theta$ ) decreases with photon energy ( $E$ )

Probability of occurenece (or cross-section)

Almost independant of $Z$ & decreases with $E$
Energy conservation

\frac{E}{E_{n e w}} = \frac{E}{m_{e} c^{2}} (1 - \cos (θ))

$\frac{E}{E_{new}}=\frac{E}{m_ec^2}(1-\cos(\theta))$

Klein-Nishina coefficient

σ_{c o m p t o n} = f_{K - N} (E, θ)

$\sigma_{compton}=f_{K-N}(E,\theta)$

Rayleigh (Coherent Scatter)

Electromagnetic wave resonance:

The incident electric wave makes electrons to oscillate in phase and emit radiation
Energy is conserved $E_{new}=E$
Photon is deflected with a certain angle
Scatter angle $\theta$ decreases with photon energy $(E)$

Probability of occurence (or cross-section):

Mainly for large $Z$
Decreases rapidly with $E$
Atomic Form Factor (AFF)

σ_{r a y l e i g h} - f_{A F F} (E, θ)

$\sigma_{rayleigh}-f_{AFF}(E,\theta)$

Total attenuation

δ I = - σ \cdot ρ \cdot I_{0} \cdot Δ T

$\delta I = -\sigma\cdot\rho\cdot I_0\cdot\Delta T$

Total Attenuation Cross Section, $\sigma$ $[cm^2/g]$

σ (E) = σ_{p h o t o} (E) + σ_{c o m p t o n} (E) + σ_{r a y l e i g h} (E) + \dots

$\sigma(E)=\sigma_{photo}(E)+\sigma_{compton}(E)+\sigma_{rayleigh}(E)+\dots$

Linear Attenuation Coefficient, $\mu$ $[cm^{-1}]$

\begin{aligned} μ (E) = σ (E) \cdot ρ & for a single atom \\ (\frac{μ}{ρ}) (E) = \sum_{Z} w_{z} \cdot (\frac{μ}{ρ})_{Z} & for all atoms \end{aligned}

$\begin{aligned} \mu(E)=\sigma(E)\cdot\rho &\text{for a single atom}\\ (\frac{\mu}{\rho})(E)=\sum_Z w_z\cdot(\frac{\mu}{\rho})_Z&\text{for all atoms} \end{aligned}$

X-Ray Detection

Primary X-ray image

Photographic Film & Phosphor Plates

Solid State Detectors: Indirect Detection

Summary

S i g n a l (i) = \underset{Gain}{\underset{⏟}{k}} \int \underset{detector technology}{\underset{⏟}{ξ (E)}} \underset{efficiency}{\underset{⏟}{η (E)}} [\underset{grid}{\underset{⏟}{G}} \cdot I_{s c a t t e r} (E, i) + I_{0} (E, i) \cdot e^{- \int μ (E) \cdot d l}] d E

$Signal(i)=\underbrace{k}_{\text{Gain}}\int\underbrace{\xi(E)}_{\text{detector technology}}\underbrace{\eta(E)}_{\text{efficiency}}\biggr[\underbrace{G}_{\text{grid}}\cdot I_{scatter}(E,i)+I_0(E,i)\cdot e^{-\int\color{red}{\mu}(E)\cdot dl}\biggr]dE$

Overview

What characterizes an Imaging System ?

Tube output (spectra, power)
Beam geometry (narrow or wide beam)
Detector technology (integration, electronics, …)
2D vs 3D imaging

What system Design vs Imaging Target ?

Spatil resolution for specific diagnostic value
Radiation dose vs image nois

Digital Image Formation

Projection Image

Disregarding scatter & non-idealities:

Object signal (i) = k \int [E \cdot I_{0} (E, i) \cdot e^{- \int μ (E) \cdot d l}] d E Air signal (i) = k \int E \cdot I_{0} (E, i) d E

$\text{Object signal}(i) = k\int\biggr[E\cdot I_0(E,i)\cdot e^{-\int\mu(E)\cdot dl}\biggr]dE\\ \text{Air signal}(i)=k\int E\cdot I_0(E,i)dE$

Image formation:

\begin{aligned} Image (i) & = - \log (\frac{Signal (i)}{Air Signal (i)}) \end{aligned}

$\begin{aligned} \text{Image}(i)&=-\color{red}{\log}(\frac{\text{Signal}(i)}{\text{Air Signal}(i)}) \end{aligned}$

3D Reconstruction

Projection (Mono-E):

ρ (β, t) = \int_{L_{i - s}} μ (x, y) d l

$\rho(\beta, t)=\int_{L_{i-s}}\mu(x,y)dl$

If $l=x\cos(\beta)+y\sin(\beta)$ then we have the Radon tranform

Numeric Approximation (Filtered Back Projection, FPB)

μ (x, y) = \frac{Δ β}{2 π} \sum_{β} \underset{weight (e.g. beam geometry)}{\underset{⏟}{w (β, t)}} \cdot (\underset{high-pass filter e.g., H (f) = | f |}{\underset{⏟}{h (t)}} * p (β, t))

$\mu(x,y)=\frac{\Delta\beta}{2\pi}\sum_{\beta}\underbrace{w(\beta, t)}_{\text{weight} \\ \text{(e.g. beam geometry)}}\cdot(\underbrace{h(t)}_{\text{high-pass filter} \\ \text{e.g., } H(f)=\vert f\vert} * p(\beta, t))$

Optimization problem (Iterative Recon)

arg min_{μ (x, y)} ‖ p (β, t) \underset{projection matrix with w (β, t) (e.g. beam geometry)}{\underset{⏟}{R}} \cdot μ (x, y) ‖

$\text{arg}\min_{\mu(x,y)}\Vert p(\beta, t)\underbrace{R}_{\text{projection matrix with }w(\beta,t) \\ \text{(e.g. beam geometry)}}\cdot\mu(x,y)\Vert$

Practical Issues

Beam quantity and quality

Beam Hardening

Anti-scatter grids

Image Noise

Quantum noise:

Discrete nature of photon production (“rain drops”)
Visible effects when Nb of particles are small
Poisson distribution (Gaussian for large numbers)

P {k} = P (X = k) = \frac{λ^{k}}{k!} e^{- λ} S N R = \frac{Av. Signal}{noise} = \frac{N}{\sqrt{N}}

$\mathcal P\{k\}=P(X=k)=\frac{\lambda^k}{k!}e^{-\lambda}\\ SNR=\frac{\text{Av. Signal}}{\text{noise}}=\frac{N}{\sqrt{N}}$

S i g n a l (i) = k \int E \cdot P {η (E) I (E} d E + N (σ)

$Signal(i)=k\int E\cdot\mathcal P\{\eta(E)I(E\}dE+\mathcal N(\sigma)$

Modulation Transfer Function (MTF)

M t = \frac{Modulation of Output Signal}{Modulation of Input Signal} = \frac{M_{o} (f)}{M_{i} (f)} = F c t (f)

$Mt=\frac{\text{Modulation of Output Signal}}{\text{Modulation of Input Signal}} = \frac{M_o(f)}{M_i(f)} = Fct(f)$

Imaging System Optimization

Noise Power Spectrum (NPS)

System Performance

Imaging Systems & Applications

Mammography

Spectral mammography

On trouve un rassemblement de beaucoup de vaisseaux montres par l’iode, etant une indication d’un cancer.

IMED2: X-Ray Imaging

Basic concepts

Exponential Behavior

X-Ray Production

Coolidge Tube

Interaction with Anode Material

Bremsstrahlung

Emitted Spectra

X-Ray Interaction with Matter

Photoelectric Absorption

Compton (Incoherent Scatter)

Rayleigh (Coherent Scatter)

Total attenuation

X-Ray Detection

Primary X-ray image

Photographic Film & Phosphor Plates

Solid State Detectors: Indirect Detection

Summary

Overview

Digital Image Formation

Projection Image

3D Reconstruction

Practical Issues

Beam quantity and quality

Beam Hardening

Anti-scatter grids

Image Noise

Modulation Transfer Function (MTF)

Imaging System Optimization

Noise Power Spectrum (NPS)

System Performance

Imaging Systems & Applications

Mammography

Spectral mammography

Chest X-Ray

Computed Tomography

Wrap-up

X-ray physics

Radiology

IMED2: X-Ray Imaging

Basic concepts

Exponential Behavior

X-Ray Production

Coolidge Tube

Interaction with Anode Material

Bremsstrahlung

Emitted Spectra

X-Ray Interaction with Matter

Photoelectric Absorption

Compton (Incoherent Scatter)

Rayleigh (Coherent Scatter)

Total attenuation

X-Ray Detection

Primary X-ray image

Photographic Film & Phosphor Plates

Solid State Detectors: Indirect Detection

Summary

Overview

Digital Image Formation

Projection Image

3D Reconstruction

Practical Issues

Beam quantity and quality

Beam Hardening

Anti-scatter grids

Image Noise

Modulation Transfer Function (MTF)

Imaging System Optimization

Noise Power Spectrum (NPS)

System Performance

Imaging Systems & Applications

Mammography

Spectral mammography

Chest X-Ray

Computed Tomography

Wrap-up

X-ray physics

Radiology

Further Reading

IMED2: TP3

IMED2: Project

IMED2: 3D/3D registration with labeled atlas