IMED2: X-Ray Imaging

Basic concepts

Exponential Behavior

Exponential Decay/Growth:

\[\frac{\delta N}{\delta t} = \pm\lambda\]

X-Ray Production

Coolidge Tube

1. High-voltage Generator for heating ($U_h$) and cathode/anode ($U_a$)
2. Filaments is heated and gives off electrons
3. Electrons are accelerated from cathode to anode
4. Electrons collide with the anode material and accelerate other electrons
5. 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:

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

Emitted Spectra

Pour un tungstene:

X-Ray Interaction with Matter

X-Ray photon life span:

1. Photon is transmitted through the matter
2. Photon is absorbed (end of life)
3. 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):

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

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

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

• Klein-Nishina coefficient

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

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)

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

Total attenuation

\[\delta I = -\sigma\cdot\rho\cdot I_0\cdot\Delta T\]

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

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

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

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

X-Ray Detection

Primary X-ray image

Photographic Film & Phosphor Plates

Solid State Detectors: Indirect Detection

Summary

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

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:

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

Image formation:

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

3D Reconstruction

Projection (Mono-E):

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

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

1. Numeric Approximation (Filtered Back Projection, FPB)

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

1. Optimization problem (Iterative Recon)

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

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)

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

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

Modulation Transfer Function (MTF)

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

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.

Chest X-Ray

Computed Tomography

Wrap-up

X-ray physics

• X-ray production: Coolidge Tube, Bremsstrahlung, Characterisics X-Rays
• Interaction with matter: photoelectric, compton, Rayleigh
• X-ray detectors: films, image intensifiers, solid state detectors

Radiology

• Image formation
• Image quality
• 3D reconstruction
• Clinical application examples