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X-Ray Physics. Assumptions: Matter is composed of discrete particles (i.e. electrons, nucleus) Distance between particles >> particle size X-ray photons are small particles Interact with body in binomial process Pass through body with probability p

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slide1

X-Ray Physics

Assumptions:

Matter is composed of discrete particles (i.e. electrons, nucleus)

Distance between particles >> particle size

X-ray photons are small particles

Interact with body in binomial process

Pass through body with probability p

Interact with body with probability 1-p (Absorption or scatter)

No scatter photons for now (i.e. receive photons at original energy or not at all.

slide2

N |∆x|N + ∆N

The number of interactions (removals)  number of x-ray photons and ∆x

∆N = -µN∆x

µ = linear attenuation coefficient (units cm-1)

slide3

Id (x,y) = ∫ I0 () exp [ -∫ u (x,y,z,) dz] d 

Integrate over  and depth.

If a single energy I0() = I0 ( - o),

If homogeneous material, then µ (x,y,z, 0) = µ0

Id (x,y) = I0 e -µ0l

slide4

Notation

I0

I = I0 e - l

l

Often to simplify discussion in the book or

problems on homework, the intensity transmission, t, will be

given for an object instead of the attenuation coefficient 

t = I/Io = e-µl

slide5

X-ray Source

Accelerate electrons towards anode. Three types of events can happen.

  • Collision events -> heat
  • Photoelectric effect
  • Braking of electron by nucleus creates an x-ray (Bremstrahlung effect)

Typically Tungsten Target

High melting point

High atomic number

Andrew Webb, Introduction to Biomedical Imaging, 2003, Wiley-Interscience.

slide6

Thin Target X-ray Formation

There are different interactions creating X-ray photons between the accelerated electrons and the target. Maximum energy is created when an electron gives all of its energy, 0 , to one photon. Or, the electron can produce n photons, each with energy 0/n. Or it can produce a number of events in between. Interestingly, this process creates a relatively uniform spectrum. Power output is proportional to 02

Intensity

= nh

0

Photon energy spectrum

slide7

Thick Target X-ray Formation

We can model target as a series of thin targets. Electrons

successively loses energy as they moves deeper into the target.

Gun 

X-rays

Relative

Intensity

0

Each layer produces a flat energy

spectrum with decreasing

peak energy level.

slide8

Thick Target X-ray Formation

In the limit as the thin target planes get thinner, a wedge

shaped energy profile is generated.

Relative

Intensity

0

Again, 0 is the energy of the accelerated electrons.

slide9

Thick Target X-ray Formation

Andrew Webb, Introduction to Biomedical Imaging, 2003, Wiley-Interscience.

(

Lower energy photons are absorbed with

aluminum to block radiation that will be absorbed

by surface of body and won’t contribute to image.

The photoelectric effect(details coming in attenuation section)

will create significant spikes of energy when

accelerated electrons collide with tightly bound electrons, usually in the K shell.

slide10

How do we describe attenuation of X-rays by body?

µ = f(Z, ) Attenuation a function of atomic number Z and energy 

Solving the differential equation suggested by the second slide of this lecture,

dN = -µNdx

Ninx Nout

µ

Noutx

∫ dN/N = -µ ∫ dx

Nin 0

ln (Nout/Nin) = -µx

Nout = Nin e-µx

slide11

If material attenuation varies in x, we can write attenuation as u(x)

Nout = Nin e -∫µ(x) dx

Io photons/cm2

(µ (x,y,z))

Id (x,y) = I0 exp [ -∫ µ(x,y,z) dz]

Assume: perfectly collimated beam ( for now),

perfect detector

no loss of resolution

Actually recall that attenuation is also a function of energy ,

µ = µ(x,y,z, ). We will often assume a single energy source, I0 = I0(). After analyzing a single energy, we can add the effects of other energies by superposition.

Detector Plane

Id (x,y)

slide12

Diagnostic Range

50 keV < E < 150 keV

 ≈ 0.5%

Rotate anode to prevent melting

What parameters do we have to play with?

  • Current
    • Units are in mA
  • Time
    • Units · sec

3. Energy ( keV)

slide13

Mass Attenuation Coefficient

Since mass is providing the attenuation, we will consider the

linear attenuation coefficient, µ, as normalized to the density of the object first. This is termed the mass attenuation coefficient.

µ/p cm2/gm

We simply remultiply by the density to return to the linear

attenuation coefficient. For example:

t = e- (µ/p)pl

Mixture

µ/p = (µ1/p1) w1 + (µ2/p2) w2 + …

w0 = fraction weight of each element

slide14

Mechanisms of Interaction

1. Coherent scatter or Rayleigh (Small significance)

2. Photoelectric absorption

3. Compton Scattering – Most serious significance

slide15

Physical Basis of

Attenuation Coefficient

Coherent Scattering - Rayleigh

Coherent scattering varies

over diagnostic energy range as:

µ/p  1/2

••

••

slide16

Photoelectric Effect

Andrew Webb, Introduction to Biomedical Imaging, 2003, Wiley-Interscience.

Photoelectric effect varies

over diagnostic energy range as:

log /r

  1

p 3

log  ( Photon energy)

K-edge

slide17

Photoelectric Effect

Longest photoelectron range 0.03 cm

Fluorescent radiation example:

Calcium 4 keV Too low to be of interest.

Quickly absorbed

Items introduced to the body:

Ba, Iodine have K-lines close to region of diagnostic interest.

slide18

We can use K-edge to dramatically increase absorption in areas where material is injected, ingested, etc.

Photoelectric linear attenuation varies by Z4/ 3

ln /r

Log () Photon energy

K edge

µ/r 1/ 3

slide19

Compton Scatter

Interaction of photons and electrons produce scattered photons of reduced energy.

When will this be a problem?

Is reduced energy a problem?

Is change in direction a problem?

E’ photon

E

a

Outer

Shell

electron

v Electron (“recoil”)

slide20

Satisfy Conservation of Energy and Momentum

(m-mo = electron mass : relativistic effects)

Conservation of Momentum

2)

3)

slide21

Energy of recoil or Compton electron

can be rewritten as h = 6.63 x 10-34 Jsec

eV = 1.62 x 10-19 J

mo = 9.31 x 10-31 kg

∆ = h/ moc (1 - cos ) = 0.0241 A0 (1 - cos )

∆ at  = π = 0.048 Angstroms

Energy of Compton photon

slide22

Greatest effect ∆/  occurs at high energy

At 50 kev, x-ray wavelength is .2 Angstroms

Low energy  small change in energy

High energy higher change in energy

slide23

µ/r electron mass density

Unfortunately, almost all elements have electron mass density ≈ 3 x 10 23 electrons/gram

Hydrogen (exception) ≈ 6.0 x 1023 electrons/gram

Mass attenuation coefficient (µ/r) for Compton scattering is

Z independent

Compton Linear Attenuation Coefficient µ  p

Avg atomic number for Bone ~ 20

Avg atomic number for body 7 or 8

slide24

Rayleigh, Compton, Photoelectric are independent sources of attenuation

t = I/I0 = e-µl = exp [ -(uR + up + uc)l]

µ () ≈ pNg { f() + CR (Z2/ 1.9) + Cp (Z3.8/ 3.2)}

Compton Rayleigh Photoelectric

Ng electrons/gram ( electron mass density)

So rNg is electrons/cm3

Ng = NA (Z/A) ≈ NA /2 (all but H) A = atomic mass

f() = 0.597 x 10-24 exp [ -0.0028 (-30)]

for 50 keV to 200 keV  in keV

attenuation mechanisms
Attenuation Mechanisms

Andrew Webb, Introduction to Biomedical Imaging, 2003, Wiley-Interscience.

Curve on left shows how photoelectric effects dominates at

lower energies and how Compton effect dominates at higher energies.

Curve on right shows that mass attenuation coefficient varies little over

100 kev. Ideally, we would image at lower energies to create contrast.

photoelectric vs compton effect
Photoelectric vs. Compton Effect

Macovski, Medical Imaging Systems, Prentice-Hall

The curve above shows that the Compton effect dominates at higher energy values

as a function of atomic number.

Ideally, we would like to use lower energies to use the higher contrast available with

The photoelectric effect. Higher energies are needed however as the body gets thicker.