1. Mathematics for Computed Tomography

2. Intensity measurements
Scanning
Patient
X-Ray beam
X-Ray detector
Intensity measurements
Computer Memory

3. Scanning X-ray tube & detectors rotate around patient
(All recent scanners)
Detectors measure radiation transmitted through patient for various pencil beam projections
Relative transmission calculated
Fraction of beam exiting patient
Patient
X-Ray beams

4. CT Detectors electronic / quantitative extremely sensitive
small radiation input differences measureable
output digitized & sent to computer

5. Photon Phate
What can happen to an x-ray photon passing through a material (tissue)?
Material
???
Incoming X-ray
Photon

6. Photon Phate #1: Nothing
Photon exits unaffected
same energy
same direction
Good
These photons form the CT image
Material
Incoming X-ray
Photon
Outgoing X-ray
Photon

7. Photon Phate #2: Absorption
Photon disappears
Its energy is absorbed by material
Good
Creates differential absorption which forms CT image
Bad
Source of patient dose
Material
Incoming X-ray
Photon

8. Photon Phate #3: Scatter
Lower energy photon emerges
energy difference deposited in material
Photon usually emerges in different direction
Bad
Degrades image
Material
Outgoing X-ray
Photon
Incoming X-ray
Photon

9. Photon Beam Attenuation
Anything which removes original photon from beam
absorption
scatter
Material
Incoming X-ray
Photon
Absorption
Material
Incoming X-ray
Photon
Outgoing X-ray
Scatter

10. Example Beam Attenuation (Mono-energy source)
Each cm of material reduces beam intensity 20%
exiting beam intensity 80% of incident for 1 cm absorber
1cm
1cm
1cm
1cm
100
100 * .8 = 80
80 * .8 = 64
64 * .8 = 51
51 * .8 = 41

11. Attenuation Equation for Mono-energetic Photon Beams
I = Ioe-mx
For photons which are neither absorbed nor scattered
I = Exiting beam intensity
Io = Incident beam intensity
e = constant (2.718…)
m = linear attenuation coefficient
property of
absorber material
beam energy
x = absorber thickness
Material I Io x

12. More Realistic CT Example Beam Attenuation for non-uniform Material
4 different materials
4 different attenuation coefficients x
#1 ?
#2 ? #3 ?
#4 ? Io I m1 m2 m3 m4
I = Ioe-(m1+m2+m3+m4)x

13. Effect of Beam Energy on Mono-energetic Beam Attenuation
Low energy photons more easily absorbed
High energy photons more penetrating
All materials attenuate a larger fraction of low than high energy photons
Material
Material 80
<80
100
100
Higher-energy
mono-energetic
beam
Lower-energy
mono-energetic
beam

14. Attenuation Coefficient & Beam Energy
m depends on beam energy as well as material x
#1 ?
#2 ?
#3 ?
#4 ? Io I m1 m2 m3 m4
I = Ioe-mx
I = Ioe-(m1+m2+m3+m4)x

15. Mono-energetic X-ray Beams
Available from radionuclide sources
Not used in CT
Radionuclide intensity much lower than that of x-ray tube

16. Mono-energetic beam equation!
X-ray Tube Beam x
High intensity
Produces poly-energetic beam
Characteristic radiation
Bremsstrahlung #1 #2 #3 #4 Io I m1 m2 m3 m4
Mono-energetic beam equation!
I = Ioe-(m1+m2+m3+m4)x

17. Beam Hardening Complication
Beam quality changes as it travels through absorber
greater fraction of low-energy photons removed from beam
Average beam energy increases
1cm
1cm
1cm
1cm A B C D E
Fewer Photons but kVavg(B) > kVavg(A)
Fewer Photons but kVavg(C) > kVavg(B)
Fewer Photons but kVavg(D) > kVavg(C)
Fewer Photons but kVavg(E) > kVavg(D)

18. Beam Hardening Complication
Attenuation coefficients mn depend on beam energy!!!
Beam spectrum incident on each block unknown
Four m’s, each for a different & unknown energy
1cm
1cm
1cm
1cm m1 m2 m3 m4
I = Ioe-(m1+m2+m3+m4)x

19. I = Ioe-(m1+m2+m3+m4 +...)x Reconstruction
Scanner measures “I” for thousands of pencil beam projections
Computer calculates tens of thousands of attenuation coefficients
one for each pixel
Computer must correct for beam hardening
effect of increase in average beam energy from one side of projection to other
I = Ioe-(m1+m2+m3+m4 +...)x

20. Why is CT done with High kV’s?
Less dependence of attenuation coefficient on photon energy
Attenuation coefficient changes less at higher kV’s
High kV provides high radiation flux at detector

21. Image Reconstruction IA = Ioe-(mA1+mA2+mA3+mA4 +...)x
One of these equations for every projection line
IA = Ioe-(mA1+mA2+mA3+mA4 +...)x
Projection #A
Projection #B
IB = Ioe-(mB1+mB2+mB3+mB4 +...)x
IC = Ioe-(mC1+mC2+mC3+mC4 +...)x
Projection #C

22. Reconstruction Calculates:
Image Reconstruction *
The equations
IA, IB, IC, ...
What We Measure:
Projection #A
IA = Ioe-(mA1+mA2+mA3+mA4 +...)x
mA1, mA2, mA3, ...
Reconstruction Calculates:
mB1, mB2, mB3, ...
mC1, mC2, mC3, ...
Etc.
Projection #B
IB = Ioe-(mB1+mB2+mB3+mB4 +...)x
Projection #C
IC = Ioe-(mC1+mC2+mC3+mC4 +...)x

23. CT (Hounsfield) Number
Calculated from reconstructed pixel attenuation coefficient
(mt - mW)
CT # = 1000 X mW
Where:
ut = linear attenuation coefficient for tissue in pixel
uW = linear attenuation coefficient for water

24. CT Numbers for Special Stuff
Bone: +1000
Water: 0
Air: -1000
(mt - mW)
CT # = 1000 X mW