Mathematics for
Computed
Tomography
Scanning
Patient
X-Ray beam
X-Ray detector
Intensity
measurements
Computer
Memory
Scanning
 X-ray tube & detectors rotate around
Patient
patient
 (All recent scanners)
 Detectors measure radiation
transmitted through patient for
various pencil beam projections
 Relative transmission calculated

Fraction of beam exiting patient
X-Ray beams
CT Detectors
 electronic / quantitative
 extremely sensitive
 small radiation input
differences measureable
 output digitized & sent to
computer
Photon Phate
 What can happen to an x-ray photon passing
through a material (tissue)?
Incoming X-ray
Photon
Material
???
Photon Phate #1: Nothing
 Photon exits unaffected
 same energy
 same direction
 Good
 These photons form the CT image
Incoming X-ray
Photon
Material
Outgoing X-ray
Photon
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
Incoming X-ray
Photon
Material
Photon Phate #3: Scatter
 Lower energy photon emerges
 energy difference deposited in material
 Photon usually emerges in different direction
 Bad
 Degrades image
Incoming X-ray
Photon
Material
Outgoing X-ray
Photon
Photon Beam Attenuation
 Anything which removes original photon from
beam
 absorption
 scatter
Incoming X-ray
Photon
Incoming X-ray
Photon
Material
Material
Outgoing X-ray
Photon
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
100
1cm
100 * .8 =
80
1cm
80 * .8 =
64
1cm
64 * .8 =
51
51 * .8 =
41
Attenuation Equation for
Mono-energetic Photon Beams
I = Io
-mx
e
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
For photons
which are neither
absorbed nor
scattered
Material
Io
x
I
More Realistic CT Example Beam
Attenuation for non-uniform Material
 4 different materials
 4 different attenuation coefficients
x
Io
#1 #2 #3 #4
? ? ? ?
I
m1 m2 m3 m4
I = Io
-(m
+m
+m
+m
)x
1
2
3
4
e
Effect of Beam Energy on Monoenergetic 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
100
Higher-energy
mono-energetic
beam
80
100
Material
Lower-energy
mono-energetic
beam
<80
Attenuation Coefficient & Beam Energy
 m depends on beam energy as well as material
x
Io
#1 #2 #3 #4
? ? ? ?
I
m1 m2 m3 m4
I = Io
-mx
e
I = Io
-(m
+m
+m
+m
)x
1
2
3
4
e
Mono-energetic X-ray Beams
 Available from radionuclide sources
 Not used in CT
 Radionuclide intensity much lower than
that of x-ray tube
X-ray Tube Beam
 High intensity
 Produces poly-energetic beam
 Characteristic radiation
 Bremsstrahlung
x
#1 #2 #3 #4
Io
I
m1 m2 m3 m4
I = Io
-(m
+m
+m
+m
)x
1
2
3
4
e
Beam Hardening Complication
 Beam quality changes as it travels through absorber
 greater fraction of low-energy photons removed from beam
 Average beam energy increases
A
1cm
B
Fewer Photons but
kVavg(B) > kVavg(A)
1cm
C
1cm
Fewer Photons but
kVavg(C) > kVavg(B)
D
1cm
Fewer Photons but
kVavg(D) > kVavg(C)
E
Fewer Photons but
kVavg(E) > kVavg(D)
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 = Io
-(m
+m
+m
+m
)x
1
2
3
4
e
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 = Io
-(m
+m
+m
+m
+...)x
1
2
3
4
e
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
Image Reconstruction
Projection #A
One of these equations for
every projection line
IA = Ioe-(mA1+mA2+mA3+mA4 +...)x
Projection #B
IB = Ioe-(mB1+mB2+mB3+mB4 +...)x
Projection #C
IC = Ioe-(mC1+mC2+mC3+mC4 +...)x
Image Reconstruction
*
Projection #A
What We Measure:
IA, IB, IC, ...
IA = Ioe-(mA1+mA2+mA3+mA4 +...)x
Projection #B
IB = Ioe-(mB1+mB2+mB3+mB4 +...)x
Projection #C
IC = Ioe-(mC1+mC2+mC3+mC4 +...)x
Reconstruction
Calculates:
mA1, mA2, mA3, ...
mB1, mB2, mB3, ...
mC1, mC2, mC3, ...
Etc.
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
CT Numbers for Special Stuff
 Bone: +1000
 Water: 0
 Air: -1000
(mt - mW)
CT # = 1000 X -----------mW