prince&links-medical imaging signals&systems allslides 2009

Medical Imaging Signals and Systems

Jerry L. Prince

Johns Hopkins University

August 20, 2009

Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 1 / 412

AcknowledgementsThese notes are intended to be used with the thetextbook:

▶ Jerry L. Prince and Jonathan M. Links,“Medical Imaging Signals and Systems,” UpperSaddle River: Pearson Prentice Hall, 2006.

Images and figures lacking a specific bibliographiccitation are either taken from this book and thecopyright is owned by Pearson Prentice Hall or werehand drawn by Jerry Prince.

Images and figures with specific bibliographiccitations are used with permission of the copyrightholder.


Outline

Outline I

1 Introduction to Medical Imaging Systems

2 Multidimensional Signal Processing

3 Image Quality

4 Physics of Radiography

5 Projection Radiography


Outline

Outline II6 Computed Tomography

7 Physics of Nuclear Medicine

8 Planar Scintigraphy

9 Emission Tomography

10 Ultrasound Physics

11 Ultrasound Imaging


Outline

Outline III12 Physics of Magnetic Resonance

13 Magnetic Resonance Imaging


Introduction to Medical Imaging Systems

1 Introduction to Medical Imaging SystemsOverall PerspectivePossible objectivesSignals Examples


Introduction to Medical Imaging Systems Overall Perspective

Overall PerspectiveCourse breakdown

▶ 1/3 physics▶ 1/3 instrumentation▶ 1/3 signal processing

Understand systems from a “signals” viewpoint:

input signal→ system or process→ output signal


Introduction to Medical Imaging Systems Overall Perspective

A Signal Example

ExampleInput signal: �(x , y) is the linear attenuationcoefficient for x-rays

Process (integration over x variable):

g(y) =

∫�(x , y)dx

Output signal: g(y)


Introduction to Medical Imaging Systems Possible objectives

Possible objectivesunderstand “noise” or “artifacts” created by system

understand “contrast” in input and output

process output to create a “picture” of input


Introduction to Medical Imaging Systems Signals Examples

Examples of Signals in Medical Imaging�(x , y , z), linear attenuation coefficient in x-rays

h(x , y , z), CT numbers in computed tomography

A(x , y , z), radioactivity in nuclear medicine

Chest X-ray Abdominal CT Cardiac SPECT



More ExamplesPD(x , y , z), proton density in MRI imaging

T1(x , y , z), longitudinal relaxation time in MRI

T2(x , y , z), transverse relaxation time in MRI

PD-weighted T2-weighted T1-weighted



More ExamplesR(x , y , z), reflectivity in ultrasound imaging

vR(x , y , z), range component of velocity in Dopplerultrasound

11-week Embryo Fetus Heart


Multidimensional Signal Processing

2 Multidimensional Signal ProcessingMultidimensional SignalsDelta FunctionsSystemsFourier TransformRect and SincHankel TransformSamplingAliasingArea Detectors


Multidimensional Signal Processing Multidimensional Signals

1D, 2D, and 3D SignalsA 1D signal is:

▶ f (t), a function of one variable, or▶ a waveform, or▶ a graph (a collection of points in a 2D space)



A 2D signal is:▶ f (x , y), a function of two variables, or▶ an image, or▶ a graph (a collection of points in a 3D space)



A 3D signal is:▶ f (x , y , z), a function of three variables, or▶ a “volumetric image,” or▶ a graph (a collection of points in a 4D space)

We focus (mostly) on 2D signals in this course

Separable signals:▶ f (x , y) = f1(x)f2(y)▶ f (x , y , z) = f1(x)f2(y)f3(z)


Multidimensional Signal Processing Delta Functions

Delta FunctionsThe 1D delta or impulse “function” is defined bytwo properties:

�(x) = 0 , x ∕= 0∫∞−∞ f (x)�(x)dx = f (0)



Properties of the Delta FunctionThe area of �(x) is unity∫ ∞

−∞�(x)dx = 1

A 2D delta function �(x , y) is defined by

�(x , y) = 0 , (x , y) ∕= 0∫∞−∞∫∞−∞ f (x , y)�(x , y)dx dy = f (0, 0)

A 3D delta function is analogous.



More PropertiesProperties of delta functions:

�(−x) = �(x) even

�(x , y) = �(x)�(y) separable∫ ∞−∞

f (�)�(� − x)d� = f (x) sifting

2D sifting property∫ ∞−∞

∫ ∞−∞

f (�, �)�(� − x , � − y)d�d� = f (x , y)



Point Source Modeldelta function models a point source

▶ metal bead in x-ray▶ vial of radioactivity in nuclear medicine▶ vitamin E pill in magnetic resonance imaging▶ small bubble or microcalcification in ultrasound


Multidimensional Signal Processing Systems

Transformations of SignalsComponents of a transformation:

▶ Input: f▶ System: ℋ[⋅]▶ Output: g

The impulse response or point spread function dueto an impulse at (�, �) is

h(x , y ; �, �) = ℋ[�(x − �, y − �)]

h(x , y ; �, �) is a 2D signal parameterized by a 2Dvector



A linear system satisfies:

ℋ[w1f1 + w2f2] = w1ℋ[f1] + w2ℋ[f2]

for all signals f1 and f2 and weights w1 and w2.

A linear system satisfies the superposition integral

g(x , y) =

∫ ∞−∞

∫ ∞−∞

h(x , y ; �, �)f (�, �)d�d�

We model most medical imaging systems as linear.



Shift-Invariant SystemsA system is shift-invariant is

g(x − x0, y − y0) = ℋ[f (x − x0, y − y0)]

for every (x0, y0) and f (⋅, ⋅).

A linear shift-invariant (LSI) system yields

h(x , y ; �, �)→ h(x − �, y − �)

[Watch out for abuse of notation]



Convolution IntegralAn LSI system satisfies the convolution integral

g(x , y) =

∫ ∞−∞

∫ ∞−∞

h(x − �, y − �)f (�, �)d�d�

which is abbreviated as

g(x , y) = h(x , y) ∗ f (x , y)

We model most medical imaging systems as LSI


Multidimensional Signal Processing Fourier Transform

LSI Systems and Complex ExponentialsA 2D complex exponential signal is

e j2�(ux+vy) = e j2�uxe j2�vy i.e., separable

wheree j2�ux = cos 2�ux + j sin 2�ux

The response of an LSI system to

f (x , y) = e j2�(ux+vy)

isg(x , y) = H(u, v)e j2�(ux+vy)



The function

H(u, v) =

∫ ∞−∞

∫ ∞−∞

h(x , y)e−j2�(ux+vy)dxdy

H(u, v) is called the Fourier transform of h(x , y).

The inverse Fourier transform of H(u, v) is

h(x , y) =

∫ ∞−∞

∫ ∞−∞

H(u, v)e+j2�(ux+vy)dudv



Magnitude of the 2D Fourier Transform



Comments on the Fourier TransformNotation:

F (u, v) = ℱ{f }

=

∫ ∞−∞

∫ ∞−∞

f (x , y)e−j2�(ux+vy)dxdy

f (x , y) = ℱ−1{F}

=

∫ ∞−∞

∫ ∞−∞

F (u, v)e+j2�(ux+vy)dudv

e j2�(ux+vy) is a complex sinusoid “oriented” in the(u, v) direction

2�ux has units of radians



⇒ ux is unitless

⇒ x has units of length, e.g., cm or mm

⇒ u has units of inverse length, e.g., cm−1 ormm−1.

u is referred to as (cyclic) spatial frequency

The 1D Fourier transform pair is given by

F (u) =

∫ ∞−∞

f (x)e−j2�uxdx

f (x) =

∫ ∞−∞

F (u)e+j2�uxdu



Properties of the Fourier Transform[Refer to text for complete list]

Linearity:

ℱ{w1f1 + w2f2} = w1F1 + w2F2

Scaling:

ℱ{f (�x , �y)} =1

∣��∣F (

u

�,v

�)

Shifting:

ℱ{f (x − �, y − �)} = F (u, v)e−j2�(u�+v�)

ℱ{f (x , y)e+j2�(�x+�y)} = F (u − �, v − �)



Convolution:

ℱ{f1 ∗ f2} = F1F2

Correlation:

ℱ{∫ ∞−∞

∫ ∞−∞

f1(�, �)f ∗2 (x + �, y + �)d�d�

}= F1(u, v)F ∗2 (u, v)

Separable input: If f (x , y) = f1(x)f2(y) then

ℱ{f (x , y)} = F1(u)F2(v)



Parseval’s theorem:∫ ∞−∞

∫ ∞−∞∣f (x , y)∣2dxdy

=

∫ ∞−∞

∫ ∞−∞∣F (u, v)∣2dudv

Product:

ℱ{f1(x , y)f2(x , y)} = F1(u, v) ∗ F2(u, v)

Impulse:ℱ{�(x , y)} = 1

Constant:ℱ{1} = �(u, v)



Sinusoid (1D):

ℱ{sin 2�u0x} =1

2j[�(u − u0)− �(u + u0)]

ℱ{cos 2�u0x} =1

2[�(u − u0) + �(u + u0)]

Sinusoid (2D):

ℱ{sin 2�(u0x + v0y)}

=1

2j[�(u − u0, v − v0)− �(u + u0, v + v0)]

ℱ{cos 2�(u0x + v0y)}

=1

2[�(u − u0, v − v0) + �(u + u0, v + v0)]


Multidimensional Signal Processing Rect and Sinc

Rect and SincRect function: (“gate” or “pedestal”)

rect(x) =

{1 ∣x ∣ ≤ 1/20 otherwise

Sinc function:

sinc(x) =sin �x

�x


Multidimensional Signal Processing Rect and Sinc

Fourier transform relationship:

ℱ{rect(x)} = sinc(u)

x

sinc( )x

1

−1 1 2 3 40−2−3−4

rect( )x

x1 / 2−1 / 2

1

0

(a) (b)


Multidimensional Signal Processing Hankel Transform

RotationRotation:

f�(x , y) = f (x cos � − y sin �, x sin � + y cos �)

Fourier transform rotates also

ℱ2D(f�)(u, v)

= F (u cos � − v sin �, u sin � + v cos �)



Circular Symmetry2D signal is circularly symmetric if

f�(x , y) = f (x , y) , for every �

ℱ2D(f�)(u, v) is also circularly symmetric

f (x , y) and F (u, v) are functions of radii only

f (x , y) = f (r)

andF (u, v) = F (q)



Hankel TransformFourier transform of circularly symmetric objects isdescribed by the Hankel transform

F (q) = 2�

∫ ∞0

f (r)J0(2�qr) r dr

J0(r) is zero-order Bessel function of the first kind

J0(r) =1

�

∫ �

0

cos(r sin�) d�

Example pair:

ℋ{exp{−�r 2}} = exp{−�q2}Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 38 / 412

Multidimensional Signal Processing Sampling

Sampling

x

y

∆x

∆y

∆x

∆y

x

y

Point sampling:

f [m, n] = f (mΔx , nΔy)



Impulse TrainsImpulse train or comb or shah function:

comb(x) =∞∑

n=−∞�(x − n)

Fourier transform relationship

ℱ{comb(x)} = comb(u)



Sampling FunctionThe sampling function:

�s(x ; Δx) =∞∑

n=−∞�(x − nΔx)

Impulse scaling property:

�(ax) =1

∣a∣�(x)

Relation to shah/comb function:

�s(x ; Δx) =1

Δxcomb

( x

Δx

)Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 41 / 412


Sampling Model (see text for 2D)Sampled signal

fs(x) = f (x)�s(x ; Δx)

fs(x) contains the same information as

f [k] = f (kΔx)

Fourier transform of fs(x):

Fs(u) = F (u) ∗ ℱ{�s(x ; Δx)}



Sampled SpectrumFourier transform of sampling function:

ℱ{�s(x ; Δx)} = comb(Δxu)

=1

Δx

∞∑−∞

�(u − n

Δx)

Sampled spectrum is therefore:

Fs(u) =1

Δx

∞∑−∞

F (u) ∗ �(u − n

Δx)



Sampled Spectrum in 2D

u

vF u v( , )

V

−V

U−U

v

u

F u vs ( , )

(a) (b)

1 2/ ∆y

−1 2/ ∆y

−

1

2∆x

1 / ∆y

−1 / ∆y

−1 / ∆x 1 / ∆x

U−U

V

−V

1

2∆x

v

u

F u vs ( , )

−1 /∆y

−1 / ∆x 1 / ∆x

1 /∆y

U

−V

V

−U

aliasing



Sampling TheoremThe (spatial) sampling frequency is:

us =1

Δx

Let U be the highest frequency in F (u).

Then sampled spectra do not overlap if

us > 2U

2U is called the Nyquist rate


Multidimensional Signal Processing Aliasing

AliasingAliasing occurs if us < 2U .

▶ Overlapping sampled spectra.▶ Corruption of high frequencies▶ Artifacts are high frequency patternsv

u

F u vs ( , )

−1 /∆y

−1 / ∆x 1 / ∆x

1 /∆y

U

−V

V

−U

aliasing


Multidimensional Signal Processing Aliasing

Anti-aliasing FiltersSuppose:

▶ us = 1/Δx▶ Highest frequency in f (x) is U .

Filter f (x):▶ before sampling▶ Use low pass filter with cutoff frequency us/2.


Multidimensional Signal Processing Area Detectors

Area Detector AnalysisShape of detector: p(x) [maybe rect(x/D)]

Area detector sampling model:

fs(x) = [p(x) ∗ f (x)]�s(x ; Δx)

Fourier domain:

Fs(u) = [P(u)F (u)] ∗ comb(Δxu)

= [P(u)F (u)] ∗ 1

Δx

∞∑n=−∞

�(u − n

Δx

)


Image Quality

3 Image QualityBasic NotionsContrastResolutionNoiseArtifactsAccuracy


Image Quality Basic Notions

What is Quality?What makes a good medical image?

▶ physics-oriented answer:faithful representation of the truth

▶ task-oriented answer:discrimination of healthy vs. diseasedtissues


Image Quality Basic Notions

Measures of QualityPhysics-oriented issues:

▶ contrast, resolution▶ noise, artifacts, distortion▶ accuracy

Task-oriented issues:▶ sensitivity, specificity▶ diagnostic accuracy


Image Quality Contrast

Contrast or ModulationSinusoidal image brightness function:

f (x , y) =fmax + fmin

2

+fmax − fmin

2sin(2�u0x)

Contrast = modulation =

mf =amplitude

average=

fmax − fmin

fmax + fmin



Sinusoidal Signals with Different Contrast



Sinusoid Input/Output in a Linear SystemInput: (assume f ≥ 0)

f (x , y) = A + B sin(2�u0x)

Assume impulse response h(x , y) is real

Output:

g(x , y) = H(0, 0)A

+ ∣H(u0, 0)∣B sin[2�u0x + ∠H(u0, 0)]



Contrast Change in a Linear SystemInput contrast: mf = B/A

Output contrast:

mg =∣H(u0, 0)∣BH(0, 0)A

=∣H(u0, 0)∣H(0, 0)

mf

A B+

A B-

A

A B H u+ | ( , )|0

A B H u- | ( , )|0

mB

Af = m

B

AH ug = | ( , )|0

medical

imaging

system

input f x y( , ) output g x y( , )



Modulation Transfer FunctionModulation transfer function:

MTF(u) =mg

mf=∣H(u, 0)∣H(0, 0)

spatial frequency u

0 8. mm-1

0

1 0.

MTF( )u

0 6.0 2. 0.4 1 0. 1 2. 1.4

0 5.



More on Modulation Transfer FunctionGeneral case

MTF(u, v) =∣H(u, v)∣H(0, 0)

MTF is partial characterization of real system

Rule of thumb:▶ ∣H(u, v)∣ holds 1/8 of info▶ ∠H(u, v) hold 7/8 of info



Contrast is Related to Resolution

decreasing contrast

MTF

outp

ut si

gna

l



Local ContrastNon-sinusoidal signals: identify

▶ target intensity: ft▶ background intensity: fb

Local contrast:

C =ft − fbfb

Optional:▶ C (%) = C × 100%▶ C (abs) = ∣C ∣


Image Quality Resolution

Resolution: Bar Phantom



Bar Phantom Properties50% duty cycle

Material depends on modality▶ metal or plexiglass bars▶ tubes of radioactivity

resolution defined as the highest line density suchthat lines can be distinguished

units: line pairs (lp) per distance▶ gamma camera: 2–3 lp/cm▶ CT: 2 lp/mm▶ chest x-ray: 6–8 lp/mm



Resolution: Line ResponseLine “function”:

f (x , y) = �(x)

Line response:

l(x) = S{�(x)} =

∫ ∞−∞

h(x , �)d�

Relation to MTF:

MTF(u) =∣L(u)∣L(0)



Resolution: Line Separation



Resolution: FWHMFull Width at Half Maximum (FWHM)


Image Quality Noise

Noise: Random VariablesTypical imaging model:

g(x , y) = f (x , y) ∗ h(x , y) + N(x , y)

N(x , y) is noise

N(x , y) is a random variable at each (x , y)

N(x , y) could be continuous or discrete

Probability Distribution Function (PDF)

PN(�) = Pr[N ≤ �]


Image Quality Noise

Continuous Random VariablesProbability density function (pdf):

pN(�) =dPN(�)

d�

Mean:

�N =

∫ ∞−∞

�pN(�)d�

Variance:

�2N =

∫ ∞−∞

(� − �)2pN(�)d�

Standard deviation:

�N =√�2N


Image Quality Noise

Gaussian Random Variablepdf

pN(�) =1√

2��2e−(�−�)2/2�2

mean:�N = �

variance:�2N = �2

standard deviation:

�N = �


Image Quality Noise

Discrete Random VariablesProbability mass function (PMF):

pN(�i) = Pr[N = �i ]

Mean:�N =

∑all �i

�ipN(�i)

Variance:

�2N =

∑all �i

(�i − �N)2pN(�i)

Standard deviation:

�N =√�2N


Image Quality Noise

Poisson Random VariablePMF

pN(k) =ake−a

k!, for k = 0, 1, . . .

mean:�N = a

variance:�2N = a

standard deviation:

�N =√a


Image Quality Noise

Sum of Independent Random VariablesLet N and M be joint random variables

Let Q = N + M

Then�Q = �N + �M

If N and M are independent then

�2Q = �2

N + �2M


Image Quality Noise

Images Degrade with NoiseNoise level depends on modality and acquisitionparameters

Fast imaging is almost always noisier

Low dose imaging is almost always noisier


Image Quality Noise

Signals in NoiseSignal is f

Noise is N

Signal-to-noise ratio

SNRa =amplitude(f )

amplitude(N)

SNRp =power(f )

power(N)

Hint: Units must match


Image Quality Noise

More on Signal-to-noiseSNR in decibels

SNR(dB) = 20 log10 SNRa

SNR(dB) = 10 log10 SNRp

Common example of SNR▶ signal height is A▶ noise standard deviation is �N▶ SNR is then

SNRa =A

�N


Image Quality Noise

Noise and Blurring Degrade Quality


Image Quality Artifacts

Nonrandom EffectsArtifacts: image features that do not correspond toa real object, and are not due to noise

▶ star artifact, beam hardening artifact▶ ring artifact, ghosts

Distortion: geometric or intensity changes notcorresponding to the real object

▶ magnification▶ barrel or pincushion distortion▶ quantization, saturation


Image Quality Artifacts

Common Artifacts

(a) motion(b) star(c) hardening(d) ring


Image Quality Accuracy

AccuracyAccuracy:

▶ conformity to truth→ quantitative accuracy

▶ clinical utility→ diagnostic accuracy

Quantitative accuracy:▶ numerical accuracy: bias, precision▶ geometric accuracy: dimensions



Diagnostic QualityContingency table:

Disease

+ −

+ a b

Test

− c d

Variables:

a = # w/ disease & test says disease

b = # w/o disease & test says disease

c = # w/ disease & test says normal

d = # w/o disease & test says normal



Diagnostic Accuracy

sensitivity =a

a + c

specificity =d

b + d

diagnostic accuracy =a + d

a + b + c + d

Disease

+ −

+ a b

Test

− c d



Disease Prevalence

positive predictive value =a

a + b

negative predictive value =d

c + d

prevalence =a + c

a + b + c + d

Disease

+ −

+ a b

Test

− c d


Physics of Radiography

4 Physics of RadiographyX-ray ModalitiesAtomic StructureIonizing RadiationEnergetic ElectronsElectromagnetic RadiationEM StrengthEM AttenuationEM Dose


Physics of Radiography X-ray Modalities

X-ray ModalitiesChest x-rays

Mammography

Dental x-rays

Fluoroscopy

Angiography

Computed tomography

These do not involve radioactivity


Physics of Radiography Atomic Structure

Atomic Structurenucleons = {protons, neutrons}mass number A is # nucleons

atomic number Z is # protons

element symbol X is redundant with Z

nuclide is particular combination of nucleons▶

ZAX

▶ X -A



ElectronsOrbit in shells

Shell Number n Shell Label # Electrons 2n2

1 K ≤ 22 L ≤ 83 M ≤ 184 N ≤ 32



Electron Binding EnergyBasic principle:

bound energy < unbound energy + electron energy

Binding energy is difference

Binding energy of hydrogen electron: 13.6 eV

1 eV is the kinetic energy gained by an electron thatis accelerated across a one (1) volt potential



Ionization and ExcitationIonization is “knocking” an electron out of atom

▶ creates electron + ion

Excitation is “knocking” an electron to a higherorbit



Characteristic RadiationWhat happens to ionized or excited atom?

Return to ground state by rearrangement ofelectrons

Causes atom to give off energy

Energy given off as characteristic radiation▶ infrared▶ light▶ x-rays


Physics of Radiography Ionizing Radiation

Ionizing RadiationRadiation with energy > 13.6 eV is ionizing

Energy required to ionize:

▶ air ≈ 34 eV▶ lead ≈ 1 keV▶ tungsten ≈ 4 keV

These are average binding energies.

Radiation energies in medical imaging30 keV–511 keV

can ionize 10–40,000 atoms



Particulate RadiationConcerned with electron here (x-ray tube)(positron in later chapters)

Relativistic theory required (see text)

An electron accelerated across 100 kVpotential difference yields a 100 keV electron



Electromagnetic EM RadiationMany types of EM radiation:

▶ radio, microwaves,▶ infrared, visible light, ultraviolet▶ x-rays, gamma rays

electric and magnetic wave at right angles▶ waves with frequency �, or▶ “particles” (photons) with energy E

E = h–�

Planck’s constant h– = 4.14× 10−15 eV-sec


Physics of Radiography Energetic Electrons

Energetic Electron InteractionsTwo primary interactions:

▶ collisional transfer▶ radiative transfer

Collisional transfer:▶ Electron hits other electrons▶ Occasionally produces delta ray



Energetic Electrons: Radiative TransferTwo types of radiative transfer:

▶ characteristic x-rays▶ bremsstrahlung x-rays

Characteristic x-rays:▶ electron ejects a K-shell electron▶ reorganization generates x-ray



Energetic Electrons: BremsstrahlungBremsstrahlung x-rays

▶ Electron “grazes” nucleus, slows down▶ Energy loss generates x-ray



X-ray Spectrum


Physics of Radiography Electromagnetic Radiation

EM InteractionsTwo important interactions:

▶ Photoelectric effect▶ Compton scattering



Photoelectric effectAtom completely absorbs incident photon

All energy is transferred

Atom produces▶ characteristic radiation, and/or▶ energetic electron(s)

Characteristic radiation might be▶ x-ray, or▶ light ← very important



Illustration of Photoelectric Effect



Compton ScatteringPhoton collides with outer-shell electron

Photon is deflected, angle �

Deflected photon has lower energy:

E ′ =E

1 + E (1− cos �)/(m0c2)

m0 is rest mass of electron

m0c2 = 511 keV



Illustration of Compton Scattering

When E higher▶ more Compton events scatter forward▶ Compton more of a problem



Probability of EM InteractionsPhotoelectric effect:

Prob[photoelectric event] ∝Z 4

eff(h–�)3

Photons are more penetrating at higherfrequencies/energies

Compton scattering:

Prob[Compton event] ∝ ED

ED approximately constant over diagnostic range


Physics of Radiography EM Strength

Beam Strength: Photon CountsPhoton fluence:

Φ =N

APhoton fluence rate:

� =N

AΔt



Beam Strength: Energy FlowEnergy fluence:

Ψ =Nh–�

AEnergy fluence rate:

=Nh–�

AΔt

Intensity: (= )

I (E ) =NE

AΔt



Polyenergetic Beam StrengthX-ray spectrum S(E ):

▶ S(E ) is the number of photons per unit energyper unit area per unit time

Photon fluence rate from spectrum:

� =

∫ ∞0

S(E ′) dE ′

Intensity from spectrum:

I =

∫ ∞0

E ′S(E ′) dE ′


Physics of Radiography EM Attenuation

EM Attenuation Geometries



“Good Geometry”, MonoenergeticNon-homogeneous slab:

dN

N= −�(x)dx

Integration yields:

N(x) = N0 exp{−∫ x

0

�(x ′)dx ′}

For intensity:

I (x) = I0 exp{−∫ x

0

�(x ′)dx ′}



Homogeneous SlabHomogeneous slab thickness Δx

Fundamental photon attenuation law

N = N0e−�Δx

� is linear attenuation coefficient

In terms of intensity:

I = I0e−�Δx

This is known as Beer’s Law



Half-value LayerHomogeneous slab (shielding)

HVL = thickness that willstop half the photons

1

2= exp{−� HVL}

Relation to �

HVL =0.693

�



“Good Geometry”, PolyenergeticMust deal with x-ray spectrum S0(E )

Abandon photon counting: use intensityFor heterogeneous materials

I (x) =

∫ ∞0

S0(E ′)E ′ exp

{−∫ x

0

�(x ′;E ′)dx ′}dE ′

Not very useful

Better to define effective energy,use monoenergetic approximation



Mass Attenuation Coefficientmass attenuation coefficient �/�


Physics of Radiography EM Dose

EM Radiation DoseHow many photons? → fluence

How much energy? → energy fluence

What does radiation do to matter?→ dose



Exposure: (the creation of ions)How many ions are created?

Exposure X , the number of ion pairs produced in aspecific volume of air by EM radiation

SI Units: C/kg

Common Units: roentgen, R

1 C/kg = 3876 R



Dose: (the deposition of energy)How much energy is deposited into material?

Dose, D, the energy deposited per unit volume

SI unit: Gray (Gy) 1 Gy = 1 J/kg

Common unit: rad

1 Gy = 100 rads

When X = 1 R soft tissue incurs 1 rad absorbeddose.



KermaHow much energy is deposited intothe electrons?

Kerma, K , is the energy deposited into the electronsof a material

SI units: Gray (Gy) = 1 J/kg = 100 rads

At diagnostic energies in the body, K = D

(In general, K ≥ D. Some electrons can causebremsstrahlung and their energy irradiated away →no dose. Not likely in body.)


Projection Radiography

5 Projection RadiographyRadiographic SystemsX-ray TubesFiltration, Restriction, and Contrast AgentsScatter ControlScreen and CassetteImaging EquationFilmNoise


Projection Radiography Radiographic Systems

Projection RadiographySystems:

▶ chest x-rays,mammography

▶ dental x-rays▶ fluoroscopy, angiography

Properties▶ high resolution▶ low dose▶ broad coverage▶ short exposure time


Projection Radiography Radiographic Systems

Radiographic System


Projection Radiography X-ray Tubes

X-ray Tube Diagram



X-ray Tube Components

Filament controls tube current (mA)

Cathode and focussing cup

Anode is switched to high potential▶ 30–150 kVp▶ Made of tungsten▶ Bremsstrahlung is 1%▶ Heat is 99%▶ Spins at 3,200–3,600 rpm

Glass housing; vacuum



Exposure ControlkVp applied for short duration

▶ fixed timer (SCR), or▶ automatic exposure control (AEC), 5 mm thick

ionization chamber triggers SCR

Tube current mA controlled by▶ filament current, and▶ kVp

mA times exposure time yields mAs

mAs measures x-ray exposure



X-ray Spectra


Projection Radiography Filtration, Restriction, and Contrast Agents

FiltrationInherent filtration

▶ Within anode▶ Glass housing

Added filtration▶ Aluminum▶ Copper/Aluminum

Note: Cu has 8keV characteristic xrays▶ Measured in mm Al/Eq



RestrictionGoal: To direct beam toward desired anatomy



Compensation FiltersGoal: to even out film exposure



Contrast AgentsGoal: To create contrast where otherwise none

10 20 30 40 50 60 70 80 100 150 200150.1

1.0

10

100Li

ne

ar A

tte

nua

tion C

oe

ffic

ient (c

m)

-1

Photon Energy (keV)

Hypaque

Kedge

37.4muscle

soft tissue

Bone

Fat

Kedge

33.2

BaSO

mix4


Projection Radiography Scatter Control

Scatter ControlIdeal x-ray path: a line!

Compton scattering causes blurring

How to reduce scatter?▶ airgap▶ scanning slit▶ grid



Grids

Effectiveness in scatter reduction?

grid ratio =h

b

6:1 to 16:1 (radiography) or 2:1 (mammo)



Problems with GridsRadiation is absorbed by grid

▶ grid conversion factor

GCF =mAs w/ grid

mAs w/o grid

▶ Typical range 3 < GCF < 8

Grid visible on x-ray film▶ move grid during exposure▶ linear or circular motion


Projection Radiography Screen and Cassette

Intensifying ScreenFilm stops only 1–2% of x-rays

Film stops light really well

Phosphor = calcium tungstate

Flash of light lasts 1× 10−10 second

∼1,000 light photons per 50 keV x-ray photon


Projection Radiography Screen and Cassette

Radiographic Cassette

Cassette holds two screens; makes “sandwich”

One side is leaded


Projection Radiography Imaging Equation

Basic Imaging Equation

I (x , y) =

∫ ∞0

S0(E ′)E ′ exp

{−∫ r(x ,y)

0

�(s;E ′, x , y)ds

}dE ′



Geometric EffectsX-rays are diverging from source

Undesirable effects:

▶ cos3 � falloff across detector▶ anode heel effect▶ pathlength irregularities▶ magnification

I0 is intensity at (0, 0)

r is distance from (x , y) to x-ray origin

� is angle between (0, 0) and (x , y)



Inverse Square LawNet flux of photons decrease as 1/r 2.Therefore

I0 =IS

4�d2Ir =

IS4�r 2

Eliminate source intensity IS

Ir = I0d2

r 2

Since cos � = d/r

Ir = I0 cos2 �



Obliquity

Intensity isId = I0 cos �



Beam Divergence and Flat DetectorInverse square law and obliquity combine

Id(xd , yd) = I0 cos3 �

Can usually be ignored. Why?

▶ Detector is far away▶ Field of view (FOV) is often small



Anode Heel EffectIntensity within the x-ray cone

▶ Not uniform▶ stronger in the cathode direction▶ 45% variation is typical

Compensate, use to advantage, or ignore

We will ignore in math



Path Length of SlabUniform slab yields different intensities



Effect of Pathlength on IntensityIntensity on detector

Id(x , y) = I0 exp{−�L/ cos �}

Including inverse square law and obliquity:

Id(x , y) = Ii cos3 � exp{−�L/ cos �}

If d ≈ r all effects can be ignored



Object MagnificationSize on detector depends on distance from source



Magnification FormulaObject at position z from source

Height of object is w .

Height wz on detector is

wz = wd

z

Magnification is

M(z) =d

zCan lead to edge blurring and misleading sizes



Thin Slab Imaging EquationThin slab at z of �(x , y)Let “transmittivity” be

tz(x , y) = exp{−�(x , y)Δz}

On detector, intensity is

Id(x , y) = I0 cos3 � tz

(x

M(z),

y

M(z)

)After substitution

Id(x , y) = I0

(d√

d2 + x2 + y 2

)3

tz(xzd,yz

d

)Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 140 / 412


Sources of BlurringExtended source

Intensifier screen



Extended Source

z

d

ExtendedX-raySource

Image ofExtendedSource

PointHole

DetectorPlane

s x y( , )D

D´

Source spatial distribution: s(x , y)



Source MagnificationSource diameter on detector:

D ′ =d − z

zD

Source magnification:

m(z) = −d − z

z= 1−M(z)



Source BlurringImage of source through pinhole at z

Id(x , y) =1

4�d2m2s( xm,y

m

)Intensity at detector:

Id(x , y) =cos3 �

4�d2m2tz( x

M,y

M

)∗ s( xm,y

m

)



Film-Screen BlurringFilm

X-rayPhoton

r

x

L

Light Photons

Phosphors

Film-screen impulse response: h(x , y)



Overall Imaging EquationInclude all geometric effects

Id(x , y) = cos3 �1

4�d2m2s( xm,y

m

)∗ tz

( x

M,y

M

)∗ h(x , y)


Projection Radiography Film

FilmDeveloped film

Optical transmissivity

T =ItIi

Optical density

D = log10

IiIt

Note: O = 1/T is optical opacity

Usable densities 0.25 < D < 2.25

Best densities 1.0 < D < 1.5



H & D CurveOptical density from x-ray exposurefor film-screen combination:

10-1

100

101

102

103

0

0.5

1

1.5

2

2.5

3

3.5

4

(a) High Speed

Film with

CaWO

Screens4

(b) Direct X-ray

Film

(c) High Speed

Film Without

Screens

Fog Level

Exposure, mR

Op

tica

l De

nsi

ty

Toe

Linear

Region

Shoulder



X-ray Exposure to Film DensityX-ray exposure yields optical density

D = Γ log10

X

X0

Γ is film gamma

Typical ranges: 0.5 < Γ < 3.0

Latitude is range exposures where relationship islinear

Speed is inverse of exposure at which

D = 1 + fog level


Projection Radiography Noise

NoiseLocal contrast

C =It − IbIb

Signal is It − IbNoise is due to Poisson behavior

Variance of noise in background: �2b

Signal to noise

SNR =It − Ib�b

=CIb�b



Signal-to-noiseModel x-ray burst as monoenergetic

▶ effective energy is h�▶ background intensity is

Ib =Nbh�

AΔt

Signal-to-noise is

SNR = C√

Nb

More photons is better



Detective Quantum EfficiencyHow good is a detector?

Consider:▶ Potential SNR before detection▶ Actual SNR upon detection

Detective Quantum Efficiency

DQE =

(SNRoutSNRin

)2

Degradation of SNR during detection



Compton ScatterCompton adds intensity “fog”: IsResulting contrast

C ′ =C

1 + Is/Ib

Resulting SNR

SNR′ =SNR√

1 + Is/Ib


Computed Tomography

6 Computed TomographyOverviewCT GenerationsSystem ComponentsCT MeasurementsRadon TransformReconstructionProjection-Slice TheoremResolutionNoiseFan Beam Reconstruction


Computed Tomography Overview

Computed TomographyTomography:

▶ image of slice▶ removes “overlaying structure”▶ improves contrast within slice

Computed:▶ requires computer▶ reconstruction algorithm



1-D Projection“fan beam” collimation

row of electronic detectors



Premise of CTA single 1-D projection is not informative

Many 1-D projections▶ permit slice reconstruction▶ many angular views is the key

http://www.gehealthcare.com


Computed Tomography CT Generations

1G CT Scanner



2G CT Scanner



3G CT Scanner



4G CT Scanner



Electron Beam (5G) CT Scanner

http://radiology.rsnajnls.org



Gantry, Slipring, and Table

http://www.gehealthcare.com http://www.cissincorp.com

1–2 revolutions per secondJerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 163 / 412


Helical (6G) CT ScannerStep-wise table movement yields stack of 2D slices

Continuous table movement yields stream of 1Dprojections

3D volume is reconstructed from helical acquisition

http://imaging.cancer.gov



Multi-slice (7G) CT ScannerFeatures:

▶ 16–64 parallel detectorrows

▶ 14,336–57,344 detectorelements

▶ 20–80 mm detector“height”

▶ 16–64 0.5mm slices witheach second gantryrevolution

CT is becoming “cone beam”


Computed Tomography System Components

X-ray Tubes in CTUse only one tube

▶ exception: EBCT▶ exception: dual-source CT

80kVp–140kVp, continuous excitation▶ dual-energy is possible

fan-beam (1–10 mm thick), or

thin-cone collimation 20–80 mm

More filtering than projection radiography▶ copper followed by aluminum▶ Better approximation to monoenergetic



CT DetectorsMost are solid-state:

▶ scintillation crystal▶ solid state photo-diode



CT Detector SpecificationsSingle-slice scanners:

▶ Area: 1.0 mm × 15.0 mm▶ Thick in 3G, thin in 4G & EBCT

Multi-slice scanners:▶ Area: 1.0 mm × 1.25 mm▶ Grouped in multiples of 1.25 mm

Xenon gas detectors for less expensive scanners


Computed Tomography CT Measurements

CT Measurement ModelMonoenergetic model:

Id = I0 exp

{−∫ d

0

�(s; E )ds

}E is effective energy

E is that energy which in a given materialwill produce the same measured intensityfrom a monoenergetic source as from theactual polyenergetic source.



CT MeasurementObserve IdRearrange monoenergetic model:

gd = − lnIdI0

=

∫ d

0

�(s; E )ds

gd is a line integral of the linear attenuationcoefficient at the effective energy

Note: Requires calibration measurement of I0



CT NumbersConsistency across CT scanners desired

CT number is defined as:

h = 1000× �− �water�water

h has Hounsfield units (HU)

Usually rounded or truncated to nearestinteger

Range: −1,000 to ∼3,000


Computed Tomography Radon Transform

Describing LinesPossible descriptions of lines:

▶ Functional: y = ax + b▶ Parametric: (x(s), y(s))▶ Set: {(x , y)∣(x , y) are on a line}

Critique:▶ Functional: what about vertical lines???▶ Parametric: good for model of process▶ Set: good for theory of reconstruction



Picture of a Line

l

l

L( , )l θ

θ

f(x,y)

x

y

0



Line ParametersDescribed by:

▶ Orientation or angle, �▶ Lateral translation or position, ℓ

Written as L(ℓ, �)

L(ℓ, �) = {(x , y)∣(x , y) are on the line

with position ℓ

and angle �}



Line Integral: parametric formWhat is integral of f (x , y) on L(ℓ, �)?

Step 1: Parameterize L(ℓ, �):

x(s) = ℓ cos � − s sin �

y(s) = ℓ sin � + s cos �

Step 2: Integrate f (x , y) over parameter s

g(ℓ, �) =

∫ ∞−∞

f (x(s), y(s))ds

Use this form for the forward problem



Line Integral: set formIntegrate over whole plane;non-zero only on L(ℓ, �)

Key is sifting property

q(ℓ) =

∫ ∞−∞

q(ℓ′)�(ℓ′ − ℓ)dℓ′

Use line impulse on L(ℓ, �)

g(ℓ, �) =∫ ∞−∞

∫ ∞−∞

f (x , y)�(x cos � + y sin � − ℓ) dxdy



Physical Meanings of f (x , y) and g(ℓ, �)Recall monoenergetic model:

Id = I0 exp

{−∫ d

0

�(s; E )ds

}Rearrange:

− lnIdI0

=

∫ d

0

�(x(s), y(s); E )ds

Relationship is:

f (x , y) = �(x , y ; E )

g(ℓ, �) = − lnIdI0



What is g(ℓ, �)?Fix ℓ and �: line integral of f (x , y)

Fix �: projection of f (x , y) at angle �

Function of � and ℓ:g(ℓ, �) is the Radon transform of f (x , y)

g(ℓ, �) = ℛ{f (x , y)}

Transform . . . hmmm . . . can we find aninverse transform?

f (x , y) = ℛ−1{g(ℓ, �)}



SinogramCT data acquired for collection of ℓ and �

CT scanners acquires a sinogram


Computed Tomography Reconstruction

BackprojectionGoal: find f (x , y) from g(ℓ, �)Strategy: “smear” g(ℓ, �) into planeFormally:

b�(x , y) = g(x cos � + y sin �, �)

b�(x , y) is a laminar image



Backprojection Summation“Add up” all the backprojection images:

fb(x , y) =

∫ �

0

b�(x , y)d�

=

∫ �

0

g(x cos � + y sin �, �)d�

=

∫ �

0

[g(ℓ, �)]ℓ=x cos �+y sin � d�

fb(x , y) is called a laminogram orbackprojection summation image



Properties of Laminogram“Bright spots” tend to reinforce

Problem:fb(x , y) ∕= f (x , y)

What is wrong?



Convolution BackprojectionCorrect reconstruction formula:

f (x , y) =

∫ �

0

[c(ℓ) ∗ g(ℓ, �)]ℓ=x cos �+y sin � d�

wherec(ℓ) = ℱ−1{∣%∣}

is called the ramp filter.

Three steps: ← know/understand these!!▶ 1. convolution▶ 2. backprojection▶ 3. summation



Step 1: ConvolutionConvolve every projection with c(ℓ)

the horizontal direction in a sinogram



Step 2: Backprojection1D projection → 2D laminar function



Step 3: SummationAccumulate sum of backprojection images


Computed Tomography Projection-Slice Theorem

Projection-Slice TheoremRadon transform:

g(ℓ, �) = ℛ{f (x , y)}

Fourier transforms:

G (%, �) = ℱ1D {g(ℓ, �)}F (u, v) = ℱ2D {f (x , y)}

Projection-slice theorem:

G (%, �) = F (% cos �, % sin �)



Illustration of Projection-Slice Theorem

x

y

u

v

ρ

l

θ

θ

f(x,y) F(u,v)

2D Fourier Transform

1D Fourier Transform



Exact Reconstruction FormulasFourier reconstruction:

f (x , y) = ℱ−1

2D {G (%, �)}

Filtered backprojection:

f (x , y) =

∫ �

0

[∫ ∞−∞∣%∣G (%, �)e+j2�%ℓd%

]ℓ=x cos �+y sin �

d�

Convolution backprojection:

f (x , y) =

∫ �

0

∫ ∞−∞

g(ℓ, �)c(x cos � + y sin � − ℓ)dℓd�



Ramp Filter Design∣%∣ is not integrable

⇒ c(ℓ) does not exist

Actual ramp filter is designed as

c(ℓ) = ℱ−1

1D{W (%)∣%∣}

Simplest window function is

W (%) = rect

(%

2%0

)


Computed Tomography Resolution

Factors Affecting CT ResolutionDetector width ∼ area detectors

detector indicator function = s(ℓ)

Window function W (%)Approximate CBP:

f (x , y) =∫ �

0

[∫ ∞−∞

G (%, �)S(%)W (%)∣%∣e j2�%ℓ d%]ℓ=x cos �+y sin �

d�

whereS(%) = ℱ{s(ℓ)}



Blurry ReconstructionBlurry projection:

g(ℓ, �) = g(ℓ, �) ∗ s(ℓ) ∗ w(ℓ)

= g(ℓ, �) ∗ h(ℓ)

Radon transform convolution theorem

ℛ{f ∗2 h} = ℛ{f } ∗1 ℛ{h}

Leads to

f (x , y) = f (x , y) ∗ ℛ−1{h(ℓ)}



Circular Symmetry of BlurringCT image blurred by convolution kernel

h(x , y) = ℛ−1{h(ℓ)}

Fourier transform of h(ℓ)

H(%) = ℱ1{h(ℓ)} = S(%)W (%)

which is independent of �.

Therefore, H(u, v) is circularly symmetric

H(q) = ℱ2{h(x , y)} = S(q)W (q)



PSF Given by Hankel TransformPSF is circularly symmetric and given by

h(r) = ℋ−1{S(%)W (%)}

Reconstructed image given by

f (x , y) = f (x , y) ∗ h(r)


Computed Tomography Noise

Noise in CT MeasurementsBasic measurement is:

gij = − ln

(Nij

N0

)

▶ line Lij▶ angle i▶ position j

Noise is “in” Poisson random variable Nij

▶ mean Nij

▶ variance Nij



Functions of Random VariablesIt follows that gij is a random variable

gij ≈ ln

(N0

Nij

)Var(gij) ≈

1

Nij

�(x , y) is approximate reconstruction

It follows that �(x , y) is a random variable

What are the mean and variance of �?



CBP ApproximationConvolution backprojection (CBP):

�(x , y) =

∫ �

0

∫ ∞−∞

g(ℓ, �)c(x cos � + y sin � − ℓ)dℓd�

Approximations:▶ M angles; Δ� = �/M▶ N + 1 detectors; Δℓ = T▶ c(ℓ) ≈ c(ℓ)

Discrete CBP:

�(x , y) =( �M

) M∑j=1

T

N/2∑i=−N/2

g(iT , j�/M)c(x cos �j+y sin �j−iT )



More Definitions and ApproximationsNij is mean for i-th detector and j-th angle

Nij is independent for different measurements

Nij = N , an “object uniformity” assumption

c(ℓ) is created using rectangular window W (%) withcutoff %0.



ConclusionsMean(�) is desired result

Var(�) = �2� is inaccuracy

�2� ≈

2�2

3%3

0

1

M

1

N/T

Be cautious on conclusions: not all variables areindependent in a real physical system



Signal-to-noise RatioDefinition (usual)

SNR =C �

��

After substitution:

SNR =C �

�

√3M

2%30

N

T



SNR in a Good DesignWhat should %0 be?

Let dectector width = w

%0 should be anti-aliasing filter:

%0 =k

wwhere k ≈ 1

In 3G scanner w = T

ThenSNR ≈ 0.4kC �w

√NM



SNR in Fan-Beam CaseDefinitions:

▶ Nf is mean photon count per fan▶ D is number of detectors▶ L is length of detector array

Then

SNR ≈ 0.4kC �L

√NfM

D3

Strange: In 3G, increasing D decreases SNR.

Reason: This analysis ignores resolution



Rule of ThumbVariables:

▶ D is number of detectors▶ M is number of angles▶ J2 is number of pixels in image

Very approximate “rule”:

D ≈ M ≈ J

Typical numbers:

Lo: D ≈ 700 M ≈ 1, 000 J ≈ 512

Hi: D ≈ 900 M ≈ 1, 600 J ≈ 1, 024


Computed Tomography Fan Beam Reconstruction

Fan Beam Geometry



Sinogram Rebinning



Fan-Beam Variables



Fan-Beam Convolution BackprojectionFormula

f (x , y) =

∫ 2�

0

1

(D ′)2

∫ m

− mp( , �)c ′( ′ − )d d�

D ′ depends on (x , y)

c ′ is a different filter than c


Physics of Nuclear Medicine

7 Physics of Nuclear MedicineBinding EnergyRadioactivityRadiotracers


Physics of Nuclear Medicine Binding Energy

NomenclatureAtomic number: Z , number of protons in nucleus

Mass number: A, number of nucleons in nucleus

Nuclide: unique combination of protons andneutrons in nucleus

Radionuclide: a nuclide that is radioactive

Isotope: atoms with same Z , different A

Isobar: atoms with same A, different Z



Mass Defect and Binding EnergyMass defect =

Mass of constituents of atom

− actual mass of atom

unified mass unit, u, = 1/12 mass of C-12 atom

Binding energy = mass defect ×c2

One u is equivalent to 931 MeV

Generally, more massive atom, more binding energy



Binding Energy per Nucleon


Physics of Nuclear Medicine Radioactivity

What is Radioactivity?Radioactive decay: rearrangement of nucleii tolower energy states = greater mass defect

Parent atom decays to daughter atom

Daughter has higher binding energy/nucleon thanparent

A radioatom is said to decay when its nucleus isrearranged

A disintegration is a radioatom undergoingradioactive decay.

Energy is released with disintegration.



“Line” of StabilityNuclides divide into two groups:

▶ Non-radioactive — i.e., stable atoms▶ Radioactive — i.e., unstable atoms

“Line” of stability:



Decay ModesFour main modes of decay:

▶ alpha particles (2 protons, 2 neutrons)▶ beta particles (electrons)▶ positrons (anti-matter electrons)▶ isomeric transition (gamma rays produced)

Medical imaging is only concerned with:▶ positrons (PET), and▶ gamma rays (scintigraphy, SPECT)



Measurement of RadioactivityRadioactivity, A, # disintegrations per second

1 Bq = 1 dps

1 Ci = 3.7× 1010 Bq



Radioactive Decay LawTime evolution of radioactivity:

At = A0e−�t

� is the decay constant

Half-life t1/2 is defined by

At1/2

A0=

1

2= e−�t1/2

It follows that

t1/2 =0.693

�



Statistics of DecayOver “short” time Δt relative to t1/2:

# radioatoms N0 approximately constant

Statistics are Poisson:

P[ΔN = k] =(�N0Δt)ke−�N0Δt

k!

Interpretation: �N0Δt is probability of having onedisintegration from N0 radioatoms in time intervalΔt

�N0 is called Poisson rate, units aredisintegrations per second, it is activity


Physics of Nuclear Medicine Radiotracers

RadiotracersRadionuclides in the body must be safe

By themselves:▶ Iodine-123,▶ Iodine-131

Labeled: Chemically attached to natural substances:

▶ Technetium-99m labeled DTPA,▶ Oxygen-15 labeled O2,▶ Fluorine-18 labeled glucose



Radiotracer PropertiesEmit gamma rays or positrons

Half life: minutes to a few hours

Positron emission:

▶ positrons annihilate▶ produces two 511 keV gamma rays,▶ gamma rays are 180-degrees apart

Gamma ray emission:

▶ monoenergetic gamma rays (desirable)▶ high energy gamma rays (desirable)



Some RadiotracersGamma Ray Emitters:

▶ Iodine-123 (13.3 h, 159 keV)▶ Iodine-131 (8.04 d, 364 keV)▶ Iodine-125 (60 d, 35 keV) (Bad. Why?)▶ Thallium-201 (73 h, 135 keV)▶ Technetium-99m (6 h, 140 keV)

Positron Emitters:▶ Fluorine-18 (110 min, 202 keV)▶ Oxygen-15 (2 min, 696 keV)


Planar Scintigraphy

8 Planar ScintigraphyScintigraphy SystemsGamma CameraAcquisition ModesImage EquationResolution and SensitivityArtifacts and Noise


Planar Scintigraphy Scintigraphy Systems

Broad PurposeGamma emitter in body; where is it?

Planar camera; like radiography

2D projection of 3D concentration

X-ray Image Bone Scintigram



A SPECT/Scintigraphy/CT System




Example PreparationGallium-67 citrate

half-life is 78 hr

93 keV (40%), 184 keV (24%), 296 keV (22%), and388 keV (7%).

150-220 MBq (4-6 mCi) intravenously

48 hr after injection, about 75% remains in body

equally distributed among the liver, bone and bonemarrow, and soft tissues.

Scintigrams 24–72 hrs after injection



Whole Body Image



Planar Scintigraphy Gamma Camera

Gamma/Anger Camera Components



Collimators

(a) (b)

(c)(d)

(a) Parallel hole

(b) Converging hole (magnifies)

(a) Diverging hole (minifies)

(a) Pin-hole (2–5 mm)



DetectorSingle large-area NaI(Tl) crystal

Diameters:▶ 30–50 cm in diameter▶ Mobile units: 30 cm▶ Fixed scanners: 50 cm

Thickness:▶ High-E emitters: 1.25 cm thick▶ Low-E emitters: 6–8 mm thick



Photomultiplier Tube Array



Photomultiplier Tube

Dynodes

Focussing

Grid

Photocathode

Light Photons

1,200 V

Output Signal

Anode

e-

e-

e-

e-



Pulse HeightResponse to single gamma ray photon

PMT responses, ak , k = 1, . . . ,K

Total response of cammera is Z -pulse

Z =K∑

k=1

ak

Height of Z pulse is important▶ Can remove Compton photons▶ Can reject multiple hits



Pulse Height Analysis

Discriminator circuit rejectsnon-photopeak events



Event Positioning LogicTube centers at (xk , yk) k = 1, . . . ,K

Center of mass of pulse responses is

X =1

Z

K∑k=1

xkak

Y =1

Z

K∑k=1

ykak

This is pulse location


Planar Scintigraphy Acquisition Modes

Acquisition ModesHow to use the camera to make images?

▶ List mode▶ Static frame mode▶ Dynamic frame mode▶ Multiple-gated acquisition▶ Whole body mode



List Mode

Complete information, but memory hog



Static Frame Mode

Matrix sizes: 64 × 64, 128 × 128, 256 × 256



Dynamic Frame Mode

Useful for imaging transient physiological processes



Multiple Gated Acquisition

Cardiac (ECG) gated. Data resorted using ECG



Whole Body Mode

Common in bone scans and tumor screening


Planar Scintigraphy Image Equation

Imaging Geometry and Assumptions

Lines defined by (parallel) collimator holesIgnore Compton scatteringRadioactivity is A(x , y , z)Monoenergetic photons, energy E



Imaging EquationPhoton fluence on detector is

�(x , y) =∫ 0

−∞

A(x , y , z)

4�z2e−∫ 0

z

�(x , y , z ′;E )dz ′

dz

Depth-dependent effects from:▶ inverse square law, and▶ object-dependent attenuation

Consequences:▶ Near activity brighter▶ Front and back are different



Planar SourcesAz0

(x , y) has radioactivity on z = z0

A(x , y , z) = Az0(x , y)�(z − z0)

Detected photon fluence rate

�(x , y) = Az0(x , y)

1

4�z20

exp

{−∫ 0

z0

�(x , y , z ′;E )dz ′}

Two terms attenuate desired result▶ inverse square law: constant for (x , y)▶ �: not constant for (x , y)


Planar Scintigraphy Resolution and Sensitivity

Collimator Resolution



Collimator ResolutionCollimator Resolution = FWHM =

RC (∣z ∣) =d

l(l + b + ∣z ∣)

Gaussian approximation

hc(x , y ; ∣z ∣) = exp{−4(x2 + y 2) ln 2/R2

C (∣z ∣)}

Planar source is blurred

�(x , y) = Az0(x , y)

1

4�z20

×

exp

{−∫ 0

z0

�(x , y , z ′;E )dz ′}∗ hc(x , y ; ∣z0∣)



Instrinsic ResolutionWhere did the x-ray photon hit?

▶ Compton in crystal spreads out light▶ Crystal thickness▶ Noise in light, PMTs, and electronics

Gaussian approximation

hI (x , y) = exp{−4(x2 + y 2) ln 2/R2

I

}Planar source is further blurred

�(x , y) = Az0(x , y)

1

4�z20

exp

{−∫ 0

z0

�(x , y , z ′;E )dz ′}

∗hC (x , y ; ∣z0∣) ∗ hI (x , y)



Collimator SensitivityCollimator Efficiency = Sensistivity =

� =

(Kd2

l(d + h)

)2

where K ≈ 0.25.

� is the fraction of photons (on average) that passthrough the collimator for each emitted photondirected at the camera



Resolution vs. Sensitivity

Table: Resolution and Sensitivity for Several Collimators

collimator d (mm) l (mm) h (mm) resolution relative@ 10 cm (mm) sensitivity

LEUHR 1.5 38 0.20 5.4 12.1LEHR 1.9 38 0.20 6.9 20.5LEAP 1.9 32 0.20 7.8 28.9LEHS 2.3 32 0.20 9.5 43.7

LEUHR = low energy ultra-high resolutionLEHR = low energy high resolutionLEAP = low energy all purposeLEHS = low energy high sensitivity



Detector EfficiencyDepends on crystal thickness

▶ thicker ⇒ more efficient▶ 100% at 100keV; 10-20% at 511keV

Tradeoff:▶ If E low ⇒ use thinner crystal

★ better intrinsic resolution▶ If E high ⇒ use thicker crystal

★ poorer intrinsic resolution

▶ Higher E , less abosorption in body


Planar Scintigraphy Artifacts and Noise

Geometry and NonuniformityGeometric distortion

▶ pincushion distortion▶ barrel distortion▶ wavy line distortion

Image nonuniformity▶ variation as much as 10%▶ non-uniform detector efficiencies▶ geometric distortions → “hot spot”▶ edge packing



Image SNRSuppose N photons are detected

Then intrinsic SNR of frame mode is

SNR(intrinsic) =

√N

J

J2 is number of pixels in image

For similar areas of target and background:

SNR = C√Nb

Just like in projection radiography



Energy ResolutionEnergy resolution = FWHM of photopeak



Pulse PileupPulse pileup = two simultaneous -rays

Event rejected▶ because of energy discrimination▶ wasted photons

Cannot improve image using larger dose

Instead, keep dose low and image longer


Emission Tomography

9 Emission TomographyOverviewSPECT System ComponentsSPECT Imaging EquationSPECT ReconstructionPrinciple of PETPET System ComponentsPET Imaging EquationPET ReconstructionResolution and Artifacts


Emission Tomography Overview

OverviewSPECT

▶ uses gamma ray emitters▶ uses Anger camera▶ 3-D volume reconstruction

PET▶ uses positron emitters▶ requires coincidence detectors▶ multiple 2-D slices


Emission Tomography SPECT System Components

SPECT HardwareRotating gamma camera

Each “row” is separate slice

Multiple heads (2 or 3) are common

High-performance cameras used▶ < 1% nonuniformity required▶ need good mechanical alignment



Typical SPECT System




Multiple Head Tradeoffs

Table: Comparison of acquisition times and relative sensitivities forsingle- and multi-head systems with identical camera heads andcollimation.

360∘ 180∘

Acq Time Rel Sens Acq Time Rel Sens

Single 30 1 30 1Double (heads@180∘) 15 2 30 1Double (heads@90∘) 15 2 15 2Triple 10 3 20 1.5


Emission Tomography SPECT Imaging Equation

SPECT Coordinate System

“Home position:” x → z , y → ℓ, z → y



Basic Imaging EquationParallel hole collimators

Camera fixed distance R from origin(origin in patient)

Imaging equation in “home” position:

�(z , ℓ) =

∫ R

−∞

A(x , y , z)

4�(y − R)2

exp

{−∫ R

y

�(x , y ′, z ;E )dy ′}dy



Tomographic Imaging Geometry

z is irrelevant

Line described by

L(ℓ, �) = {(x , y)∣ x cos � + y sin � = ℓ}



Tomographic Imaging Equation

�(ℓ, �) =

∫ R

−∞

A(x(s), y(s))

4�(s − R)2

exp

{−∫ R

s

�(x(s ′), y(s ′);E )ds ′}ds

Two unknowns: A(x , y) and �(x , y)

Generally intractable ⇒▶ ignore attenuation (often done)▶ assume constant▶ measure and apply atten correction



Approximate SPECT Imaging EquationBold approximations: ignore attenuation, inversesquare law, and scale factors:

�(ℓ, �) =

∫ ∞−∞

A(x(s), y(s))ds

Using line impulse:

�(ℓ, �) =∫ ∞−∞

∫ ∞−∞

A(x , y)�(x cos � + y sin � − ℓ) dx dy


Emission Tomography SPECT Reconstruction

SPECT ReconstructionRecognize:

f (x , y) = A(x , y)

g(ℓ, �) = �(ℓ, �)

Use convolution backprojection

A(x , y) =∫ �

0

∫ ∞−∞

�(ℓ, �)c(x cos � + y sin � − ℓ) dℓ d�

Approximate ramp filter:

c(ℓ) = ℱ−11D {∣%∣W (%)}


Emission Tomography Principle of PET

PET PrinciplesPositron emitters

Positron annihilation:▶ short distance from emission▶ produces two 511 keV gamma rays▶ gamma rays 180∘ opposite directions

Principle: detect coincident gamma rays



A PET Scanner

Used with permission of GE Healthcare



Positron Annihilation



Annihilation Coincidence Detection (ACD)Event occurs if detections are coincident

Time window is typically 2–20 ns

12 ns is common setting

No detector collimation required

Dual-head SPECT systems can be used


Emission Tomography PET System Components

PET Detector Block

Crystals plus PMTs

BGO = Bismuth Germanate

BGO has 3x stopping power than NaI(Tl)



Typical PET Detector Arrangement2 mm × 2 mm elements

8 by 8 elements per blocks; 2 by 2 PMTs per block

48 blocks per major ring; 3 major rings

⇒ 24 detector rings; 384 detectors per ring

⇒ 8216 crystals total



2-D or 3-D PET GeometrySepta or no septa between rings?

Septa: ⇒ multiple 2-D PET rings▶ Reconstruction like 2-D CT

No septa: ⇒ 3-D PET▶ Need 3-D reconstruction algorithms

We focus on 2-D PET


Emission Tomography PET Imaging Equation

2-D PET Geometry



Lines of Response (LORs)



Imaging EquationLine integrals of activity

On line L(ℓ, �)

�(ℓ, �) = K exp

{−∫ R

−R�(x(s), y(s);E ) ds

}×∫ R

−RA(x(s), y(s)) ds

Unknowns �(x , y) and A(x , y) separate



Attenuation CorrectionCorrected sinogram

�c(ℓ, �) =�(ℓ, �)

K exp{−∫ R

−R �(x(s), y(s);E ) ds}

�(x , y) found from CT (transmission PET)


Emission Tomography PET Reconstruction

PET ReconstructionConvolution backprojection yields A(x , y)

Ac(x , y) =∫ �

0

∫ ∞−∞

�c(ℓ, �)c(x cos � + y sin � − ℓ) dℓd�


Emission Tomography Resolution and Artifacts

Resolution in Emission TomographyApproximation:

f (x , y) = f (x , y) ∗ h(r)

In SPECT, h(r) includes:▶ collimator and intrinsic resolutions▶ ramp filter window effect

In PET, h(r) includes:▶ the positron range function▶ detector width effects▶ ramp filter window effect



PET Events



Coincidence TimingThree classes of events

▶ true coincidence▶ scattered coincidence▶ random coincidence

Sensitivity in PET▶ measures capability of system to detect

“trues” and reject “randoms”


Ultrasound Physics

10 Ultrasound PhysicsAn Ultrasound SystemWave EquationsWave PropagationField Patterns


Ultrasound Physics An Ultrasound System

Ultrasound Image

http://www.gehealthcare.com http://www.gehealthcare.com


Ultrasound Physics Wave Equations

UltrasoundUltrasound is sound with f > 20 kHz

Medical ultrasound imaging uses f > 1 MHz

Same physics = physics of longitudinal waves▶ Wave equations▶ Snell’s laws (reflection and refraction)▶ Attenuation and absorption▶ The Doppler effect▶ Vibrating plates and field patterns▶ Scattering



3-D Wave EquationAcoustic pressure: p(x , y , z , t)

3-D wave equation

∇2p(x , y , z , t) =1

c2ptt(x , y , z , t)

where∇2p = pxx + pyy + pzz

and c is the speed of sound

General solution is very complicated

We go after plane waves and spherical waves



Plane WavesPlane wave in z direction:

p(z , t) = p(x , y , z , t)

Plane wave equation:

pzz(z , t) =1

c2ptt(z , t)

General solution:

p(z , t) = �f (t − c−1z) + �b(t + c−1z)

where �f (t) and �b(t) are arbitrary



Harmonic Waves“Harmonic” plane wave

p(z , t) = cos[k(z − ct)]

Definitions:▶ wavenumber: k▶ frequency: f = kc/2�▶ period: T = 1/f▶ wavelength: � = c/f



Spherical Waves3-D spherical wave:

p(r , t) = p(x , y , z , t)

where r =√x2 + y 2 + z2.

Spherical wave equation:

1

r

∂2

∂r 2(rp) =

1

c2

∂2p

∂t2

General solution (outward expanding):

p(r , t) =1

r�o(t − c−1r)


Ultrasound Physics Wave Propagation

Characteristic ImpedanceCharacteristic impedance

Z = �c

where � is density

Why impedance?p = Zv

where v is particle velocity v ∕= c

▶ p is “like” voltage▶ v is “like” current



Acoustic EnergyKinetic energy density:

wk =1

2�0v

2

Potential energy density:

wp =1

2�p2

where � is compressibility.

Acoustic energy density:

w = wk + wp



Acoustic PowerAcoustic Intensity:

I = pv =p2

Z

(like electrical power p = vi)

Propagation of acoustic power (plane wave):

∂I

∂z+∂w

∂t= 0



Reflection and Refraction

Snell’s Laws:

�r = �isin �isin �t

=c1

c2



Reflected and Refracted WavesPressure reflectivity:

R =prpi

=Z2 cos �i − Z1 cos �tZ2 cos �i + Z1 cos �t

Pressure transmittivity:

T =ptpi

=2Z2 cos �i

Z2 cos �i + Z1 cos �t

At normal incidence:

R =Z2 − Z1

Z2 + Z1



Attenuation and AbsorptionPhenomenological model:

p(z , t) = A0e−�az f (t − c−1z)

�a is amplitude attenuation factor [cm−1]

Absorption coefficient:

� = 20(log10 e)�a [dB/cm]

In range 1 MHz ≤ f ≤ 10 MHz

� ≈ af and a ≈ 1 dB/cm-MHz



ScatteringParticle at (0, 0, d), reflection coefficient R

Generates spherical wave

ps(r , t) =Re−�arA0e

−�ad

rf (t − c−1d − c−1r)

r is distance from (0, 0, d)


Ultrasound Physics Field Patterns

Field PatternsGeometric approximation

Diffraction formulation (book)

Simple model:



Far Field = Fraunhofer PatternTransducer face indicator function:

s(x , y) =

{1 (x , y) in face0 otherwise

Far field pattern:

q(x , y , z) ≈ 1

ze jk(x2+y2)/2zS

( x

�z,y

�z

)S(u, v) is Fourier transform of s(x , y).

Pulse-echo sensitivity: q2(x , y , z).



FocusingFocal length field pattern:

q(x , y , d) ≈ 1

de jk(x2+y2)/2dS

( x

�d,y

�d

)


Ultrasound Imaging

11 Ultrasound ImagingUltrasound System ComponentsTransducersDisplay ModesEffects of AbsorptionPhased ArraysImaging EquationResolutionSpeckle


Ultrasound Imaging Ultrasound System Components

Block Diagram


Ultrasound Imaging Transducers

Transducerslead zirconate titantate (PZT)

▶ piezoelectric crystal▶ good transmit and receive efficiencies▶ different shapes:



Piezoelectric Effect



ResonanceShock excite yields resonant pulse

Resonant frequency:

fT =cT

2dT

Damps out after 3–5 cycles



Typical Transmit Pulse



Ultrasound Probe



Mechanical Scanners



Electronic Scanner

Linear arrays:▶ 64–256 elements, fire in groups▶ each element ≈ 2 mm by 10 mm

Phased arrays:▶ 30–128 elements; electronically steered▶ each element ≈ 0.2 mm by 8 mm


Ultrasound Imaging Display Modes

A-mode Display

The Range Equation

z =ct

2



M-mode Display


fast time vs. slow timeJerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 306 / 412


B-mode Display


Ultrasound Imaging Effects of Absorption

Depth of PenetrationSignal is “lost” from absorption

Total travel before “lost” is

d =L

�

where L is system sensitivity in dB

depth of penetration is

dp =d

2=

L

2�≈ L

2af

Rule-of-thumb: dp ≈ 40/f (MHz) cm



Pulse RepetitionSignal “dies”; then repeat

Pulse repetition interval:

TR ≥2dpc≈ L

afc

Pulse repetition frequency/rate:

fR =1

TR

@2 MHz, fR ≤ 3,850 Hz.



Image Frame RateN scan lines to make image

Image frame rate:

fF ≤fRN

How to increase frame rate?▶ restrict field-of-view▶ increase frequency (why?)


Ultrasound Imaging Phased Arrays

Phased Arrays: Transmit Steering



Phased Arrays: Transmit Focussing



Delays for Transmit FocussingFocal point at (xf , zf )

Ti is at (id , 0).

Then range from Ti to focal point is:

ri =√

(id − xf )2 + z2f

Assume T0 fires at t = 0. Then Ti fires at

ti =r0 − ri

c

=

√x2f + z2

f −√

(id − xf )2 + z2f

c



Phased Arrays: Receive Beamforming



Phased Arrays: Receive DynamicFocussing



Dynamic Focussing Time DelaysT0 fired at t = 0 (focussed or steered)

spherical wave originates at (x , z)

Distance from (x , z) to Ti is

ri =√

(id − x)2 + z2

Dynamic time delays are (requires derivation)

�i(t) = t −√

(id)2 + (ct)2 − 2ctid sin �

c+

Nd

c


Ultrasound Imaging Imaging Equation

Complex SignalComplex signal:

n(t) = ne(t)e j�e−j2�f0t

Complex envelope is n(t) = ne(t)e j�

The pulse isn(t) = Re{n(t)}

The envelope is

ne(t) = ∣n(t)∣

(This will form the A-mode signal.)



Complex Pressure in SpaceAcoustic dipole (complex) pressure

p(x , y , z ; t) =1

r0

z

r0n(t − c−1r0)

Superposition over transducer face

p(x , y , z ; t) =

∫ ∫s(x0, y0)

z

r 20

n(t − c−1r0)dx0dy0



Echo from Point ScattererPoint scatterer with reflectivity R(x , y , z)

Pressure at (x ′0, y′0) on face

ps(x′0, y′0; t) = R(x , y , z)

1

r ′0p(x , y , z ; t − c−1r ′0)

Integrated dipole response (voltage)

r(x , y , z ; t) =

K

∫ ∫s(x ′0, y

′0)z

r ′0ps(x

′0, y′0; t)dx ′0dy

′0



Total Response from Point Scatterer

r(x , y , z ; t) = KR(x , y , z)

⋅∫ ∫

dx ′0dy′0 s(x ′0, y

′0)

z

r ′20

⋅∫ ∫

dx0dy0 s(x0, y0)z

r 20

⋅ n(t − c−1r0 − c−1r ′0)

Now apply a series of approximations:▶ plane wave, paraxial▶ Fresnel, Fraunhofer



Plane Wave ApproximationExcitation pulse envelope arrives at all points at agiven range simultaneously.

Mathematically,

n(t − c−1r0 − c−1r ′0) ≈n(t − 2c−1z)e jk(r0−z)e jk(r ′0−z)

where wavenumber is

k = 2�f0c−1

and range equation gives

ct = 2z



Received Signal with Field PatternDefine field pattern as

q(x , y , z) =

∫ ∫s(x0, y0)

z

r 20

e jk(r0−z)dx0dy0

Then received signal (from single scatterer) is

r(x , y , z ; t) =

KR(x , y , z)n(t − 2c−1z)[q(x , y , z)]2



Basic Pulse-echo Imaging EquationSpatial distribution of scatterers

Assume superposition holds

Include attenuation

Total response is

r(t) = K

∫ ∫ ∫R(x , y , z)

n(t − 2c−1z)e−2�az [q(x , y , z)]2dxdydz



Paraxial ApproximationPattern is large near the transducer axis

Then r0 ≈ z

Field pattern becomes

q(x , y , z) ≈ 1

z

∫ ∫s(x0, y0)e jk(r0−z)dx0dy0



Fresnel and Fraunhofer ApproximationsBoth involve phase approximations

Fresnel field pattern

q(x , y , z) ≈ 1

zs(x , y) ∗ e jk(x2+y2)/2z

Fraunhofer field pattern

q(x , y , z) ≈ 1

ze jk(x2+y2)/2zS

( x

�z,y

�z

)for z ≥ D2/�.



General Pulse-echo EquationDefine

q(x , y , z) = zq(x , y , z)

Fresnel or Fraunhofer satisfies

r(t) = Ke−�act

(ct)2∫ ∫ ∫R(x , y , z)n(t − 2c−1z)q2(x , y , z)dxdydz



Time-gain CompensationAmplitude of r decays predictably

Compensate with time-varying gain

rc(t) = g(t)r(t) =∫ ∫ ∫R(x , y , z)n(t − 2c−1z)q2(x , y , z)dxdydz

g(t) =(ct)2e�act

K



Envelope Detection: A-modeComplex signal model n(t) throughout

Linear system model (superposition)

Therefore, gain-compensated A-mode signal is

ec(t) =

∣∣∣∣∫ ∫ ∫ R(x , y , z)

n(t − 2c−1z)q2(x , y , z)dxdydz∣∣

=

∣∣∣∣∫ ∫ ∫ R(x , y , z)

ne(t − 2c−1z)e j2kz q2(x , y , z)dxdydz∣∣



Transducer Motion and Range EquationMove transducer to (x0, y0); yields ec(t; x0, y0).

Use range equation as z0 = ct/2.

Then ec(⋅) estimates reflectivity

R(x0, y0, z0) = ec(2z0/c ; x0, y0)

=

∣∣∣∣∫ ∫ ∫ R(x , y , z)e j2kzne(2(z0 − z)/c)

⋅ q2(x − x0, y − y0, z)dxdydz∣∣


Ultrasound Imaging Resolution

Resolution CellWhere is the acoustic energy in space?

resolution cell(x , y , z ; x0, y0, z0) =

∣ne(2(z0 − z)/c)q2(x − x0, y − y0, z)∣

For geometric approximation

resolution cell(x , y , z ; x0, y0, z0) =

ne(2(z0 − z)/c)s(x − x0, y − y0)


Ultrasound Imaging Speckle

Origin of SpeckleUnder geometric assumption

R(x , y , z) =

K

∣∣∣∣R(x , y , z)e j2kz ∗ s(x , y)ne

(z

c/2

)∣∣∣∣Term e j2kz is “fast-changing” sinusoid in resolutioncell

Gives rise to essentially “random” constructive anddestructive interference


Physics of Magnetic Resonance

12 Physics of Magnetic ResonanceSpin SystemsMagnetizationNMR SignalExcitationRelaxationBloch EquationsSpin EchoesContrast


Physics of Magnetic Resonance Spin Systems

NucleiNMR is concerned with nuclei

... but not radioactivity

All nuclei have charge

Some nuclei have angular momentum Φ

Angular momentum + charge ≡ spin

Nuclei with spin are NMR-active



Visualization of Nuclear Spin



Nuclear Spin SystemsNuclear spin systems =

collections of identical nuclei▶ regardless of chemical environment▶ Examples: 1H, 13C, 19F, 31P

Whole-body MRI uses 1H▶ prevalent in the body (water, fat)▶ strong NMR signal▶ misnomer: “proton” imaging


Physics of Magnetic Resonance Magnetization

Microscopy Magnetic FieldMicroscopic magnetic moment vector:

� = Φ

is gyromagnetic ratio [radians/s-T]

– has more convenient units [Hz/T]

– =

2�

For 1H – = 42.58 MHz/T



Nuclear MagnetismPut sample in external magnetic field

B0 = B0z

Spins align in one of two directions▶ 54∘ off z “up”▶ 180− 54∘ off z “down”

Slight preference for “up” direction

Sample becomes magnetized



Macroscopic MagnetizationMagnetization vector:

M =

Ns∑n=1

�n



Equilibrum MagnetizationEquilibrium value: M0

▶ same direction as B0

▶ depends on x = (x , y , z) only

Magnitude: M0

M0 =B0

2ℏ2

4k–TPD

▶ k– is Boltzmann’s constant▶ T is temperature▶ PD is proton density



Evolution of MagnetizationM = M(x, t)

Relation to bulk angular momentum J

M = J

Focus on small sample → voxel

▶ M = M(t)▶ Equations of motion = Bloch equations



Torque on “Current Loop”Current loop in magnetic field

▶ magnetic (dipole) moment M▶ magnetic field B▶ torque is � = M× B



PrecessionTorque acts on rotating body in “funny” way

Torque is related to angular momentum

� =dJ

dt

Eliminate J to yield

dM(t)

dt= M(t)× B(t)

Equation describes precession

Valid for “short” times



Larmor FrequencyLet B(t) = B0; M(0) angle � with z

Then

Mx(t) = M0 sin� cos (− B0t + �)

My(t) = M0 sin� sin (− B0t + �)

Mz(t) = M0 cos�

whereM0 = ∣M(0)∣ � arbitrary

Precession with Larmor frequency

!0 = B0 or �0 = –B0



Isochromats!0 not constant for spin system due tomagnetic field inhomogeneities

Main magnetic field, shimming ⇒ ignore

magnetic susceptibility:▶ diamagnetic, paramagnetic, ferromagnetic▶ body/air interface strong change

chemical shift▶ chemical environment ⇒ shielding▶ fat is 3.35 ppm down from water

Isochromats: nuclei with same Larmor freq



Magnetization ComponentsMagnetization

M(t) = (Mx(t),My(t),Mz(t))

Think of M(t) with two components

▶ Longitudinal magnetization

Mz(t)

▶ Transverse magnetization

Mxy(t) = Mx(t) + jMy(t)


Physics of Magnetic Resonance NMR Signal

Origin of NMR SignalPrinciple of Reciprocity

Br(r) is field produced at r by unit directcurrent in coil around sample.

⇒ Now reverse scenario ⇐Voltage produced in coil by changing magnetic fieldis (by Faraday’s law of induction)

V (t) = − ∂

∂t

∫object

M(r, t) ⋅ Br(r) dr



NMR SignalLongitudinal magnetization changes too slow

Transverse magnetization dominates

Mxy(t) = M0 sin�e−j(!0t−�)

Final expression

V (t) = −!0VsM0 sin�B r sin(−!0t + �− �r)



Rotating FrameCoordinate transformation

x ′ = x cos(!0t)− y sin(!0t)

y ′ = x sin(!0t) + y sin(!0t)

z ′ = z

Transverse magnetization in rotating frame

Mx ′y ′(t) = M0 sin�e j�

Magnitude M0 sin�

Phase angle �


Physics of Magnetic Resonance Excitation

RF ExcitationCircularly polarized RF excitation pulse

B1(t) = Be1 (t)e−j(!0t−')

Yields forced precession

M(t) motion is spiral


Physics of Magnetic Resonance Excitation

Tip Anglez-magnetization magnitude after excitation

Mz = M0 cos�

Tip angle is

� =

∫ �p

0

Be1 (t)dt

where �p is pulse duration

For rectangular pulse:

� = B1�p


Physics of Magnetic Resonance Relaxation

RelaxationMagnetization cannot precess forever

Two independent relaxation processes

Transverse relaxation▶ ≡ spin-spin relaxation

Longitudinal relaxation▶ ≡ spin-lattice relaxation

Detailed properties differ in tissues▶ Gives rise to tissue contrast



Transverse Relaxation

Transverse relaxation decays

Mxy(t) = M0 sin�e−j(!0t−�)e−t/T2



Free Induction DecayWhat RF signal is produced?

Called a free induction decay (FID)



T ∗2 DecayIn fact RF signal decays faster

T ∗2 < T2

Underlying T2 relaxation is preservedJerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 354 / 412


Longitudinal RelaxationMz(t) behaves as rising exponential

Mz(t) = M0(1− e−t/T1) + Mz(0+)e−t/T1

Mz(0+) is value after RF excitation pulseM0 is final (equilibrium) value


Physics of Magnetic Resonance Bloch Equations

Bloch EquationsEquation(s) of “motion” for M(t)

dM(t)

dt= M(t)× B(t)− R{M(t)−M0}

Includes RF excitation

B(t) = B0 + B1(t) ,

Includes relaxation

R =

⎛⎝ 1/T2 0 00 1/T2 00 0 1/T1

⎞⎠Jerry L. Prince (Johns Hopkins University) Medical Imaging Signals and Systems August 20, 2009 356 / 412

Physics of Magnetic Resonance Spin Echoes

Concept of Spin EchoesPure transverse relaxation T2 is random

So why faster decay T ∗2 ?

▶ Fixed, local perturbations in magnetic field▶ Local dephasing from faster & slower spins

Echoes produced by “re-phasing”

▶ Make slower spins “jump” to front▶ Make faster spins “jump” to rear

TE is echo time

Multiple echoes are possible until about 3T2



Formation of a Spin Echo



Pulse Sequence for Spin Echo


Physics of Magnetic Resonance Contrast

Source of MR ContrastMR Contrast: why tissues “look” different in MRI

Intrinsic MR parameters:▶ T1, T2, and PD

Pulse sequence parameters:▶ tip angle �▶ echo time TE

▶ pulse repetition interval TR



Contrast Manipulation

PD-weighted T2-weighted T1-weighted

Weighted means “primarily influenced by”

Weighted does not mean “a picture of”



PD-weighted ContrastShould be proportional to # 1H nuclei in voxel

Procedure:▶ Start with sample in equilibrium▶ Apply excitation pulse▶ Image quickly

Practical parameters:▶ TR = 6000 ms (long)▶ TE = 17 ms (relatively quick)▶ � = �/2 (max signal)



T2-weighted ContrastEcho must be used because of T ∗2Procedure:

▶ Start with sample in equilibrium▶ Apply excitation pulse▶ Image at approx T2

Practical parameters:▶ TR = 6000 ms (long)▶ TE = 102 ms (moderate)▶ � = �/2 (max signal)

Trick: get PD and T2 in two echoes



Principle of T1-weighted Contrast



T1-weighted ContrastUses TR to capture T1 differences

Procedure:▶ Reach a steady-state, not equilibrium▶ Apply excitation so that TR ≈ T1

▶ Image quickly

Possible parameters:▶ TR = 600 ms (moderate)▶ TE = 17 ms (fast)▶ � = �/2 (max signal)


Magnetic Resonance Imaging

13 Magnetic Resonance ImagingMR Scanner ComponentsFrequency EncodingSlice SelectionSignal ModelsScanning Fourier SpaceGradient EchoesPhase EncodingSpin EchoesRealistic Pulse SequencesImage ReconstructionSampling, Resolution, and Noise


Magnetic Resonance Imaging MR Scanner Components

Five System Components1 Main magnet2 Gradient coils3 RF resonators or coils4 Pulse sequence electronics5 Computer and viewing console



MR Scanner Components



MR Scanner Photograph

http://www.gehealthcare.com http://www.gehealthcare.com



Superconducting Magnet1 meter niobium-titanium wiresuperconducting coils4∘ K liquid helium cryostat



Magnet SpecificationsField strengths from 0.5T to 12.0T

Most common field strength: 1.5T

Shimming to maintain homogeneous field▶ passive shimming▶ active shimming▶ better than ±5 ppm required

Minimize fringe field (outside the bore)

▶ nuisance and dangerous▶ passive: iron shield, or▶ active: second superconducting wires



Purpose of Gradient CoilsFit just inside the bore

Role: change B0 as a function of position

Three coils:▶ x , y , and z directions▶ Gx , Gy , and Gz strengths

Modify main field as follows

B = (B0 + Gxx + Gyy + Gzz)z

This is the key to MR imaging



Gradient Coilsx and y are saddle coilsz is opposing coils



Specifications of Gradient CoilsMaximum gradient 1–6 Gauss/cm

Switching times 0.1–1.0 ms

slew rates 5–250 mT/m/msec

Additional shielding outside to reduceeddy currents

FDA limit 40 T/s



RF CoilsTwo purposes:

▶ Exciting spin systems▶ Listening for FIDs and echoes

Two basic types:▶ volume coils▶ surface coils

Volume coils have uniform response

Surface coils are more sensitive buthave spatially-dependent response



RF Coil Designs

(a) saddle coil: head

(b) birdcage coil: body, head

(c) surface loop: peripherals



Scanning Console and ComputerControl scanner

Acquire images

Coordinate with EKG and breathing

Reconstruct images (10–50 images/s)

View, store, and manipulate images



Laboratory Coordinates


Magnetic Resonance Imaging Frequency Encoding

(Larmor) Frequency EncodingGradient G = (Gx ,Gy ,Gz) produces B-field:

B = (B0 + G ⋅ r)z

where r = (x , y , z)

Spatially varying Larmor frequency

�(r) = –(B0 + G ⋅ r)


Magnetic Resonance Imaging Slice Selection

Principle of Slice SelectionLet G = (0, 0,Gz)

Then�(r) = �(z) = –(B0 + Gzz)



Slice Selection ExcitationExcite frequencies � ∈ [�1, �2]

Causes “slab” excitation of spin system



Slice Selection ParametersRF parameters:

� =�1 + �2

2center frequency

Δ� = ∣�2 − �1∣ frequency range

Slice parameters:

z =� − �0

–Gzslice position

Δz =Δ�

–Gzslice thickness



Ideal Slice Selection RF ExcitationExcite frequencies in range [�1, �2] HzExcitation signal has Fourier transform

S(�) = A rect

(� − �

Δ�

)Signal is s(t) = AΔ� sinc(Δ�t)e j2��t



Practical Slice Selection RF ExcitationTruncated sinc

s(t) =[AΔ� sinc(Δ�t)e j2��t

]rect(t/�p)

Corresponding tip angle profile:

�(z) = A�prect

(z − z

Δz

)∗ sinc (�p Gz(z − z))



Slice Dephasing and RefocussingDifference Larmor frequencies across slice:

▶ “slow” on low side▶ “fast” on high side

Phase difference is

�(z) = Gz(z − z)�p/2

Refocus with negative gradient pulse▶ strength −Gz

▶ duration �/2



A Simple Pulse Sequence


Magnetic Resonance Imaging Signal Models

Basic Signal ModelSlice selection FID is

s(t) = e−j2��0t

∫ ∞−∞

∫ ∞−∞

f (x , y) dx dy

Where effective spin density is

f (x , y) = AM(x , y ; 0+)e−t/T2(x ,y)

Baseband signal is

s0(t) =

∫ ∞−∞

∫ ∞−∞

f (x , y) dx dy

Note: T ∗2 always decays the FID signal



Frequency Encoding



Frequency Encoding SignalLarmor frequency is function of x

�(x) = –(B0 + Gxx)

Baseband signal becomes

s0(t) =

∫ ∞−∞

∫ ∞−∞

f (x , y)e−j2� –Gxxt dx dy

Recognize Fourier transform frequencies:

u = –Gxt

v = 0


Magnetic Resonance Imaging Scanning Fourier Space

Relation to Fourier transform

F (u, 0) = s0

(u

–Gx

)



Polar Scanning

u = –Gxt v = –Gy t



Scanning Fourier SpaceIgnore readout/ADC

Applied gradients “drive” us around in Fourier space

This is concept of Fourier trajectory

Fourier trajectories underly all MRI▶ spin echoes▶ gradient echoes▶ frequency encoding▶ polar scanning▶ phase encoding


Magnetic Resonance Imaging Gradient Echoes

Gradient Echoes

Note: T ∗2 decay overall


Magnetic Resonance Imaging Phase Encoding

Phase Encoding



Phase Encoding SignalAccumulated phase after phase encode

�y(y) = − GyTpy

Baseband signal during readout

s0(t) =

∞∫−∞

∞∫−∞

f (x , y)e−j2� –Gxxte−j2� –GyTpy dx dy

Recognize Fourier transform frequencies:

u = –Gxt

v = –GyTp



Gradient Echo Pulse Sequence


Magnetic Resonance Imaging Spin Echoes

Concept of a Spin Echo


Magnetic Resonance Imaging Spin Echoes

Basic “Spin Echo” Pulse Sequence


Magnetic Resonance Imaging Realistic Pulse Sequences

Realistic Gradient Echo Pulse Sequence



Realistic Spin Echo Pulse Sequence



Realistic Spin Echo Polar Pulse Sequence


Magnetic Resonance Imaging Image Reconstruction

Acquired Rectilinear DataAcquire data for all phase encode areas

Ay = GyTp

Baseband signal

s0(t,Ay) =

∫ ∫f (x , y)e−j2� –Gxxte−j2� –Ayydxdy

Identify Fourier frequencies

u = –Gxt

v = –Ay



Image Reconstruction: Rectilinear DataFourier transform is built over repetitions

F (u, v) = s0

(u

–Gx,v

–

)0 ≤ u ≤ –GxTs

Inverse Fourier transform

f (x , y) =

∫ ∫s0

(u

–Gx,v

–

)e+j2�(ux+vy)dxdy

This is a fundamental equation in MRI



Acquired Polar DataGx and Gy identify readout parameters

% = –t√

G 2x + G 2

y

� = tan−1 Gy

Gx

Projection slice theorem connects 2D Fourier spaceto Fourier transform of 1D projection

G (%, �) = F (% cos �, % sin �)



Image Reconstruction: Polar DataBaseband signal is s0(t, �)

Relation to 1-D projection

G (%, �) = s0

⎛⎜⎝ %

–√G 2x + G 2

y

, �

⎞⎟⎠Filtered backprojection is the answer

f (x , y) =∫ �

0

[∫∣%∣G (%, �)e j2�%ℓ d%

]ℓ=x cos �+y sin �

d�


Magnetic Resonance Imaging Sampling, Resolution, and Noise

SamplingDuration of readout

Ts = NaT

Receiver bandwidth (sampling rate)

fs =1

T

Antialiasing filter chops outside [−fs/2, fs/2]



Readout Field of ViewAntialiasing filter chops Larmor frequencies,leading to

FOVx =fs –Gx

=1

–GxT

Step in Fourier space is

Δu = –GxT

Relation to field of view

FOVx =1

Δu



Phase Encode Field of ViewStep size in phase encode direction:

Δv = – ΔAy

Field of view

FOVy =1

– ΔAy

=1

Δv

Lack of antialising filter could cause wrap-around



ResolutionFourier space coverage

U = Nx –GxT

V = Ny –ΔAy

Implied lowpass filter is

H(u, v) = rect( uU

)rect

( vV

)Spatial PSF is

h(x , y) = UV sinc(Ux)sinc(Vy)



Full Width Half Max’sFWHMs are

FWHMx =1

U=

1

Nx –GxT=

1

NxΔu

FWHMy =1

V=

1

Ny –ΔAy=

1

NyΔv

The Fourier resolutions of MRI



NoiseJohnson (thermal) noise dominates

�2 =2k–T RTA

k– = Boltzmann’s constant

T = temperature ⇒ colder is better

R = effective resistance ⇒ use small coils

TA = total acquisition time ⇒ scan longer



Signal-to-Noise RatioRecall magnitude of signal is

∣V ∣ = 2��0VsM0 sin�B r

Signal-to-noise Ratio is

SNR =∣V ∣√�2

= h2

√4�k–

2��0PD√�

r 20

√LT 3

Vs sin�√TA


prince&links-medical imaging signals&systems allslides 2009

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