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The University of Nottingham

SCHOOL OF PHYSICS & ASTRONOMY

A LEVEL 3 MODULE, SPRING SEMESTER 2010-2011

FUNCTIONAL MEDICAL IMAGING

Time allowed ONE Hour THIRTY Minutes

Candidates may complete the front cover of their answer book and sign their desk card butmust NOT write anything else until the start of the examination period is announced.

Answer THREE out of Four Questions

Only silent, self contained calculators with a Single-Line Display or Dual-Line Display arepermitted in this examination.

Dictionaries are not allowed with one exception. Those whose first language is not Englishmay use a standard translation dictionary to translate between that language and

English provided that neither language is the subject of this examination.Subject specific translation dictionaries are not permitted.

No electronic devices capable of storing and retrieving text, including electronic dictionaries,

may be used.

An indication is given of the approximate weighting of each part of a question by means of abold figure enclosed by curly brackets, e.g. {2}, immediately following that part.

DO NOT turn examination paper over until instructed to do so

Speed of light in free space c 3.00x108 m s-1Gravitational constant G 6.67x10-11 N m2 kg-2Plancks constant h 6.63x10-34 J s

1.055x10-34 J sElementary charge e 1.60x10-19 CMass of electron me 9.11x10

-31 kgMass of proton mp 1.6726x10

-27 kgMass of neutron mn 1.6749x10

-27 kgBoltzmanns constant kB 1.38x10

-23 J K-1Gas constant R 8.31 J K-1 mol-1Permittivity of free space 0 8.85x10

-12 F m-1Permeability of free space 0 4x10

-7 H m-1Bohr magneton B 9.27x10

-24 J T-1Stefan-Boltzmann constant 5.67x10-8 W m-2 K-4Avogadros number NA 6.02x10

23 mol-1

Gyromagnetic ratio for protons H 42.57x106 Hz T-1

Gyromagnetic ratio for carbon-13 C13 10.71x106 Hz T-1

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You must answer 3 out of 4 questions.You should aim to spend about 30 minutes on each question.

1. (a) A sample has a T2* of 100 ms. Sketch the Free Induction Decay when the signal is(i) on resonance and (ii) off resonance by 10 Hz. {3}

(b) Explain how slice selection is achieved in MRI. Describe how to excite a 3 mm slicein the presence of a 20 mT/m gradient. {2} Calculate the width of the first zerocrossing of the RF pulse, T. {3}

(c) Explain how a gradient echo is formed from an FID. {2}(d) Spin warp imaging and echo-planar imaging are both gradient echo imaging

methods. For each, describe how an image is formed including the k-space

diagram. {10}

(e) Discuss the advantages and disadvantages {3}, and applications of each ofthese imaging methods. {2}

2. (a) Show that in the high temperature limit the population difference of an ensembleof spin nuclei is given by:

Tk

BNn

B

z

2

1

whereBz is the main magnetic field, Nthe total number of spins in the system,

and Tthe temperature of the system. {3} Evaluate this population difference for1Hwhen the temperature is 300 K and the applied magnetic field is 3 T. Expressyour answer in ppm, as a fraction of the total number of spins in the system.{2}

(b) Write down and explain the equation of motion for M in the absence ofrelaxation {2}. Give the expression for the Larmor precession frequency. {2}

(c) Explain the advantages of using the rotating frame. {1} Show thatthe equationof motion for M in the rotating frame (in the absence of relaxation) when thesystem is subjected to rf radiation with angular frequency polarised along the

x-axis and a static magnetic field aligned along thez-axis, is:

eff

zyxdt

dBM

M

''',,

and derive the form ofBeff. {6}

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(d) Given that a 1H nucleus resonates at 500 MHz, determine the resonancefrequency of

13C for the same spectrometer {1}. In MR spectroscopy, what is

the origin of the chemical shift and state the equation defining the chemical

shift. {2} In a1Hspectra recorded at 500 MHz the chemical shift of two species

of interest are 1.5 ppm and 2.7 ppm, by how much in Hz are their resonancesseparated? {2}

(e) Contrast the use of 1H and 13CMRS in the study of human brain metabolism.{5}

3. (a) Solve the Bloch equation to produce an expression for the recovery of longitudinalmagnetisation after an inversion pulse, and give the equation for the transversemagnetisation following a 900 pulse. {3} Describe the origin of longitudinal and

transverse relaxation. {4}

(b) Grey matter (GM) and white matter (WM) at 3 T have the following longitudinaland transverse relaxation times (T1,GM= 1200 ms T1,WM = 800 ms ; T2*GM= 40

ms and T2*WM =60 ms ). You may assume the spin density (S0) is the same forgrey and white matter. Calculate:

i) the inversion time which maximises the contrast between grey matter andwhite matter in a T1-weighted image produced using an inversion recoverysequence; {5}

ii) the echo time which maximises the contrast between grey matter andwhite matter in a T2* weighted image produced using a gradient echosequence. {5}

(c) Image contrast can be enhanced using intravenous MR contrast agents. What

properties should these have? {3}How much will the T1 of grey matter at 3 Tchange if Gd-DTPA is taken up in the tissue to a concentration of 0.1 mM,

assuming T1 relaxivity of Gd-DTPA of 4.52 mM-1s-1.{2}

(d) Haemoglobin is used as an endogenous contrast agent for fMRI. Explain theunderlying contrast mechanism and how the optimum echo time is chosen. {3}

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4. (a) Explain why it is thought that synchronised, post synaptic current flow in thedendrites of pyramidal neurons is responsible for generating EEG and MEGsignals. {5}

(b) Write down the equation for the magnitude of a current dipole moment in asingle dendrite. {1} The conductivity of a neuron is ~

111

m , the dendriticdiameter is m1 . Assuming a voltage change of 25 mV across the dendrites

length, show that the dipole moment for a single dendrite is approximately 20fAm. {4}

(c) Briefly state the EEG/MEG forward and inverse problems {2}. Why is the inverseproblem ill-posed? {1} What is the crucial difference between the MEG and EEGforward problems, which explains the main advantage of MEG compared to EEG?{2}

(d) A current dipole,Q , is located at position Qr inside a spherical conductor. Aradially-orientated, superconducting pickup coil is located at position, r , outside

the sphere. The radial component of the magnetic field,r

B , is given by:

3

0

4

.

Q

rQ

rBrr

erQ

wherere is a unit vector pointing in the radial direction at position r .

If cmQ 7r , cm14r , and the angle between r and Qr is 200, and we assume

that Q arises from synchronous current flow in 100,000 dendrites, calculate themagnitude of the detected magnetic field. You may assume that Q is oriented in

the tangential plane. Note you will need to use the answer to part (b) {5}.

Note cos2222

QQQ rrrrrr

is the angle between r and Qr .

27

0 104 NA

(e) Write down the matrix equation representing MEG measurements, and labeleach of the terms. {2} Explain what is meant by a lead field vector {1} and howit can be used in a dipole fitting algorithm. {2}

End