Basic Concept of MRI Chun Yuan

Magnetic Moment Magnetic dipole and magnetic moment Nuclei with an odd number of protons or neutrons have a net magnetic moment (spin) Most common nuclei which have magnetic moments are: – 1 H, 2 H, 7 Li, 13 C, 19 F, 23 Na, 31 P, and 127 I Electron Proton Neutron _ _

No External Magnetic Field In the absence of an external magnetic field The nuclei align randomly The nuclei produce no net magnetization

External Magnetic Field (B 0 ) The nuclei align in 1 of 2 positions depending on energy state Low energy nuclei align with the field in parallel position High energy nuclei align against the field in antiparallel position B0B0

Increasing B 0 As B 0 increases more nuclei align in the parallel low energy position B0B0

Net Magnetization Vector A net magnetization vector is formed – Pairs of parallel and antiparallel nuclei cancel – The magnetic moments of the unpaired nuclei create a sum effect called net magnetization vector – Only the unpaired nuclei participate in the MR signal B0B0

Net Magnetization Vector The net vector is the sum of all of the parallel, unpaired, low energy protons – The strength is the SUM of the magnetic strengths of the individual protons – The direction is the SUM of the polar directions of the individual protons – In the low energy state the net vector aligns along the longitudinal or Z axis and is called Mz B0B0 Mz

Precession in B 0 They wobble like gyroscope – Thermal agitation prevents the nuclei from aligning perfectly with B0 so the nuclei actually align at an angle – As B0 attempts to pull the nuclei into perfect alignment the conflicting forces cause the nuclei to precess B0B0

The Larmor Equation The larmor equation calculates the frequency of precession – Precessional frequency depends on The type of nucleus The strength of the external magnetic field   Omega or Precessional Frequency Gamma or Gyromagnetic Ratio External Magnetic Field Strength

Gyromagnetic Ratio  The gyromagnetic ratio yields frequency at 1 Tesla The GMR is unique for each type of nucleus GMR in MHz Nucleus n 1H 2H 13C 14N 19F 23Na 27Al 31P

Example Most MR scanners operate at 1.5 T. What is the Larmor Frequency of protons at this field strength? f 0 =  0 / 2  =  B 0 / 2  = (42.58 MHz/T)(1.5 T) = MHz

There’s No Signal Yet M z CANNOT BE MEASURED WHEN ALIGNED WITH B0 M z must be moved away from B 0 in order to generate a signal How do we move M z away from B 0 ? B0B0 Mz

RF Excitation The frequency of the RF energy must match the frequency of the precessing nuclei in order to transfer energy The magetic field exerted by the RF energy is called B1 B1 must be transmitted perpendicular to B0 B0B0 =  B1B1

Resonance In the presence of B1, low energy nuclei absorb energy and shift to high energy state B0B0 B1B1 Mz B0B0 B1B1

Shift of the Net Magnetization The direction of net vector shifts as the individual nuclei transition to high energy – The RF pulse is labelled according to shift it creates in the net magnetization – A 90 degree pulse moves the net magnetization 90 degrees – How far does a 180 degree pulse move the net magnetization? – When the net magnetization is in the transverse plane it is called Mxy B0B0 B1B1 Mxy

Flip Angle Magnetization is tipped using a radiofrequency pulse – Frequency of RF pulse is ω 0 – Magnitude of RF pulse is B 1 (t) – Total tip angle is α=γ∫B 1 (t) dt – α=90º (π/2) maximizes signal – α=180º (π) called an inversion pulse M essentially precesses around B1(t) with an instantaneous frequency of  =  B(t) x y z M B 1 (t) M0M0

Example An RF field B 1 exp{-j  0 t} is applied to a sample where B 1 = 50 milligauss. How long must it be applied to produce a tip of 90º?  B 1  t =  /2  t =  /(2  B 1 ) =  /(2 (2  x MHz/T)(0.05 Gauss x T/Gauss)) = 1.17 milliseconds

Relexation WHEN B1 IS REMOVED THE NUCLEI EMIT ENERGY AND SHIFT BACK TO LOW ENERGY STATE THE TRANSITION BACK TO LOW ENERGY STATE IS CALLED RELAXATION AFTER EMITTING ENERGY THE NUCLEI RETURN TO PARALLEL ALIGNMENT B0B0 Mxy

Faraday’s Law of Induction 3 CRITERIA MUST BE MET TO GENERATE A SIGNAL – A conductor – A magnetic field – Motion of the magnetic field in relation to the conductor IN MR – The RF coil provides the conductor – And Mxy provides the moving magnetic field because it precesses x y z M M xy MzMz Antenna s(t)  M xy (t)

Free Induction Decay (FID) In the 90-FID pulse sequence, net magnetization is rotated down into the XY plane with a 90 o pulse. The net magnetization vector begins to precess about the +Z axis. The magnitude of the vector also decays with time.

Bloch Equation The Block equation relate the time evolution of magnetization to – the external magnetic fields, – relaxation times (T1 and T2), – the molecular self-diffusion coefficient (D).  is the gyromagnetic ratio –depends on nucleus –For proton  = MHz/Tesla

Rotation Reference Frame xyx'y'

External Magnetic Fields Static Magnetic Field B 0 RF Magnetic Field B 1 x y x' y' z

T1 Relaxation T1 relaxation is also known as thermal or spin-lattice relaxation T1 relaxation involves an energy exchange--excited nuclei release energy and return to equilibrium T1 relaxation causes recovery of the net magnetization to the longitudinal axis MzMz t Short T 1 Long T 1 M0M0 63%

Example For a sample with T 1 = 1 second, how long after a 180 degree pulse will the net magnetization be 0? z M z M z M0M0 z M M z (t) = M 0 (1 - e -t/T1 ) + M z (0) e -t/T1 0 = M 0 (1 - e -t/T1 ) - M 0 e -t/T1 0 = 1 - 2e -t/T1 t = T 1 ln2 = 0.69 seconds

T2 Relaxation T2 relaxation is also known as thermal or spin-spin relaxation T2 relaxation involves the loss of phase coherence and is caused by the local magnetic field T2 relaxation causes dephasing of the net magnetization in the transverse plane T2T2 37%

T2* (Star) Relaxation Two factors contribute to the decay of transverse magnetization. – molecular interactions (said to lead to a pure T 2 molecular effect) – variations in B o (said to lead to an inhomogeneous T 2 effect The combination of these two factors is what actually results in the decay of transverse magnetization. The combined time constant is called T 2 star and is given the symbol T 2 *. The relationship between the T 2 from molecular processes and that from inhomogeneities in the magnetic field is as follows.

Relaxation and contrast Relaxation time T1, T2 and T2* vary with – Field strength – Temperature – Tissue types – In vitro vs. in vivo – Age Fundamentally important for generating contrast At 1.5T: Gray matterWhite MatterCSF T 1 (ms) T 2 (ms) proton density (relative)

Images with Different Contrast

Example Suppose an  degree RF pulse is applied every TR seconds for a long time. What is the steady-state magnitude of M xy immediately after excitation assuming TR >> T 2 Let M(n - ) be the magnetization just before the n th RF pulse and M(n + ) be the magnetization just after the pulse. Because TR >> T 2, we know M xy (n - ) = 0. Therefore, M xy (n + ) = M z (n - ) sin   and M z (n + ) = M z (n - ) cos  T1 relaxation gives M z ([n+1] - ) = M 0 (1 - e -TR/T1 ) + M z (n + ) e -TR/T1...  TR RF

Solution At steady state, M(n - ) = M([n+1] - ) M z (n - ) = M 0 (1 - e -TR/T1 ) + M z (n + ) e -TR/T1 M z (n - ) = M 0 (1 - e -TR/T1 ) + M z (n - ) cos  e -TR/T1 Thus, M z (n - ) = M 0 (1 - e -TR/T1 ) / (1- cos  e -TR/T1 ) and M xy (n + ) = M z (n - ) sin  = M 0 sin  (1 - e -TR/T1 ) / (1- cos  e -TR/T1 ) (This equation comes in handy for analyzing MR imaging because images require multiple RF excitations and this equation is useful for optimizing  )

Spin Echo The basic MRI sequence is called “spin echo”. The RFexcitation for spin echo is as follows: Sketch its response, where TE is on the order of several times T 2 * We know we get an FID in response to the 90 degree pulse: But, what does the 180 degree pulse do? 90º180º TE/2 RF T2*T2*

Spin Echo Recall dephasing gives: After the 180 degree pulse, the faster spins trail the slower ones: Thus, the spins “rephase”, then dephase again: (Note: Only dephasing due to T2* can be rephased. T2 relaxation is affected by random processes. Thus, the echo is lower in amplitude than the original FID) x y slower faster x y slower faster 180º T2*T2* 90º180º T2*T2* T2*T2* T2T2 TE s(t) RF

Spin Echo

Spin echo signal for  =90  From previous slide, with  =90: M xy (0) = M 0 (1 - e -TR/T1 ) Adding T2 relaxation gives: M xy (TE) = M 0 (1 - e -TR/T1 ) e -TE/T2 “proton density (PD)” T 1 weighting T 2 weighting PD weighting T 1 weighting T 2 weighting T 1 long short long T 2 short short long

MR Image Formation Magnetic Field Gradient Three key concepts in MRI formation: – slice selection – frequency encoding – phase encoding

Slice Selection Goal: Excite (M z -> M xy ) in a well defined slice of tissue Application of RF pulse and gradient field – Energy deposition at selective frequencies Excite this slice only RF GzGz

Diagram of Slice Selection GzGz RF bandwidth slice thickness depends on RF pulse bandwidth slice thickness depends on Gradient strength wider BW wider slice steeper gradient narrower slice B0B0 BW = (  /2  )Gz  z

Pulse shape Slice profile  Fourier Transform of the RF pulse shape Square pulse: Better choice: sinc pulse FT  RF PulseSlice Profile FT  RF PulseSlice Profile BW ~ 1/  T

Slice Profile and RF Pulse

Example What duration should an RF pulse be to excite a 1 mm slice of tissue using a gradient strength of 5 Gauss/cm (assume bandwidth (Hz)  1/duration (sec)). Required bandwidth is BW = (  /2  )G z  z = (42.58 MHz/T)(5 x 10-4 T/cm)(0.1 cm) = 2.1 kHz  T= 1/BW = 0.47 msec

Slice Dephasing Total dephasing roughly equivalent to half the area of the gradient Can be fixed with a negative gradient with half the area: GzGz

Phase Encoding Phase encoding gradient is imposed before acquisition While the gradient is on the nuclei precess at different frequencies When the gradient is turned off the nuclei return to precessing at the same frequency but their phase has been shifted relative to their gradient position GpGp GpGp GpGp

Phase Encoding Equation

Frequency Encoding Goal: Map M xy (x,y) within the slice or “image plane” Application of gradient field G x after slice selection – Position along x axis encoded by frequency – applied during data acquisition – Centered at echo GfGf

Frequency Encoding Equation Note: Signal acquired in k x, k y space is a Fourier transform of M(x,y), so image M(x,y) can be reconstructed with inversed Fourier transform.