# Presentation: 546444

## (F)MRI PhysicsWith Hardly Any Math*

**Robert W Cox, PhD**
- Scientific and Statistical Computing Core
- National Institute of Mental Health
- Bethesda, MD  USA

- *Equations will be supplied to the inquiring student

## MRI  Cool (and Useful) Pictures

- 2D slices extracted from a 3D image
- [resolution about 111 mm]

- axial

- coronal

- sagittal

## Synopsis of MRI

-  1) Put subject in big magnetic field (leave him there)
-  2) Transmit radio waves into subject    [about 3 ms]
-  3) Turn off radio wave transmitter
-  4) Receive radio waves re-transmitted by subject
-  Manipulate re-transmission with magnetic fields during this *readout* interval     [10-100 ms: MRI is not a snapshot]
-  5) Store measured radio wave data vs. time
-  Now go back to 2) to get some more data
-  6) Process raw data to reconstruct images
-  7) Allow subject to leave scanner (this is optional)

## Components of Lectures

-  1) Magnetic Fields and Magnetization
-  2) Fundamental Ideas about the NMR RF Signal
-  3) How to Make an Image
-  4) Some Imaging Methods
-  5) The Concept of MRI Contrast
-  6) Functional Neuroimaging with MR

- }

- NMR
- Physics

- }

- MRI
- Principles

- }

- Making
- Useful
- Images

## Part the First

_**Part the First**_
- Magnetic Fields;
- Magnetization of the Subject;
- How the Two Interact

## Magnetic Fields

-  Magnetic fields create the substance we “see”: *magnetization* of the H protons in H2O
-  Magnetic fields also let us manipulate magnetization so that we can make a map [or *image*] of its density inside the body’s tissue
-  *Static* fields change slowly (not at all, or only a few 1000 times per second)
-  Main field; gradient fields; static inhomogeneities
-  *RF* fields oscillate at Radio Frequencies (tens of millions of times per second)
-  transmitted radio waves into subject
-  received signals from subject

## Vectors and Fields

-  Magnetic field *B* and magnetization *M* are *vectors*:
-  Quantities with direction as well as size
-  Drawn as arrows ....................................
-  Another example: velocity is a vector (speed is its size)
-  A *field* is a quantity that varies over a spatial region:
-  e.g., velocity of wind at each location in the atmosphere
-  Magnetic field exerts torque to line magnets up in a given direction
-  direction of alignment is direction of *B*
-  torque proportional to size of *B* [units=*Tesla**,** Gauss=*10–4 T]

## B0 = Big Field Produced by Main Magnet

*B*0* = *Big Field Produced by Main Magnet

-  Purpose is to align H protons in H2O (little magnets)

## Slide 9

## Precession of Magnetization M

- it precesses clockwise around
- the direction of *B**.*
-  However, “normal” (*fully relaxed*)* *situation has *M* parallel to *B*, which means there won’t be any
- precession

- _Precession of Magnetization __*M*_

-  Magnetic field causes *M* to rotate (or *precess*) about the direction of *B* at a frequency proportional to the size of *B* — 42 million times per second (42 MHz), per Tesla of *B*

-  N.B.: part of *M* parallel to *B* (Mz)
-     does not precess

## A Mechanical Analogy

-  A gyroscope in the Earth’s gravitational field is like magnetization in an externally applied magnetic field

## How to Make M not be Parallel to B?

-  A way that does *not* work:
-  Turn on a second big magnetic field *B*1 perpendicular to main *B*0 (for a few seconds)
- Then turn *B*1 off;  *M* is now not parallel to magnetic field *B*0
-  This fails because cannot turn huge (Tesla) magnetic fields on and off quickly
-  But it contains the kernel of the necessary idea:
- 			A magnetic field *B*1 perpendicular to *B*0

*B*0

* **M* would drift over to be aligned  	with sum of *B*0 and *B*1

## B1 = Excitation (Transmitted) RF Field

-    to cause *M* to spiral away
-    from the direction of the
-    static *B* field
* **B*110–4 Tesla
-  This is called *resonance*
-  If *B*1 frequency is not close to
-    resonance, *B*1 has no effect

*B*1 = Excitation (Transmitted) RF Field

-  Left alone, *M* will align itself with *B* in about 2–3 s
-  _So don’t leave it alone_: apply (transmit) a magnetic field *B*1 that fluctuates at the precession frequency and points perpendicular to *B*0

- Time = 2–4 ms

## Another Mechanical Analogy: A Swingset

-  Person sitting on swing at rest is “aligned” with externally imposed force field (gravity)
-  To get the person up high, you could simply supply enough force to overcome gravity and lift him (and the swing) up
-  Analogous to forcing *M* over by turning on a huge static *B*1
-  The other way is to push back and forth with a tiny force, synchronously with the natural oscillations of the swing
- Analogous to using the tiny RF *B*1 to slowly flip *M* over

*g*

## Readout RF

-  When excitation RF is turned off, *M* is left             pointed off at some angle to *B*0 [*flip angle*]
-  Precessing part of *M* [Mxy] is like having a magnet rotating around at very high speed (at RF frequencies)
-  Will generate an oscillating voltage in a coil of wires placed around the subject — this is magnetic *induction*
* *This voltage is the *RF signal* whose measurements form the raw data for MRI
-  At each instant in time, can measure one voltage *V*(*t*), which is proportional to the sum of all transverse Mxy inside the coil
-  _Must find a way to separate signals from different regions_

## Part the Second

_**Part the Second**_
- Fundamental Ideas
- about
- the NMR RF Signal

## Relaxation: Nothing Lasts Forever

- _Relaxation: Nothing Lasts Forever_

-  In absence of external *B*1, *M* will go back to being aligned with static field *B*0 — this is called *relaxation*
-  Part of *M* perpendicular to *B*0 shrinks [Mxy]
-  This part of *M* is called *transverse magnetization*
-  It provides the detectable RF signal
-  Part of *M* parallel to *B*0 grows back [Mz]
-  This part of *M *is called *longitudinal magnetization*
* *Not directly detectable, but is converted into transverse magnetization by externally applied *B*1

## Relaxation Times and Rates

-  Times: ‘T’ in exponential laws like *e*–*t*/T
-  Rates: R = 1/T [so have relaxation like *e*–R*t*]
-  T1: Relaxation of *M** *back to alignment with *B*0
-  Usually 500-1000 ms in the brain [lengthens with bigger *B*0]
-  T2: Intrinsic decay of the transverse magnetization over a microscopic region ( 5-10 micron size)
-  Usually 50-100 ms in the brain [shortens with bigger *B*0]
-  T2*: Overall decay of the observable RF signal over a macroscopic region (millimeter size)
-  Usually about half of T2 in the brain [i.e., faster relaxation]

## Material Induced Inhomogeneities in B

-  Adding a nonuniform object (like a person) to *B*0 will make the total magnetic field *B* nonuniform
-  This is due to *susceptibility*: generation of extra magnetic fields in materials that are immersed in an external field
-  *Diamagnetic* materials produce negative *B* fields
-  *Paramagnetic* materials produce positive *B* fields
-  Size about 10–7*B*0 = 1–10 Hz change in precession ***f***
-  Makes the precession frequency nonuniform, which affects the image intensity and quality
-  For large scale (10+ cm) inhomogeneities, scanner-supplied nonuniform magnetic fields can be adjusted to “even out” the ripples in *B **— *this is called *shimming*
* *Nonuniformities in *B* bigger than voxel size affect whole image
-  Nonuniformities in *B* smaller than voxel size affect voxel “brightness”

## Frequency and Phase

-  RF signals from different regions that are at different frequencies will get *out of phase* and thus tend to cancel out
-  Phase = the *t** *in cos(*t*)   [frequency ***f *** = /2]

## Sum of 500 Cosines with Random Frequencies

- _Sum of 500 Cosines with Random Frequencies_

-  Starts off large when all phases are about equal

- High frequency gray curve is at the average frequency

-  Decays away as different
-  components get different phases

## Transverse Relaxation and NMR Signal

-  Random frequency differences inside intricate tissue environment cause RF signals (from Mxy) to *dephase*
-  Measurement = sum of RF signals from many places
-  Measured signal decays away over time [T2*40 ms at 1.5 T]
-  At a microscopic level (microns), Mxy signals still exist; they just add up to zero when observed from outside (at the RF coil)
-  Contents of tissue can affect local magnetic field
-  Signal decay rate depends on tissue structure and material
-  Measured signal strength will depend on tissue details
-  If tissue contents change, NMR signal will change
-  e.g., oxygen level in blood affects signal strength

## Hahn Spin Echo: Retrieving Lost Signal

-  Problem: Mxy rotates at different rates in different spots
-  Solution: take all the Mxy’s that are ahead and make them get behind (in phase) the slow ones
-  After a while, fast ones catch up to slow ones **** re-phased!

- Fast & slow
- runners

- Magically “beam”
- runners across track

- Let them run the
- same time as before

##  The “magic” trick:      inversion of the magnetization M 

-  The “magic” trick:      *inversion* of the magnetization *M*
-  Apply a second *B*1 pulse to produce a flip angle of 180 about the *y*-axis (say)
-  Time between first and second *B*1 pulses is called TI
-  “Echo” occurs at time TE = 2TI

## Spin Echo:

-  **Excite**
-  **Precess**
**   ****& dephase**
-  **180******** flip**
-  **Precess**
**   ****& rephase**

## Relaxation: My Last Word

-  Spin echo doesn’t work forever (TI can’t be too big)
-  Main reason: water molecules diffuse around randomly
-  About 5-10 microns during 10-100 ms readout window
-  They “see” different magnetic fields and so their precession frequency changes from fast to slow to fast to ................
-  This process cannot be reversed by the inversion RF pulse
- Time scale for irreversible decay of Mxy is called T2
-  Longitudinal relaxation of Mz back to “normal” (T1)
-  Caused by internal RF magnetic fields in matter
-  Thermal agitation of H2O molecules
-  Can be enhanced by magnetic impurities in tissue
-  Drugs containing such impurities can alter T1, T2, and T2* — contrast agents (e.g., Gd-DTPA, MION)

## Part the Third

- Localization
- of the
- NMR Signal,
*or*,
- How to Make Images

## Steps in 3D Localization

-  Can only detect total RF signal from entire 3D volume inside the “RF coil” (the detecting antenna)
-  Excite Mxy in only a thin (2D) slice of the subject
-  The RF signal we detect must come from this slice
-  Have localized from 3D down to 2D
-  Deliberately make magnetic field strength *B* depend on location within slice
-  Frequency of RF signal will depend on where it comes from
-  Breaking total signal into frequency components will provide more localization information
-  Make RF signal phase depend on location within slice

## Spatially Nonuniform B: Gradient Fields

-  Extra static magnetic fields (in addition to *B*0) that vary their intensity in a linear way across the subject
-  Precession frequency of *M* varies across subject
-  This is called *frequency encoding* — using a deliberately applied nonuniform field to make the precession frequency depend on location

***x***-axis

**f**

- 60 KHz

- Left = –7 cm

- Right = +7 cm

*G**x **= *1 Gauss/cm = 10 mTesla/m
-      = strength of gradient field

- Center
- frequency
- [63 MHz at 1.5 T]

##  Exciting Mxy in a Thin Slice of Tissue

-  Exciting Mxy in a Thin Slice of Tissue

##  Readout Localization

-  After RF pulse (*B*1) ends, acquisition (readout) of NMR RF signal begins
-  During readout, gradient field perpendicular to slice selection gradient is turned on
-  Signal is sampled about once every microsecond, digitized, and stored in a computer
- Readout *window* ranges from 5–100 milliseconds (can’t be longer than about 2T2*, since signal dies away after that)
-  Computer breaks measured signal *V*(*t*) into frequency components *v*(*f** *) — using the *Fourier transform*
-  Since frequency f* *varies across subject in a known way, we can assign each component *v*(*f** *) to the place it comes from

## Image Resolution (in Plane)

-  Spatial resolution depends on how well we can separate frequencies in the data *V*(*t*)
-  Resolution is proportional to *f*  = frequency accuracy
-  Stronger gradients **** nearby positions are better separated in frequencies **** resolution can be higher for fixed *f*
-  Longer readout times **** can separate nearby frequencies better in *V*(*t*) because phases of cos(ft) and cos([*f*+*f*]t) will have longer to separate: *f* = 1/(readout time)

## The Last Dimension: Phase Encoding

-  Slice excitation provides one localization dimension
-  Frequency encoding provides second dimension
-  The third dimension is provided by *phase encoding*:
-  We make the phase of Mxy (its angle in the *xy*-plane) signal depend on location in the third direction
-  This is done by applying a gradient field in the third direction (**** to both slice select and frequency encode)
-  Fourier transform measures phase  of each *v*(f ) component of *V*(*t*), as well as the frequency f
-  By collecting data with many different amounts of phase encoding strength, can break each *v*(f ) into phase components, and so assign them to spatial locations in 3D

## Part the Fourth

- Some Imaging Methods

## The Gradient Echo

-  Spin echo: when “fast” regions get ahead in phase, make them go to the back and catch up
-  Gradient echo: make “fast” regions become “slow” and vice-versa
-  Only works when different precession rates are due to scanner-supplied gradient fields, so we can control them
-  Turn gradient field on with negative slope for a while, then switch it to have positive slope
-  What was fast becomes slow (and vice-versa) and after a time, the RF signal phases all come back together
-  The total RF signal becomes large at that time (called TE)

## MRI Pulse Sequence for Gradient Echo Imaging

- Illustrates
- sequence of
- events during
- scanning
- As shown, this method
- (FLASH)
- takes 35 ms
- per RF shot,
- so would take
- 2.25 s for a
- 6464 image

## Why Use the Gradient Echo?

-  Why not readout without negative frequency encoding?
-  Purpose: delay the time of maximum RF signal
-  Occurs at *t* = TE after the RF pulse
-  During this time, magnetization *M* will evolve not only due to externally imposed gradients, but also due to microscopic (sub-voxel) structure of magnetic field inside tissue
-  Delaying readout makes signal more sensitive to these internal details
-  Resulting image intensity *I*(*x*,*y*) depends strongly on T2* at each location (*x*,*y*)
-  Most sensitive if we pick TE  average T2*

## MRI Pulse Sequence for Spin Echo Imaging

## Why Use the Spin Echo?

-  Purpose: re-phase the NMR signals that are lost due to sub-voxel magnetic field spatial variations
-  Resulting image intensity *I*(*x*,*y*) depends strongly on T2 at each location (*x*,*y*)
-  Most sensitive if we pick TE  average T2
-  SE images depend mostly on tissue properties at the 5 micron and smaller level (molecular to cellular sizes) = diffusion scale of H2O in tissue during readout
-  GE images depend on tissue properties over all scales up to voxel dimensions (molecular to cellular to structural)

## Echo Planar Imaging (EPI)

-  Methods shown earlier take multiple RF shots to readout enough data to reconstruct a single image
-  Each RF shot gets data with one value of phase encoding
-  If gradient system (power supplies and gradient coil) are good enough, can read out all data required for one image after one RF shot
-  Total time signal is available is about 2T2* [80 ms]
-  Must make gradients sweep back and forth, doing all frequency and phase encoding steps in quick succession
-  Can acquire 10-20 low resolution 2D images per second

## GE-EPI Pulse Sequence

- Actually have
- 64 (or more)
- freq. encodes
- in one readout
- (each one **<** 1 ms)
- [only 13 freq.
- encodes
- shown here]

## What Makes the Beeping Noise in EPI?

-  Gradients are created by currents through wires in the gradient coil — up to 100 Amperes
-  Currents immersed in a magnetic field have a force on them — the Lorentz force — pushing them sideways
-  Switching currents back and forth rapidly causes force to push back and forth rapidly
-  Force on wires causes coil assembly to vibrate rapidly
-  Frequency of vibration is audio frequency
-  about 1000 Hz = switching rate of frequency encode gradients
-  scanner is acting like a (low-fidelity) loudspeaker

## Other Imaging Methods

-  Can “prepare” magnetization to make readout signal sensitive to different physical properties of tissue
-  Diffusion weighting (scalar or tensor)
-  Magnetization transfer (sensitive to proteins in voxel)
-  Flow weighting (bulk movement of blood)
-  Perfusion weighting (blood flow into capillaries)
-  Temperature**;** T1, T2, T2***;** other molecules than H2O
-  Can readout signal in many other ways
-  Must program gradients to sweep out some region in *k*-space = coordinates of phase/frequency
-  Example: spiral imaging (from Stanford)

## Part the Fifth

- Image Contrast
- and
- Imaging Artifacts

## The Concept of Contrast (or Weighting)

-  ***Contrast*** = difference in RF signals — emitted by water protons — between different tissues
-  Example: gray-white contrast is possible because T1 is different between these two types of tissue

## Types of Contrast Used in Brain FMRI

-  T1 contrast at high spatial resolution
-  Technique: use very short timing between RF shots (small TR) and use large flip angles
-  Useful for anatomical reference scans
-  10 minutes to acquire 256256128 volume
-  1 mm resolution
-  T2 (spin-echo) and T2* (gradient-echo) contrast
-  Useful for functional activation studies
-  2-4 seconds to acquire 646420 volume
-  4 mm resolution [better is possible with better gradient system, and a little longer time per volume]

## Other Interesting Types of Contrast

-  Perfusion weighting: sensitive to capillary flow
-  Diffusion weighting: sensitive to diffusivity of H2O
-  Very useful in detecting stroke damage
-  Directional sensitivity can be used to map white matter tracts
-  Flow weighting: used to image blood vessels (MR angiography)
-  Brain is mostly WM, GM, and CSF
-  Each has different value of T1
-  Can use this to classify voxels by tissue type
-  Magnetization transfer: provides indirect information about H nuclei that aren’t in H2O (mostly proteins)

## Imaging Artifacts

-  MR images are computed from raw data *V*(*t*)
-  Assumptions about data are built into reconstruction methods
-  Magnetic fields vary as we command them to
-  The subject’s protons aren’t moving during readout or between RF excitations
-  All RF signal actually comes from the subject
-  Assumptions aren’t perfect
-  Images won’t be reconstructed perfectly
-  Resulting imperfections are called *artifacts*:
- Image distortion**;** bleed-through of data from other slices**;** contrast depends on things you didn’t allow for**;** weird “zippers” across the image**;** et cetera ........

## Part the Sixth

- Functional Neuroimaging

## What is Functional MRI?

-  1991: Discovery that MRI-measurable signal increases a few % *locally* in the brain subsequent to increases in neuronal activity (Kwong, *et al*.)

- Cartoon of MRI signal in an “activated” brain voxel

## How FMRI Experiments Are Done

-  Alternate subject’s neural state between 2 (or more) conditions using sensory stimuli, tasks to perform, ...
-  Can only measure relative signals, so must look for changes
-  Acquire MR images repeatedly during this process
-  Search for voxels whose NMR signal time series matches the stimulus time series pattern
-  Signal changes due to neural activity are small
-  Need 50+ images in time series (each slice) **** takes minutes
-  Other small effects can corrupt the results **** postprocess
-  Lengthy computations for image recon and temporal pattern matching **** data analysis usually done offline

## Some Sample Data Time Series

-  16 slices, 6464 matrix, 68 repetitions (TR=5 s)
-  Task: phoneme discrimination: 20 s “on”, 20 s “rest”

- graphs of 9 voxel time series

*t*

## Graphs vs. time of 33 voxel region

**Graphs vs. time of 3********3 voxel region**

**One Fast Image**

**Overlay on Anatomy**

**68 points in time 5 s apart; 16 slices of 64********64 images**

- This voxel did
- not respond

- Colored voxels responded to the mental
- stimulus alternation, whose pattern is shown in the
- yellow reference curve plotted in the central voxel

## Why (and How) Does NMR Signal ChangeWith Neuronal Activity?

-  There must be something that affects the water molecules *and/or* the magnetic field inside voxels that are “active”
-  neural activity changes** **blood flow
-  blood flow changes which H2O molecules are present and *also* changes the magnetic field
-  FMRI is thus *doubly* indirect from physiology of interest (synaptic activity)
-  also is much slower: 4-6 seconds after neurons
-  also “smears out” neural activity: cannot resolve 10-100 ms timing of neural sequence of events

## Neurophysiological Changes & FMRI

-   There are 4 changes currently used in FMRI:
-  Increased Blood Flow
-  New protons flow into slice
-  More protons are aligned with B0
- Equivalent to a shorter T1 (protons are realigned faster)
-  NMR signal goes up [mostly in arteries]
-  Increased Blood Volume (due to increased flow)
-  Total deoxyhemoglobin increases
-  Magnetic field randomness increases
-  NMR signal goes down [near veins and capillaries]

##  “Oversupply” of oxyhemoglobin after activation
 Total deoxyhemoglobin decreases
 Magnetic field randomness decreases
 NMR signal goes up [near veins and capillaries]

 Increased capillary perfusion
 Inflowing spins exchange to parenchyma at capillaries
 Can be detected with perfusion-weighted imaging methods
 This is also the basis for 15O water-based PET	

-  Total deoxyhemoglobin decreases
-  Magnetic field randomness decreases
-  NMR signal goes up [near veins and capillaries]
-  Increased capillary perfusion
-  Inflowing spins exchange to parenchyma at capillaries
-  Can be detected with perfusion-weighted imaging methods
-  This is also the basis for 15O water-based PET

## Deoxyhemo-globin is paramagnetic(increases B)

**Deoxyhemo-****globin is paramagnetic****(increases *****B*****)**

**Rest of tissue**
**is diamagnetic**
**(decreases *****B*****)**

**Cartoon of Veins inside a Voxel**

## BOLD Contrast

-  _**BOLD**_ = _**B**_lood _**O**_xygenation _**L**_evel _**D**_ependent
-  Amount of deoxyhemoglobin in a voxel determines how inhomogeneous that voxel’s magnetic field is at the scale of the blood vessels (and red blood cells)
-  Increase in oxyhemoglobin in veins after neural activation means magnetic field becomes more uniform inside voxel
-  So NMR signal goes up (T2 and T2* are larger)
-  Gradient echo: depends on vessels of all sizes
-  Spin echo:        depends only on smaller vessels

## BOLD Sensitivity to Blood Vessel Sizes

## Spatial Localization of Activity

-  Tradeoff : detectability (or scan time) vs. accuracy
-  Gradient echo
-  Largest signal changes, but veins draining active area will show “activity”, perhaps 10 mm away
-  Due to very short T2*, very hard to use at ultra-high *B*0
-  Spin echo
-  Smaller signal changes, but more localized to small vessels
-  Perfusion weighted imaging
-  Even smaller signal changes, but potentially best localization
-  “Difference of differences”

## Physiological Artifacts

-  Blood flow cycles up and down with cardiac cycle
-  Imaging rate slower than heartbeat means this looks like noise
-  Brainstem also moves about 0.5 mm with cardiac cycle
-  Respiration causes periodic changes in blood oxygenation and magnetic field (due to movement of chest tissue)
-  Subject movements inside gradient coil cause signal changes
-  Movements of imaged tissue are major practical problem
-  Movements of tissue outside image (e.g., swallowing, speaking) can change magnetic field inside image
-  Vasculature is different in each voxel, so BOLD response will be different even if neural activity is same
-  Hard to compare response magnitude and timing between locations and subjects

## Structural Artifacts

-  Un-shimmable distortions in *B* field cause protons to precess in ways not allowed for
-  Field is perturbed by interfaces between regions with different susceptibilities, especially air-tissue boundaries
-  Worst areas: above the nasal sinuses; near the ear canals
-  EP images will be warped in phase-encoding direction
-  Can be partly corrected by measuring *B* field and using that in reconstruction (the “VTE” method)
-  2D images will have signal dropout if through-slice field is not uniform
-  Palliatives: shorten TE; use thinner slices