biophysical techniques (bphs 4090/phys 5800) · biophysical techniques (bphs 4090/phys 5800) york...

Biophysical Techniques (BPHS 4090/PHYS 5800)

York University Winte7 2017 Lec.24

Instructors: Prof. Christopher Bergevin (cberge@yorku.ca)

Schedule: MWF 1:30-2:30 (CB 122)

Website: http://www.yorku.ca/cberge/4090W2017.html

NMR/MRI Overview

Franklin et al.

Quantum considerations: •  Angular momentum values are quantized (due to possible energy states)

•  Only consider hydrogen nucleus (i.e., single proton) •  Can only be in one of two states: ‘spin up’ or ‘spin down’

à Nuclear angular momentum quantized as quantum number I

NMR/MRI Overview

Franklin et al.

Ø  Useful classical analog? Spinning top that is precessing

à top doesn’t fall over because of it’s momentum

à Note directional conventions (i.e., longitudinal vs transverse)

NOTE: Key difference in quantum case is that possible energy states are quantized

NMR/MRI Overview

http://www.ualberta.ca/~dumberry/nutation.jpg

Earth’s motion – Three different rotational axes: 1.  Around sun 2.  About central axis 3.  Nutation of axis

NMR/MRI Overview

Franklin et al.

Ø  What is the ‘resonance’ aspect of NMR?

Ø  What is the magnetization vector (M)?

Larmor precession: By providing a photon at the right frequency, protons from the low-energy state will flip from one spin state to another

Key Idea: This frequency is proportional to the external magnetic field

where (gyromagnetic ratio; equal to 42.5 MHz/T for a single proton)

Simply add together all contributing magnetic moments

Ø  So how does ‘space’ come back in (re forming an image)? Gradients in field strength are encoded at different frequencies (another key idea we will return to in some detail)

NMR/MRI Overview

Franklin et al.

Ø  How does one produce a uniform magnetic field or gradient?

Electromagnets: Helmholtz coils for static fields, Anti-Helmholtz coils for gradients

NMR/MRI Overview

Franklin et al.

à RF pulse adds in energy so to change longitudinal magnetization

ex. consider hydrogen nuclei

NMR/MRI Overview

Franklin et al.

à RF pulse can cancel longitudinal magnetization and shift it entirely into transverse plane

à RF pulse can also be used to cause alignment in transverse plane (phase coherence)

Wikipedia (spin echo)

Spin Echo Pulse

NMR/MRI Overview

Franklin et al.

Ø  What is the difference between T1 and T2 imaging?

Protons are excited by RF pulse and relax back towards an equilibrium (thereby re-emitting RF photons) in two ways: longitudinal (along static field; T1) and transverse (perpendicular to static field; T2)

Differences are due to how protons lose energy: T1 due to proton’s surrounding environment (spin-lattice) and T2 due interactions amongst protons (spin-spin)

Hoppe

Spin Echo Pulse

Franklin et al.

Free Induction Decay

Franklin et al.

MRI Strength:

NMR/MRI Overview

Franklin et al.

NMR/MRI Overview

Franklin et al.

Ø  How is MRI used for medical imaging?

Wide variety of different imaging modalities allow for radiologists to image tissue both in terms of structure (e.g., tumor present?anomaly in shape?) and/or physiology (e.g., high region of blood uptake?)

Ø  What is fMRI?

NMR/MRI Overview

The ‘f’ stands for functional and allows for dynamics to be observed (via the BOLD signal, or Blood Oxygenation Level Dependence); very popular technique in the field of ‘neuroimaging’

NMR/MRI Overview

www.orthopaedics.com

Ø  What are the basic ingredients for NMR/MRI?

§  static field §  particles (e.g., protons as spinning tops) §  coil to perturb particles from static field and measure resulting dynamics (via ‘pulse sequences’ of RF photons) §  Fourier transforms

Ø  Why are MRI scanners so noisy? §  Pumps needed for liquid helium for superconducting magnets, so to produce higher field strengths (1.5-15 T) and thereby be more sensitive to NMR signal §  Gradient coils produce acoustic compression

Hoppe

Dobrovolny Hobbie (ch.8)

Magnetic moment (µ)

potential energy associated with the torque

torque is the rate of change of angular momentum

γ – gyromagnetic ratio

spin also creates a magnetic moment

S – spin vector (angular momentum)

nuclear angular momentum quantized as quantum number I

Note: Will use bold and diacritical arrow interchangeably to indicate vectors

Magnetization

average magnetic moment per unit volume – Simply add together all contributing magnetic moments <µ> - average moment

N – total # of dipoles

If M is parallel to B , there is no torque and the magnetization remains constant.

Dobrovolny Hobbie (ch.18)

Key Idea – individual magnetic moments are quantum mechanical quantities M is an ensemble average and thereby well described classically (e.g. spinning top analogy)

Average magnetic moment

Dobrovolny Hobbie (ch.18)

Consider a magnetic field uniform along z (i.e., B=Bz , where z is the longitudinal direction)

At thermal equilibrium, the probability of being in a certain energy state is given by the Boltzmann factor (note quantum # m)

assume that

numerator: =

denominator: =

finite series with a simple closed form

Average magnetic moment

Dobrovolny Hobbie (ch.18)

Franklin et al. Dobrovolny Hobbie (ch.18)

Precession

Consider a static magnetic field uniform along z (i.e., B=Bz)

when M and B are not entirely parallel, a torque is created such that M (and thereby <µ>) precess about B

straightforward to solve through...

Dobrovolny Hobbie (ch.18)

Precession

The magnetization in the z direction does not change; in the x and y directions, the magnetization moves in a circle

à The Larmor frequency is a fundamental quantity in the resonance aspect of NMR and is ultimately determined by virtue of discrete quantum mechanical energy states

Hoppe

Aside: Chemotaxis

The physics of eukaryotic chemotaxis (Levine & Rappel; Physics Today, 2013)

à Very different biological/physical system, but similar concepts to consider

Aside: Chemotaxis

The physics of eukaryotic chemotaxis (Levine & Rappel; Physics Today, 2013)

Idea: Sum together various contributions to get a net effective directionality

S – binding const. (0 or 1)

Aside: Chemotaxis

The physics of eukaryotic chemotaxis (Levine & Rappel; Physics Today, 2013)

We now return you to our regularly scheduled programming...

Franklin et al.

Relaxation

Dobrovolny Hobbie (ch.18) Franklin et al.

à Consider when the precessing dipole changes it’s transverse rotation and ‘relaxes’ back to the longitudinal direction (i.e., something affects the dipole other than the static field, such as thermal effects or nearby neighbor coupling)

[again] Consider a static magnetic field uniform along z (i.e., B=Bz)

Assumptions •  exponential •  uniform in transverse plane •  spatially independent •  equilibrium value along z

•  T1 is the longitudinal relaxation time (or spin–lattice relaxation time)

•  T2 is the transverse relaxation time (or spin–spin relaxation time)

Dephasing

Dobrovolny

Relaxation

Dobrovolny Hobbie (ch.18)

Need to include external field term:

Bloch equations

[again] consider a static magnetic field uniform along z (i.e., B=Bz)

Relaxation

Dobrovolny Hobbie (ch.18)

integrate to obtain

system of two linear 1st order ODEs

associated eigenvalues

Note: Axes mislabeled (should be Mx, My and Mz)

Free Induction Decay

Franklin et al.

MRI Strength:

Hobbie (ch.18)

à notion of a pulse sequence (to stimulate a magnetic field from the sample)

Hoppe