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Chemical Exchange

Spin-interactions •  External interactions�

– Magnetic field Bo, �–  RF field B1 �

•  Internal Interactions�– Molecular motions �–  Chemical shifts �–  J-coupling �–  Chemical Exchange�

�

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Outline •  Motional time scales�

• Motional line shapes�

•  Two site exchange�• Time scales�• Reaction rates: Bloch equations�• Steady state solutions�• Exchange regimes�• Examples�

•  Chemical exchange saturation transfer�• Principle�• Glutamate and Creatine CEST�

– Ex vivo and In vivo Results

Motional time scales

ms µs ns ps fs s

Macroscopic diffusion, flow

Chemical Exchange

Molecular rotations

Molecular vibrations

Slow Fast�

Very Slow Slow fast Very fast ultra fast

Relaxation Spectral Larmor

Malcolm Levitt’s Book

2/26/13

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Motinal time scales •  Three times scales of the nuclear spin system.�

–  Larmor time scale, τ �•  The time required for the spins to precess through 1

radian in the magnetic field.� |ωοτl|~1 (at 100 MHz, ωο/2π =100 ΜΗz, τl=1.6 ns)

  Spectral time scale tspec�   Spin system with two spins with CS at ω1 and ω2. If

the CS interactions are dominant, then the spectral time scale τspec: �

  |ω1- ω2| τspec~1 (at 100 MHz, two protons have CS difference δω/2π = 5ppm(=500 Hz), then, τspec= 330 µs) �

Relaxation time scale: Spin-lattice relaxation time T1 (in the order of seconds)�

�

�

Motional Lineshapes and Two-site exchange

•  Motional processes on the spectral timescale affect NMR line shapes.�

• Consider an isolated nuclear spin is transported between two different chemical environments, with different chemical shifts but identical free energies�

•  �-----> <----- k �

k �A � B �

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Chemical Exchange

-----> <----- k �

k �N �

H �

O� CH3 �

CH3 �N �

H �

O� CH3 �

CH3 �

N,N’-Dimethyl-formamide �N+ �

R �C�

CH3 �

CH3 �

O- �

Exchange regimes

•  If the chemical shift difference between two states is δω,

•  and chemical exchange rate constant is k then �–  K< |δω/2| slow int. exchange regime –  K~ |δω/2| Intermediate exchange regime –  K> |δω/2| fast int. exchange regime�

Δω=-δω/2 � Δω=δω/2 �

Δω=ο�

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Precesssion frequency and exchange process

•  Change in the precession frequency on a molecular exchange (A <--->B) process�

•  Superposition of transverse magnetization oscillations for exchanging molecules�

•  Total transverse magnetization for many exchanging molecules�

ωa/2π=10kHz, ωb/2π=11kHz, and k=500s-1

Simulations of precessing transverse magnetization

•  Slow intermediate exchange regime K< |δω/2| �

•  Faster the exchange, faster the decay�

•  Leads to broadened NMR line�•  Motional broadening �•  Simulation parameters: �ωa/2π = 2kHz, ωb/2π = 4kHz�

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Simulation of fast exchange

•  Simulation of Transverse magnetization �

•  Fast intermediate exchange regime�•  K> |δω/2| �•  As the exchange rate k increases,

magnetization decays slowly�•  Rapid molecular jumps leads to ‘motional narrowing’ �

•  Simulation parameters: �ωa/2π = 2kHz, ωb/2π = 4kHz�

Two-site exchange time scale

•  Spectral line shape change as one goes from slow to fast exchange�

K= |δω/2|

Malcolm Levitt’s Book

2/26/13

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NMR and reaction rates �

–  Consider nuclei at sites a and b �•  Ma = Mx+iMy for nuclei at site a�•  Mb = Mx+iMy "for nuclei at site b �•  M0 = Thermal equilibrium magnetization of the sum of two

sites�•  Δω=ω-ω0 ; δω= splitting between the two lines�•  Nuclei at sites a and b obey the following two differential

equations: �

�

dMa

dt= −Ma

T2+ i Δω + δω

2⎛⎝⎜

⎞⎠⎟Ma + iω1

M0

2

dMb

dt= −Mb

T2+ i Δω − δω

2⎛⎝⎜

⎞⎠⎟Mb + iω1

M0

2

-----> <----- k �

k �A � B �

Sclichter’s Book

Steady-state solutions

•  At steady state, and�•  steady state solutions give two

Lorentzian lines�•  Δω=-δω/2 and other at Δω=δω/2 �

�

dMa

dt= 0

�

dMb

dt= 0

Δω=-δω/2 � Δω=δω/2 �

Δω=0 �

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Exchanging spins

•  Consider that molecule reorients, taking spins from a to b-site and vice versa.�–  Jumping of spins from b ---> a site (in time δt)�

•  will add δMa to a magnetization �•  δMa= C1Mbδt C1 =constant �•  will diminish δMb from b magnetization �•  δMb= -C1Mbδt C1 =constant �

Exchanging spins

– Similarly, jumping of spins from a ---> b site (in time δt)�• will add δMb to b magnetization �•  δMb= C2Maδt C2 =constant �• will diminish δMa from a magnetization �•  δMa= -C2Maδt C2 =constant �

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Exchanging spins

�

dMa

dt= C1Mb − C2Ma

�

dMb

dt= C2Ma − C1Mb

Adding these rates to the rates without spin jumps�

dMb

dt= −Mb

T2+ i Δω − δω

2⎛⎝⎜

⎞⎠⎟Mb + iω1

M0

2+C2Ma −C1Mb

dMa

dt= −Ma

T2+ i Δω − δω

2⎛⎝⎜

⎞⎠⎟Ma + iω1

M0

2+C1Mb −C2Ma

Exchanging spins

•  since whenever a molecule reorients a spins ---> b spins and�

•  b spins ---> a spins�•  ----> C1=C2=C�•  The rate equations are solved in the

steady state �– and the resulting total complex

magnetization Mx+iMy is given by �

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Steady-state solutions

�

Mx + iMy = Ma + Mb = iγH1M 0τ[2 + τ(αa + αb ) /2](1+ αaτ)(1+ αbτ) −1

where

1/τ = C=k, and the absorption signal is given by My the imaginary part of the above equation

�

αa =1T2

+ i ωo −ω −δω2

⎛ ⎝

⎞ ⎠

�

αb =1T2

+ i ωo −ω +δω2

⎛ ⎝

⎞ ⎠

Simulated spectra

•  Two-site exchange in the slow intermediate regime�

•  K< |δω/2| (∼6.3kHz)�•  Simulation parameters�•  ωa/2π = -1kHz�•  ωb/2π = 1kHz�•  Crossover point is�•  K= |δω/2| (∼6.3kHz) �

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Fast int. exchange

K> |δω/2| (∼6.3kHz) �fast int. exchange

regime�

•  Simulation parameters�

•  ωa/2π = -1kHz�•  ωb/2π = 1kHz�

Two-site exchange time scale

•  Spectral line shape change as one goes from slow to fast exchange�

K= |δω/2|

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13C Spectra of N,N’-dimethylformamide

•  13C enriched spectra of N,N’-dimethylformamide gas at different temperatures in a field of 4.7T�

1H spectrum of N,N’-dimethyltrichloroacetamide •  1H spectrum of N,N’-

dimethyltrichloroacetamide (DMTCA) as a function of temperature at 60 MHz.�

-----> <----- k �

k �N �

R �

O� CH3 �

CH3 �

-R: H, CH3 or CCl3

N �R �

O� CH3 �

CH3 �

N+ �R �

C�CH3 �

CH3 �

O- �

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Chemical exchange saturation transfer (CEST)

•  Biological tissues, exchange of solute spins with water�

•  Can we exploit the CE to detect solute spins?�

H

ksw

kws H H

O H

O

H H O

H H O

H H O

H H O

H H O

H H O

Solute Pool Water Pool

Chemical Exchange Saturation Transfer (CEST) �

H

H

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MRS at 7T

Glu

Excitatory neurotransmitter�

3.00ppm

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Sensitivity comparison between CEST and MRS

GluCEST ~700 fold higher sensitivity than MRS

Cai et al. Nat. Med. (2012)

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Potential overlap of other brain metabolites with GluCEST

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Ø  Physiological concentrations of major brain metabolites at 37C�Ø  Glu & GABA contribute to GluCEST�

7T – B1rms 150Hz – 2s

CEST (%) CEST (%)

GluCEST in MCAO rat brain

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9.4T – B1rms 250Hz – 1s

Cai et al. Nat. Med. (in press)

4.5 hour�

<4 �

>24 �1.0 hour

ipsilateral Contralateral

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1H Image GluCEST (%)

0

30

2 Hr Baseline

Glu modulation in vivo

In vivo GluCEST

•  A. The axial view of the human brain.�•  B. Difference image (CEST image at -3ppm – CEST image at +3ppm); �•  C. GluCEST contrast map; �•  F. Map of distribution volumes (DVs) of metabotropic Glu receptor

subtype 5 from the PET image (With permission, extracted from Ametamey Simon M et al, J. Cereb Blood flow Metab, J Nucl Med 2007; 48:247-252.) �

Cai et al. Nat. Med. (2012)

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Enzymatic inter-conversion of PCr and ATP

CK facilitates high energy phosphate metabolism Current Methods 31P MRS: measures PCr and ATP �1HMRS: measures total creatine�

No noninvasive method �For measuring free Cr �

Myocardial and skeletal muscle energetics �

CREST SENSITIVITY AND SPECIFICITY (3T) �

CrEST has two-three orders of magnitude higher sensitivity�Than 1HMRS. �Amine/amide groups on PCr, ATP and ADP have an �order of magnitude slower exchange rate than Cr �Hence they don’t contribute to CrEST �

Haris et al., NMR in Biomed (2012)

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In vivo Human Skeletal Muscle CrEST Before and After Exercise at 7T

Kogan et al., Magn. Reson. Med. (in press)

[b] [a] (%) exercise

Plantar flexion exercise data for subject 1. (a) CrCESTasym maps of a human calf muscle before and every 48 seconds after (in order by number) 2 minutes of plantar flexion exercise. (b) MTR and (c) T2 maps before and every 56 seconds after (in order by number) the same exercise protocol. (d) plot of the average CrCESTasym as function of time in 4 different muscles of the calf segmented from anatomical images. Error bars represent the standard deviation in the CrCESTasym in each region.

31P MRS OF CALF MUSCLE- IN MAGNET EXERCISE �

Healthy human calf muscle exercise�Pressing of a pedal connected to �pneumatic piston for 1 minute. �

Kogan et al., Magn. Reson. Med. (in press)

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CEST imaging

•  Exchangeable groups on other endogenous molecules�– Myoinositol (Alzheimer’s disease)�– Creatine (CK metabolism in heart)�– Aggrecan (Osteoarthritis)�

Requirement for CEST

•  The chemical shift �

�•  A large ∆ω �

–  reduces direct water saturation �

–  Increases the CEST sensitivity�

Ward et al. J. Magn. Reson. 143 (2000): 79–87

�

Δω > k�

Δω ≅ k�

Δω < k

�

Δω > k