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Lecture #13���Paramagnetic Relaxation Enhancement

•  Topics–  Paramagnetic contributions to relaxation–  Paramagnetic metal ions–  Gd3+-based T1 contrast agents–  Research topic examples

•  References–  Kowalewski, Ch 15, pp 359-380.–  Caravan, et al., “Gadolinium(III) Chelates as MRI Contrast

Agents: Structure, Dynamics, and Applications”, Chem. Rev. 1999, 99, 2293−2352.

Magnetism

2Remember, for unpaired electrons: γe = −658γH

Paramagnetic materials effect both T1 and T2

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Paramagnetic contributions to relaxivity

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•  The addition of a paramagnetic solute increases both 1/T1 and 1/T2 relaxation rates.

•  Diamagnetic and paramagnetic contributions are additive.

•  Solvent relaxation rates are generally linearly proportional to the concentration of the paramagnetic species, [M].

1Ti=1Ti,d

+1Ti,p for  i =1, 2

1Ti=1Ti,d

+[M ]Ri  for  i =1, 2

“Relaxivity”

Water Interactions

5

1Ti,p

=1Ti

!

"#

$

%&inner

+    1Ti

!

"#

$

%&2nd

+    1Ti

!

"#

$

%&outer

 for  i =1, 2

•  Nuclear spins see the lattice as the combined electron spin system and other molecular degrees of freedom.

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Review: Chemical exchange and τc•  Chemical

exchangestochastic

modulations relaxation

•  Exchange rates (μs to ms time scales) << molecular tumbling-  Too slow to effect anisotropic interactions such as CSA or dipole coupling-  Can effect isotropic interactions such as chemical shift or J coupling

•  Typically, chemical exchange induces a relaxation term of the form:

Hence, the exchange time can look just like a rotational correlation time!

1T1,ex

∝ A2 τ ex1+ Δω( )2 τ 2ex

Strength of the interaction Difference frequency

between exchange sites

1/exchange rate

Magnevist

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Review: Nuclear-electron couplings•  In addition to chemical exchange, both J and dipolar

coupling occur.

7 unpaired electrons,I = 7/2

J coupling Dipolar coupling

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Review: Quadrupolar Coupling

•  This electrical quadrupole moment interacts with local electric field gradients

• Quadrupolar coupling contribution to spin-lattice relaxation is…

•  A spin S > ½ have a electrical quadrupolar moment due to their non-uniform charge distribution.

-  Static E-field gradients results in shifts of the resonance frequencies of the observed peaks.

-  Dynamic (time-varying) E-field gradients become a very effective relaxation mechanism.

1T1,Q

≈3π100

2S +3S2 2S −1( )

e2qQ!

#

$%

&

'(

2

J ωS( )+ 4J 2ωS( )( )

Quadrupolar coupling constant

Electric field gradient term

Review: Scalar relaxation of the 1st kind•  Consider a J-coupled spin pair for which the S spin undergoes chemical

exchange with an exchange time of τex.

•  The coupling constant between the I spin and a spin Si becomes a stochastic function of time.

1T2,I

=4π 2J 2S S +1( )

3τ ex +

τ ex1+ ω I −ωS( )2 τ ex2

"

#$$

%

&''

1T1,I

=8π 2J 2S S +1( )

3τ ex

1+ ω I −ωS( )2 τ ex2

•  As we have shown, this leads to a relaxation mechanism, known as scalar relaxation of the 1st kind given by:

Review: Scalar relaxation of the 2nd kind•  For a system of J-coupled spins, where one of the spins, S, has a very short T1=

(e.g. spin S is quadrupolar coupled).

1T1=2 2π J( )2 S S +1)( )

3T2,S

1+ ω I −ωs( )2 T2,S2

1T2=2π J( )2 S S +1)( )

3T1,S +

T2,S1+ ω I −ωs( )2 T2,S2

"

#$$

%

&''

Sz t( )Sz t +τ( ) =2π J( )2 S S +1)( )

3e−τ T1,S

•  We can analyze this system by assume the S spin is in continuous equilibrium with the lattice, in which case S magnetization becomes a perturbation in the Hamiltonian with correlation functions…

S+ t( )S− t( ) =2π J( )2 S S +1)( )

6eiωsτe−τ T2,Sand

•  The contribution to T1 and T2 relaxation (known as scalar relaxation of the 2nd kind) are:

Water Interactions

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1Ti,p

=1Ti

!

"#

$

%&inner

+    1Ti

!

"#

$

%&2nd

+    1Ti

!

"#

$

%&outer

 for  i =1, 2

•  Nuclear spins see the lattice as the combined electron spin system and other molecular degrees of freedom.

Inner-Sphere Relaxation

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•  Chemical exchange contributes to inner-sphere relaxation.

1T1

!

"#

$

%&inner

=PM

τM +T1,M

1T2

!

"#

$

%&inner

=PMτM

T2,M−2 + T2,MτM( )−1 +ΔωM

2

T2,M−1 +τM

−1( )2+ΔωM

2

!

"

##

$

%

&&

ΔωP =PMΔωM

τM T2,M +1( )2 +τM2 ΔωM2

•  Excess spin-lattice relaxation rate, spin-spin relaxation rate, and measured chemical shift for the ligand due to the paramagnetic material are given by…

where M = ligand in paramagentic complexPM =molar fraction of bound ligand nucleiτM = lifetime of ligand in complex

-  PRE normalized to 1 mM is called relaxivity.

-  inner-sphere paramagnetic relaxation enhancement (PRE)

•  Hence:

“7-term”

Solomon-Bloembergen Theory

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•  Spin-lattice relaxation rate for the bound nuclei is a combination of…-  Scalar coupling between nuclear and electron spins-  Dipolar coupling between nuclear and electron spins

1T1,M

=23ASC2 S S +1( ) τ e2

1+ ωS −ω I( )2 τ e22+215

µ04π"

#$

%

&'2γ I2γS2!2

rIS6 S S +1( ) τ c2

1+ ωS −ω I( )2 τ c22+

3τ c11+ω I

2τ c12 +

6τ c21+ ωS +ω I( )2 τ c22

(

)**

+

,--

1T1,M

=1T1,M

!

"##

$

%&&SC

+1T1,M

!

"##

$

%&&DD

•  The correlation times are quite interesting…τ e2−1 = τM

−1 +T2e−1 τ e1

−1 = τM−1 +T1e

−1 τ ci−1 = τ R

−1 +τM−1 +Tje

−1;  j =1,2

γeB0 γHB0

“3-term”

water exchange time

electron T2 electron T1 rotational correlation time

Scalar coupling Dipolar coupling

1T1,M

≈23ASC2 S S +1( ) τ e2

1+ωS2τ e2

2 +215bIS2 S S +1( ) 7τ c2

1+ωS2τ c2

2 +3τ c1

1+ω I2τ c1

2

"

#$

%

&'

ωS = electron spinωI = nuclear spinwhere

What’s happening at these two field strengths?

NMRD Curves

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•  Typically, the scalar term is small compared to the dipolar coupling term (valid for Gd3+ but not necessarily Mn2+)

•  If rotational correlation time dominates the dipolar coupling term, then the field dependence of the PRE is…

•  But what about T1e and T2e? Aren’t these relaxation times themselves field dependent?

•  The above plot is the behavior typically observed when constructing MRI phantom using e.g. MnCl2 or CuSO4.

Solomon-Bloembergen-Morgan (SBM) Theory

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•  SBM theory includes the field-dependence of the electron T1 and T2 relaxation times.

•  Calculations of ESR relaxation rates are typically quite complicated.

•  For the paramagnetic complexes using for MR contrast agents, electron relaxation rates are dominated by zero-field splitting (ZFS), the electron spin equivalent of nuclear quadrupolar coupling.

1T1,e

=Δt2

5τ v

1+ωS2τ v

2 +4τ v

1+ 4ωS2τ v

2

"

#$

%

&'

1T2,e

=Δt2

103τ v +

5τ v1+ωS

2τ v2 +

2τ v1+ 4ωS

2τ v2

"

#$

%

&'

Trace of transient ZFS tensor Correlation time for ZFS modulations, typically ~10-9 s

Are these the same as those for nuclear relaxation via

quadrupolar coupling?

SBM Theory: T1

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1T1,M

≈23ASC2 S S +1( ) τ e2

1+ωS2τ e2

2 +215bIS2 S S +1( ) 7τ c2

1+ωS2τ c2

2 +3τ c1

1+ω I2τ c1

2

"

#$

%

&'

1T1,e

=Δt2

5τ v

1+ωS2τ v

2 +4τ v

1+ 4ωS2τ v

2

"

#$

%

&'

τ e2−1 = τM

−1 +T2e−1

τ e1−1 = τM

−1 +T1e−1

τ ci−1 = τ R

−1 +τM−1 +Tje

−1;  j =1,2

1T2,e

=Δt2

103τ v +

5τ v1+ωS

2τ v2 +

2τ v1+ 4ωS

2τ v2

"

#$

%

&'

•  The complete equations for the T1 relaxation rate are…

•  The SBM theory works reasonably well, but there are multiple extensions and modifications such as…-  The Lipari-Szabo correction-  The modified Florence approach-  Swedish slow-motion theory

NMRD Curves Revisited

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•  Including the field-dependence of the electron relaxation rates can yield much more interesting relaxivity behavior.

τr = 0.1 nsExamples

τr = 1 ns

SBM Theory: T2

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1T2,M

≈23ASC2 S S +1( ) τ e1 +

τ e21+ωS

2τ e22

"

#$

%

&'+

215bIS2 S S +1( ) 4τ c1 +

3τ c11+ω I

2τ c12 +

13τ c21+ωS

2τ c22

"

#$

%

&'

•  For completeness, the equations for the T2 relaxation rate is…

1T1,e

=Δt2

5τ v

1+ωS2τ v

2 +4τ v

1+ 4ωS2τ v

2

"

#$

%

&'

τ e2−1 = τM

−1 +T2e−1

τ e1−1 = τM

−1 +T1e−1

τ ci−1 = τ R

−1 +τM−1 +Tje

−1;  j =1,2

1T2,e

=Δt2

103τ v +

5τ v1+ωS

2τ v2 +

2τ v1+ 4ωS

2τ v2

"

#$

%

&'

Outer-sphere Relaxation

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“…noninteracting, rigid, spherical particles … with the spins in the centers of the spheres.”

•  For agents with water binding sites, relaxivity contributions from 2nd and outer sphere water are typically small.

•  To compute outer sphere (intermolecular) relaxation effects, we need to use a more general correlation function which includes r changing with time due to translational diffusion.

•  Results in a modified spectral density function (see Kowalewski, Chp 3.5).

Lebduskova, et al., Dalton Trans. 2007, 493–501.

1H NMRD profile of [Gd(DO3APOEt )(H2O)]−

Paramagnetic Elements

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Representative Metal Ions

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Ion Spin

Electron configuration

Magnetic moment

Electron T1 ΔR1 (0.5 T)

ΔR2 (0.5 T)

24Cr3+3/2 3.9 10-1 -1 ns 4.36 10.1

25Mn2+5/2 5.9 1-10 ns 7.52 41.6

26Fe3+5/2 5.9 10-1 -1 ns 8.37 12.8

29Cu2+1/2 1.7 10-1 ns 0.83 0.98

63Eu3+7/2 3.4 10-4 -10-3 ns 0.38 0.41

64Gd3+7/2 7.9 1-10 ns 12.1 15.0

66Dy3+5/2 10.6 10-4 -10-3 ns 0.56 0.56

•  Magnetic moment due to both spin and orbital angular momentum•  Electron T1

-  High symmetry: electric fields largely cancel. -  Low symmetry: electric field gradients enhance quadrupolar relaxation.

Why do we need MR contrast agents?

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•  Remember, MRI has good spatial resolution but low sensitivity.

•  Idea of using paramagnetic salts to shorten water relaxation times goes all the way back to Bloch, Hansen, and Packard, Phys. Rev. 1948, vol. 70, p. 464.

•  MRI signal intensity is typically proportional to M0 1− e−TR T1( )e−TE T2 .

In Vivo Requirements

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•  MRI contrast agents must be both biocompatible pharmaceuticals and NMR relaxation probes

•  Relaxivity-  T1, T2, or T2

* shortening-  Typically need at least 10-20% increase in 1/T1 for robust detection.

•  Specific in vivo distribution•  In vivo stability, excretability, lack of toxicity (acute and chronic)

Types of MR Contrast Agents

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•  T1 shortening agents-  Clinical agents: Gd3+ based-  Research: targeted and/or responsive agents, Mn2+ agents

•  T2, T2*,shortening agents

-  Clinical agents: Super-paramagnetic iron oxide (SPIO) nanoparticles

-  Research: targeted and/or responsive agents•  PARACEST agents

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Clinically used Gd3+ chelates

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Doesn’t Gd-DTPA also shorten T2?•  Yes, but relaxation rates are additive•  In vivo tissue T2s are typically considerably shorter than T1s •  On a percentage basis, T1 agents such as Gd-DTPA increase 1/T1

much more than 1/T2

•  For a typical MRI sequence, signal is usually enhanced, unless the Gd-DTPA concentration gets very high.

Sign

al in

tens

ity

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Gd-DTPA vs Dy-DTPA

Gd3+: 7 unpaired e- s

Dy3+: 5 unpaired e- s

µ = 7.9 T1e= ~1 ns

µ = 10.6 T1e= ~10-4 ns

Useful T1 agent Not a useful T1 agent How might Dy3+ be best used as a contrast agent?

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MS-325���•  Gd-DTPA bound to albumin-  High T1 relaxivity-  Stays in the intravascular space

τM = 1 ns

τM = 0.1 nsT 1 re

laxi

vity

MRI X-ray

Lots of in vivo targeting strategies…

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•  A few examples

c) Enzyme-activated d) and many more

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•  Contrast based on field dependence of T1relaxivity: dR1dB

•  Hoelsher, et al, Magn, Reson Mater Phy, 2012.

Field-cycling insert coil

T1-weighted1.5 T + 90 mT

T2-weighted1.5 T - 90 mT

dreMR image dreMR overlay

Raspberry injected with Gadoflurine

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Next Lecture: ���T2, T2

*, and PARACEST Contrast Agents