lecture #13 paramagnetic relaxation enhancement€¦ · electron relaxation rates are dominated by...
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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
3
Paramagnetic contributions to relaxivity
4
• 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.
6
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
7
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
8
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
11
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
13
• 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
14
• 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
15
• 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
17
• 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
18
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
19
“…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
20
Representative Metal Ions
21
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?
22
• 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
23
• 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
24
• 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
25
Clinically used Gd3+ chelates
26
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
27
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?
28
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…
29
• A few examples
c) Enzyme-activated d) and many more
30
• 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
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