The Chemistry of Lanthanide MRI Contrast Agents - The Chemistry of Molecular Imaging - page 187

taBLe 8.2 relaxivities and I nner sphere hydration numbers (q) for a range of gadolinium complexes.

Complex r 1 /mmol(gd) -1 s -1 q ref

gadobenz 1.8 0 [43]

[gd.dTPA] 2- 3.8-4.3 1 [98, 99]

[gd.dOTA] - 3.5-4.8 1 [98-100]

gd.dTPA-BMA 3.9-4.6 1 [101]

gd.HP-dO3A 3.6-3.7 1 [100, 102, 103]

[gd.MS-325] 2- 6.6 1 [13]

gd.dO3A 4.8 2 [103]

[gd.AAZTA] - 7.1 2 [65]

gd.tren-1,2-HOPO 10.5 2 [61]

All values refer to r 1 at 20 MHz. All r 1 values are dependent upon observed field and temperature ( vide infra ).

The SBM equations describe the relationship between the rate of relaxation and the magnetic field. Because the Larmor
frequency ω is related to the applied field B 0 by the equation:

ω= .B 0 (8.17)

where γ is the gyromagnetic ratio, it is possible to use the SBM equations to rationalise both variations in relaxation rates
and variations in observed relaxivity with magnetic field. If we are considering contributions to T 1 , then:

22 2

12
15

γµ gSS

r

(( + 1 )) 3 τ

ωτ

7 τ

ωτ

=

I B c 1

2

+

c 2

2

(8.18)

T

DD 6

1 +

2

1 +

2

1 sc 1 sc 2

2

12
3

A τ

ωτ

e

= SS ( + 1 ) 

2
2

(8.19)

SC 2

T 1 +

1 Se 2

where γ I is the nuclear gyromagnetic ratio, g the electronic g factor (which can be obtained from EPR measurements), μ B is
the Bohr magneton, r is the apparent separation of electronic and nuclear spin centres, ω I is the nuclear Larmor precession
frequency, ω s the electron Larmor precession frequency, ( A / ħ ) is the electronic nuclear hyperfine coupling constant, S is the
total spin (for uncoupled gadolinium complexes S =7/2), τ Ci are the correlation times for dipole-dipole terms, and τ ei are the
correlation times for contact terms.
These equations look complicated when taken all of a piece, but reveal some important information. For instance, when
considering 1/ T 1 DD , it is important to note that the equation given is the sum of two terms. The first is a function of the nuclear
precession frequency, while the second is a function of the electronic precession frequency. Because the magnetogyric ratio
of an electron is much greater than that of a proton, the electronic term may dominate at low magnetic fields, while the
nuclear term dominates at higher applied fields. When the nuclear term dominates, 1/ T 1 DD approaches a maximum as the
ratio of ω I to τ Ci approaches unity. Furthermore, these equations show that the contact contribution to relaxation in gado-
linium complexes is likely to be small. Not only are the f-orbitals on gadolinium effectively core-like, but additionally the
two-bond (gd-O and O-H) coupling required to give rise to a contact interaction between the water proton and the lanthanide
centre will mean that A / ħ will be very small and that contact contributions will be very minor at high fields.
Having thus established that the dipole-dipole mechanism is likely to be dominant under the conditions of the MRI
experiment, it also becomes clear that the separation between the electron and the proton nucleus will have a profound
influence on the observed relaxation rate. However, it is difficult to engineer this separation except by accident, because the
distance of close approach of water will be determined by both ligand structure and the residual charge on the lanthanide
centre in the complex. 3 1/τ ci and 1/ τ ei can themselves be expressed as a sum of terms as shown in equations 8.20 and 8.21,
where τ m is the residence lifetime of a bound water molecule, τ R is the rotational correlation time of the complex as a whole,
and T ie (i = 1,2) is the electronic relaxation time.

1111
τ

=++

(8.20)

T τ τ

ci ie mR

3 However, see the section on optimisation below.