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New simulation approach using classical formalism to water
nuclear magnetic relaxation dispersions in presence of
superparamagnetic particles used as MRI contrast agents
Q.L. Vuong, Y. Gossuin, P. Gillis and S. Delangre
University of Mons, Place du Parc 20, 7000 Mons, Belgium.
Supplementary Material
1
1. Relaxation rates in the Redfield condition: Quantum and classical
formalisms
In this section, the usual quantum demonstration leading to the dependence of the relaxation rates
on the spectral densities is presented. An analogous demonstration in the classical formalism is then
given with an accent on the similarities with the quantum demonstration.
Quantum Redfield formalism
The Redfield theory [1] is based on the quantum formalism of the density matrix  which
fulfills the following equation
i
d
 H ,  
dt
(S1)
where H is the Hamiltonian. In our case, H is usually composed of the sum of the time independent
Zeeman energy H0 with a perturbative time-dependent Hamiltonian H1(t). The calculation is made in
the interaction picture [2] in which the corresponding operator A is written as
A*  exp  iH0t /
 A exp  iH0t /  .
(S2)
In this picture, only the time-dependent Hamiltonian explicitly appears in the density matrix equation
(S1)
i
d *
  H1* (t ),  *  .
dt
(S3)
Integrating equation (S3) gives an implicit solution for  * . Replacing this latter result in (S3) gives
d * i *
1
   (0), H1* (t )   2     * ( ), H1* ( )  , H1* (t ) d .
dt
0
t
(S4)
2
H1(t) generally depends on stochastic random variables and can thus be characterized by an
autocorrelation time  D . If this latter is much lower than the measurement time t, the upper integration
limit can be extended to infinity. Moreover, supposing that H1(t) is a weak markovian perturbation
(Redfield condition) inducing weak variations of the density matrix on a correlation time scale, i.e.
1
H12
1/2
 D  1 ,
(S5)
and that its average is zero, a master equation can be obtained from (S4)
d *
1
 2
dt

  H
*
1
(t ),  H1* (t   ),  * ( )  d
(S6)
0
where the bar means that the average on the random variables of H1* (t ) has been made. In the case of
SPM particles, condition (S5) is equivalent to
I
µ0 M S
 D  1 .
3
(S7)
The equilibrium magnetic moment is usually non-zero: the corresponding equilibrium density matrix
 0* must be introduced in (S6) such as
d *
1
 2
dt

  H
*
1
(t ),  H1* (t   ),  * ( )   0*  d .
(S8)
0
The evolution of the average nuclear spin I , proportional to the nuclear magnetic moment µI   I I ,
can be obtained by taking the trace of its product with the density matrix
d I*
dt

d
1
Tr( I  * )   2
dt


 Tr   H
0
*
1
(t ),  H1* (t   ), I   * ( ) d .


(S9)
The development of this equation leads to the same result as (S21)-(S23) [1, 3].
3
Classical formalism
In the classical formalism, the basic proton spin equation is given by [4]

dµI
  I µI  B0  B1 (t )
dt

(S10)
Rorschach demonstrated that the classical formula of the relaxation rates including the spectral
densities could be obtained with (S10). In the demonstration of Rorschach, the different components of
the magnetic moment were treated separately. It is interesting to use the vectorial notations to obtain
an analogy with the quantum demonstration. First, equation (S10) can be expressed in the rotating
frame in which all vectors become
 cos 0t  sin 0t 0 
B  OB ; O   sin 0t cos 0t 0 
 0
0
1 

*
(S11)
where 0   I B0 is the proton Larmor angular frequency, one supposes that the static field B0 aligned
with the z-axis. Precession equation (S10) then becomes
dµI*
  I µI*  B1* (t ) .
dt
(S12)
This equation doesn’t explicitly depend on the static field as in (S3). Integrating equation (S12) and
replacing the implicit solution in the right-hand side of (S10) gives
t
dµ1* (t )
  I  I* (0)  B1* (t )   I2    I* ( )  B1* ( )   B1* (t )d .
dt
0
(S13)
This is the classical equivalent of equation (S4). If the magnetic field B1 (t ) depends on random
variables and is zero-average, and, if it fulfills the “classical Redfield condition”,
4
 I B12
1/2
 D  1 ,
(S14)
the time evolution of the average nuclear magnetic moment mI (classical equivalent of (S9)) can be
written as

d *
*
mI (t )   I2   mI* ( )  meq
 B1* ( )   B1* (t )d

dt
0
(S15)
where the equilibrium magnetic moment meq has been inserted to make the system evolve to the
appropriate equilibrium state.
In the case of SPM particles, B1 is the dipolar magnetic field generated by the nanoparticle
B1 (r , µS ) 
µ0  µS  r

3 2 r  µS   B1x 1x  B1 y 1y  B1z 1z
3 
4 r  r

(S16)
where r is the vector from the proton position to the SPM center. Its components are random
variables because the proton position, modeled by a diffusion equation, varies randomly. The
corresponding Hamiltonian – also used in the quantum formalism – is usually written in the tensor
form as
H1 
2
F
(q)
A( q )
(S17)
q 2
with
1  3cos 2 
r3
sin 2  e 2i
;
F (2) 
r3
sin  cos  e  i
(1)
F 
r3
F (0) 
3µ0  3
1

  I z S z   I  S  I  S  
8  2
6

3µ
A(1)   0  I z S   I  S z 
8
3µ
A(2)   0 I  S
16
A(0)  
(S18)
5
where F (  q ) is defined as the complex conjugate of F ( q ) and A(  q ) is the adjoint operator of A( q ) ,
S  S x  iS y which are the components of the SPM spins (with µS   S S ,  S is the electronic
gyromagnetic ratio) and analogously for the proton spin I. With these notations, one can write the
components of vector (S16) as
3µ0 S
8
1 
1
(0)
(1)
( 1)
 S x F  S z  F  F   S 
2 
3
3µ   1
i

B1 y   0 S  S x F (0)  iS z  F (1)  F ( 1)    S  F (2)  S  F ( 2)  
8  3
2

B1x  
B1z  
(S19)
3µ0 S  2
(0)
(1)
( 1) 
  S z F  S F  S F  .
8  3

Supposing that the different component of B1 are not correlated – i.e. B1i (0) B1 j ( )  0 if i  j – and
using symmetries of the studied system, which leads to [5]
3
Gr ( )  F (0) (0) F (0) ( )  6 F (1) (0) F ( 1) ( )  F (2) (0) F ( 2) ( ) ,
2
(S20)
(the other functions equal zero) and formula (S19), equation (S15) can be written as
*
dm*x
1 * dm y
1
dm
1
  mx ;
  m*y ; z    mz  mzeq 
dt
T2
dt
T2
dt
T1
(S21)
with
2

1 µ 
R1    0   I2  cos(0 )Gr ( ) 7GSx ( )  7GSy ( )  6GSz ( ) d
T1  8 
0
(S22)
2
R2 

1  µ0  2
   I
T2  8 
7
7

0 Gr ( ) 3GSx ( )  3GSy ( )  4GSz ( )   cos(0t )Gr ( )  2 GSx ( )  2 GSy ( )  3GSz ( ) d
(S23)
6
where one has supposed that the autocorrelation functions are odd and GSi ( )  µSi (0)µSi ( ) are
defined as the SPM magnetic moment autocorrelation functions. The classical formalism leads to the
same results as the quantum formalism does [1, 3].
The complete expression of the relaxation rates can then be obtained with the autocorrelation
functions. Ayant and Freed independently showed [5, 6, 7] that, for the proton position part Gr ( ) , in
the case of a proton bulk diffusion around an impenetrable particle of radius RS, the Fourier transform
is

e

 i

Gr ( ) 2  cos( )Gr ( ) 
0
64 C A
J
135 RS D

2 D

(S24)
where C is the SPM particle concentration – i.e. the number of SPM particles in a volume unit – and
1  5z / 8  z 2 / 8
J ( z) 
.
1  z  z 2 / 2  z 3 / 6  4 z 4 / 81  z 5 / 81  z 6 / 648
A
(S25)
The spin S part depends on the used SPM model and is presented in the main paper (methodology
section).
Equivalence between the quantum and classical precession equations
The classical and quantum formalisms are based on two fundamental equations, respectively the
precession equation (S10) and the matrix density equation (S1). In this subsection, it is shown that
equation (S1) leads to equation (S10). The demonstration can also be found in reference 2.
The matrix density  is a convenient tool to describe a great ensemble of particles. Its evolution
can be calculated thanks to equation (S1) and the average of a general operator A is given by the trace
of the product between A and 
A  Tr  A  .
(S26)
7
Thus, the evolution of the average magnetic moment of the proton nucleus
I   I I  Tr  I I  
(S27)
can be computed from equation (S1). Multiplying it with (S1) and using the Zeeman Hamiltonian
H  I  B , one obtains
i
d µI
dt
 Tr  H ,   I   Tr  I , H   
(S28)
where the following properties
Tr  A  B, C   Tr  A, B  C  ; Tr  AB   Tr  BA 
(S29)
were used. One can simplify the commutator in equation (S28). For example, for the x-component, one
finds using the spin property  I i , I j   i  ijk I k (  ijk is the Levi-Civita tensor)
 µx , H    I2  I x ,  I x Bx  I y By  I z Bz    I2i
I z By   I2i I y Bz   I2i
I
y

Bz  I z By   i  I I  B

x
(S30)
Analogously, one finds
 µy , H   i
   B
I
y
;
 µz , H   i
   B .
I
z
(S31)
and thus equation (S28) can be written as
i
d µI
dt
  I , H   i  I I  B
(S32)
which is equivalent to the precession equation (S10) if B  B0  B1 (t ) .
8
2. Secular term
In this section, we will prove that the new methodology introduced in this work is equivalent to the
T2 simulation methodology previously developed for high fields. The main steps of this T2-simulation
will be first presented and then a short demonstration will show that the evolution of the magnetic
moment calculated from the T2 methodology is equivalent at high field to precession equation (S10).
T2 simulation methodology at high field
This method consists of three main steps [8]:
(1) Distribution of the nanoparticles in the simulation space.
(2) Proton diffusion and computation of their spin dephasing in the rotating frame during their
trajectory. This dephasing is caused by the magnetic homogeneities produced by the nanoparticles.
Supposing that the static field is aligned with the z-axis, each proton magnetic moment  j at time nt
can thus be written as
 j (nt )   cos nj ,sin nj , 0  ; nj  nj 1   I Bz (nt )t ; 0j  0
(S33)
where t is the time interval, Bz (nt ) is the z component of the total local field felt by the proton at
time nt . This field is the sum of all the dipolar fields produced by the nanoparticles of the sample.
(3) Computation of the time evolution of the average magnetic moment. The obtained FID
fitted by an exponential law makes it possible to obtain the corresponding relaxation rate.
Secular term in the classical formalism
It can be shown that the classical formalism is a generalization of the simulation methodology
used to predict transverse relaxation rates at high magnetic field – the so-called secular term. Indeed,
at high field B0, the rotational matrix in (S11) is simplified: the cosines and sine oscillate so rapidly
9
(because of their high Larmor frequency) that they can be considered equal to zero. Equations (S12)
become
dµ*y
dµ*x
dµ*z
 0;
  I  y B1z (t ) ;
  I  x B1z (t )
dt
dt
dt
(S34)
for which the solution is the integrated equivalent of equation (S26) [4]
t
 j (t )   cos  (t ),sin  (t ),0  ;  (t )   (0)    I B1z (t )dt
(S35)
0
Thus, a relaxation is expected for the transverse component while the longitudinal relaxation time is
expected to be infinite. If B1z(t) doesn’t depend on B0, the magnetic moment evolution doesn’t depend
on B0 either which explains why R2 usually saturates with increasing field B0.
10
3. Supplemental figures
Figure S1: Scheme of the three spin models (from left to right, isotropic model, weak anisotropy
model and strong anisotropy model). The static magnetic field is supposed to be vertical. Dashed
circles represent the precession of the magnetic moment. Dashed lines are the anisotropy axis.
Figure S2: NMRDs obtained from the isotropic model for a SPM radius of 10 nm.
11
Figure S3: NMRDs obtained from the isotropic model for a SPM radius of 115 nm.
12
Figure S4: Longitudinal NMRD profiles of SPM particles with radius of 70 nm for different SPM
magnetizations in the isotropic model.
13
Figure S5: Longitudinal NMRD profiles of SPM particles with radius of 200 nm for different SPM
magnetizations in the isotropic model.
14
Figure S6: Influence of the SPM magnetization on the relaxation rates for different B0 fields (1, 10-3
and 10-6 T) and for a SPM radius of 8 nm in the isotropic model. Longitudinal and transverse
relaxation rates are equal at a B0 = 10-6 T. The points corresponding to low magnetizations are fitted by
a quadratic law (dashed lines). Solid lines are linear fits of larger magnetization points.
15
Figure S7: Influence of the SPM magnetization on the transverse relaxation rate for two different B0
fields (1 and 10-6 T) and for a SPM radius of 200 nm in the isotropic model. Solid curves are obtained
by a linear fit on all the simulated points (red) or on the largest magnetizations (blue). The green
dashed lines are predictions of the static model.
16
Figure S8: NMRDs obtained from the weak anisotropy model for a SPM radius of 10 nm. Solid curves
are obtained with equations (21) and (22) of the main paper and dashed lines with equations (16) and
(17) of the main paper.
17
Figure S9: NMRDs obtained from simulations of the weak anisotropy model for a SPM radius of 20
nm. Dashed lines are the corresponding NMRDS of the isotropic model obtained by simulations.
18
Figure S10: NMRDs obtained from simulations of the weak anisotropy model for a SPM radius of 115
nm. Dashed lines are the corresponding NMRDS of the isotropic model obtained by simulations.
19
Figure S11: Dependence of the longitudinal and transverse relaxation rates on the SPM radius in the
WA model at different B0 fields (10-3, 10-6 and 1 T). Solid line is obtained with equation (21). Dashed
2
lines were obtained by fitting an RS law to the data. The static model prediction (7) is also traced for
comparison.
20
Figure S12: Influence of SPM magnetization on relaxation rates for different B0 fields (10-6, 10-3, 1 T)
and for a SPM radius of 8 nm in the WA model. Longitudinal and transverse relaxation rates are equal
at a B0 = 10-6 T. The points corresponding to low magnetizations are fitted by a quadratic law (dashed
lines) while larger magnetizations are fitted with a linear law (solid lines).
21
Figure S13: Influence of SPM magnetization on relaxation rates for different B0 fields (10-6, 10-3, 1 T)
and for a SPM radius of 200 nm in the WA model.
22
Figure S14: NMRDs obtained from the strong anisotropy model for a SPM radius of 10 nm.
Theoretical curves are obtained with equations (23) and (24) of the main paper (solid lines). Dashed
lines are obtained with equations (21) and (22) of the main paper for the weak anisotropy model.
23
Figure S15: NMRDs obtained from the strong anisotropy model for a SPM radius of 20 nm. Dashed
lines are obtained with simulations of the WA model.
24
Figure S16: NMRDs obtained from the strong anisotropy model for a SPM radius of 115 nm. Dashed
lines are obtained with simulations of the WA model.
25
Figure S17: Influence of the SPM magnetization on the relaxation rates for a 8nm particle in the SA
model. Dashed lines are square law fit of the low magnetization points.
26
Figure S18: Influence of the SPM magnetization on the relaxation rates for a 230nm particle in the SA
model.
27
Figure S19: Transverse relaxation rates obtained by a CPMG sequence and their variation with the
SPM radius and the inter-echo time. This is obtained in the SA model which is equivalent to the WA
model for high B0 fields.
28
References
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[3] A. Roch, R.N. Muller, P. Gillis, Theory of proton relaxation induced by superparamagnetic
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[4] H. E. Rorschach, A Classical Theory of NMR Relaxation Processes, J. Magn. Reson. 67 (1986),
519-530
[5] Y. Ayant, E. Belorizky, J. Alizon, J. Gallice, Calcul des densités spectrales résultant d’un
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[6] J.H. Freed, Dynamic effects of pair correlation functions on spin relaxation by translational
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