### Abstract

- Top of page
- Abstract
- PREVIOUSLY PUBLISHED THEORIES
- MATERIALS AND METHODS
- RESULTS
- A NEW MODEL
- DISCUSSION
- CONCLUSION
- REFERENCES

Computer simulations of water transverse relaxation induced by superparamagnetic particles are shown to disagree with the available theories, covering the slow diffusion domain. Understanding these new simulations, not in the slow diffusion domain, thus requires a new theoretical approach. A “partial refocusing model” is introduced for this purpose; it is based on a spatial division between an inner region where the gradients are too strong for the refocusing pulses to be efficient and an outer region where they are efficient. This model agrees with published simulations of relaxation induced by magnetic dipoles approximated as points. The validity domains of the various models are also compared. Magn Reson Med 47:257–263, 2002. © 2002 Wiley-Liss, Inc.

Susceptibility-induced *T*_{2}-shortening refers to the dephasing of magnetic moments due to field gradients created by small magnetized particles. We are concerned here with strongly magnetized particles, defined by the condition that the dephasing during an echo interval is large, viz., *τ*_{CP}*Δω*_{r} > 1, where *τ*_{CP} is half the interval between successive 180° pulses in a CPMG sequence, or half the echo time for a single (Hahn) spin-echo sequence (*τ*_{CP} = *TE*/2), and where *Δω*_{r} is the rms angular frequency shift at the particle surface (compared with a point infinitely far away). The present discussion is limited to spherical particles.

No general theory is able to describe the relaxation typical of this process over the entire range of variation of the parameters, but some theories address the problem over parts of the whole range.

First, the quantum-mechanical outer-sphere theory, which applies to relaxation induced by weakly magnetized particles (1, 2), remains valid for strongly magnetized particles, provided they are small enough to satisfy the motional averaging condition, i.e., *τ*_{D} < 1/*Δω*_{r}, where *τ*_{D} = *r*^{2}*/D* is the diffusion time, where *r* is the particle radius and *D* the water diffusion coefficient (*τ*_{D} is thus the time required for a water molecule to diffuse a distance r in any specified direction).

Second, there are two limits in which the relaxation behavior for larger particles is well known: one is the short echo limit. The boundary between the weak and the strong magnetization domains is defined by the equality *τ*_{CP}*Δω*_{r} = 1. The theories valid below the limit, especially mean gradient diffusion theory (MGDT (3)), appear thus as a useful theoretical starting point for the strong magnetization domain, even if this limit is unrealistic from a practical point of view (too-short echo times for magnetizations typical of superparamagnetic (SPM) particles, for instance).

The other is the long echo limit, bringing us to the FID and*T*, where relaxation is governed by the static dephasing regime (SDR) (4, 5). Intermediate echo times are less well explored, although recently theories have been introduced for the case of “slow diffusion” (*TE* < *τ*_{D}) around strongly magnetized particles (6, 7). The actual condition is stated differently in the two publications and will be defined more precisely below.

We present new simulations of relaxation induced by particles with a strong magnetization typical of superparamagnetic (SPM) particles, e.g., inorganic ferrites like iron oxides (Fe_{3}O_{4}) (8). These nanocrystals are ideal cores for the design of new and efficient contrast agents; they can also be used as therapeutic agents in cancer treatment, since they can act as high-frequency electromagnetic wave absorbers, able to induce a localized hyperthermia.

For radii such that *τ*_{D} < 1/*Δω*_{r}, the simulation results are consistent with outer sphere theory. However, for larger radii the transverse relaxation rates do not follow any existing theory. This is because of the small particle size, which causes the slow diffusion limit to break down.

We next introduce a new model that is based on a spatial division between two regions, one where the refocusing pulses are effective (i.e., relatively far from the particle) and one where the gradients are too large for them to be effective (relatively near the particle). By definition, the separation between the two regions is thus echo time-dependent and the boundary radius is defined as a function of echo time and particle magnetization with two adjustable parameters. Unexpectedly, the rate calculated in the outer region alone is shown to fit correctly the complete simulation results, including their dependence on echo time, particle radius, particle concentration, and particle magnetization.

This model, suitably modified, is also compared with simulation results for point dipoles (for which there is no motional-averaging region) (9) and shows good agreement.

### MATERIALS AND METHODS

- Top of page
- Abstract
- PREVIOUSLY PUBLISHED THEORIES
- MATERIALS AND METHODS
- RESULTS
- A NEW MODEL
- DISCUSSION
- CONCLUSION
- REFERENCES

*T*_{2} relaxation caused by small (radius = several nm) spherical particles was calculated by Monte Carlo simulations, as in previously published works (9–16).

The diffusion is modeled by a random walk through a tridimensional cubic lattice. Each magnetic moment makes discrete steps of length *λ* within a time *θ* defined so that:

- (8)

where *D* is the water diffusion coefficient. Magnetized impenetrable spheres of radius *r*, randomly distributed in the lattice, generate the field perturbations, and hence differences in proton Larmor frequencies. After each step the phase of 4000 diffusing magnetic moments is calculated and recorded, according to the equation:

- (9)

where *B* is the z component of the field experienced at the *i*th step of the random walk. *B*, which is proportional to *M*, the particle magnetization, is calculated from the dipolar fields of 405 neighboring particles.

The total magnetic moment is the vectorial sum of all elementary magnetic moments and the relaxation rate (corresponding to a free induction decay) is deduced from the decay of:

- (10)

where the average is taken over the phase distribution at time *t*.

Variables used in the simulations are *r*, the CPMG parameter *τ*_{CP}, the particle concentration *C* (the volume fraction being then *v* = (4π/3)*r*^{3}*C*). The parameter *Δω*_{r} is determined from *M* (Eq. [3]).

CPMG sequences are reproduced by changing the sign of the phase of all magnetic moments at times *t* = *τ*_{CP}, 3*τ*_{CP}, 5*τ*_{CP}, …. The same random walk is used to determine*T* and *T*_{2} for various values of *τ*_{CP}.

The rates shown in the graphs are deduced from monoexponential decays for the smallest spheres and from the slowest rate when the decay is multiexponential, as for the largest spheres.

Important scaling relations can be deduced from Eqs. [8], [9]. If *T*_{2} is the relaxation time calculated for echo time *τ*_{CP}, radius *r*, and magnetization *M*, then the relaxation time for echo time *τ*_{CP}.*Z*^{2}, radius *Z.r*, and magnetization *M*/*Z*^{2}, where *Z* is any real number, is *T*_{2}.*Z*^{2}.

### RESULTS

- Top of page
- Abstract
- PREVIOUSLY PUBLISHED THEORIES
- MATERIALS AND METHODS
- RESULTS
- A NEW MODEL
- DISCUSSION
- CONCLUSION
- REFERENCES

Figure 1 shows the simulated rate dependence on diffusion time (*τ*_{D} = *r*^{2}*/D*), which is the meaningful variable entering the simulations because of the scaling relations explained in the previous section and also discussed in Ref. 6. The equatorial field is 1 kG (*Δω*_{r} = 2.36 × 10^{7} rad/s), close to the magnetite value (1.3 kG). The volume fraction is 5 × 10^{−6}, independent of particle radius, so that the same quantity of magnetized material is distributed within a decreasing number of particles as the radius increases. For a diffusion coefficient *D* = 2.5 × 10^{−5} cm^{2}/s, the particle radius ranges from 10^{−3} to 10 μm. Seven values of echo-time, ranging from 0.1 ms to 20 ms, are presented.

The*T* values corresponding to large radii (open circles on right side, *τ*_{D} > 10^{−4} ms) are consistent with the SDR value (Eq. [2]): 1/*T* = 160 s^{−1}.

The left side of the curves, corresponding to small radii, is well fitted by the outer sphere theory, and thus by Eq. [1]; the results are echo time-independent, as expected.

On the right side, for large *τ*_{D}, after the rate starts decreasing from the SDR value, but for fixed echo time, the rate is inversely proportional to *τ*_{D}, in agreement with eqs. 16 and 16a in Ref. 1 and with Eq. [7]. However, the *τ*_{CP} dependence is far from quadratic: e.g., for the two extreme values of *τ*_{CP} with ratio 200, the ratio of the corresponding relaxation rates is only about 10.

Figure 2 shows the relaxation rate plotted vs. *Δω*_{r} for two volume fractions (*v* = 2 × 10^{−6}, and 5 × 10^{−4}), and several echo times. The dependence is not only not quadratic: it is less than linear.

Figure 3 shows the rate dependence on volume fraction, for 1/*T* and for 1/*T*_{2} with three different values of *τ*_{CP} (0.2 ms, 0.5 ms, and 2 ms): it is almost linear.

The right side of Fig. 1 (*τ*_{D} ≥ 0.01 ms) thus constitutes the unexplored domain. The slow diffusion condition of the Kiselev and Posse theory (6) (*τ*_{D} > *τ*_{CP}) is not satisfied because of the small radii. The modified “slow diffusion” condition of Ref. 7 (*τ*_{d} < *τ*_{D}) is also not satisfied, since *τ*_{D} is less than or equal to 5 ms, while the corresponding relaxation times range from 7–1000 ms. However, the weakened condition mentioned above, resulting from the substitution of *τ*_{D} by the interparticle diffusion time, i.e., *τ*_{D}*v*^{-2/3}, is satisfied for *τ*_{D} ≥ 0.01 ms (for a volume fraction *v* = 5 × 10^{−6}, the interparticle diffusion time is 3000 times the diffusion time). Yet the right side of Fig. 1 is not in agreement with Eq. [7]. The relaxation rates are not quadratic functions of *τ*_{CP} and *Δω*_{r}, nor are they 8/3 powers of *v* (see Fig. 3). The only agreement is the inverse proportionality to *τ*_{D}.

### A NEW MODEL

- Top of page
- Abstract
- PREVIOUSLY PUBLISHED THEORIES
- MATERIALS AND METHODS
- RESULTS
- A NEW MODEL
- DISCUSSION
- CONCLUSION
- REFERENCES

In this section we will attempt to present a new explanation for the behavior described above. We first note that, for usual echo times, the gradients near the magnetized particles are too large for the 180° pulses to be effective, but this is not the case sufficiently far from the particles. A spherical boundary of radius *r′* that separates the two regions may roughly be defined by the following condition:

- (11)

where *Δω*_{r}*′* characterizes the gradients at a distance r′ from the particle center.

#### Outer Region

The gradients in the outer region are small enough that the weak magnetization theories are valid. Calculating the relaxation rate in the outer region *alone* is thus straightforward: it is given by the corrected result (1) from Jensen and Chandra (2) for weakly magnetized particles and applied to spheres, i.e., eq. 16a in Ref. 1 (only the proportionality constant 2.25 differs from the MGDT result, where this constant is equal to 3):

- (12)

where *v′* = *v(r′/r)*^{3}*, Δω*_{r}*′* = *Δω*_{r}*(r/r′)*^{3}, and *τ*^{′}_{D} = *τ*_{D}*(r′/r)*^{2}.

Introducing these three relations into Eq. [12], one gets:

- (13)

This result retains the 1/*τ*_{D} rate dependence of MGDT (3) or of Eq. [7] only if r/r′ remains size-independent. We will show that Eq. [13] is remarkably effective, even for situations where the inner region becomes large.

Our modeling requires that we now relate *r′* to the physical parameters. This can easily be done if *r′* is small enough that the local magnetic field (at the boundary) is determined by the central particle alone, without accounting for overlapping fields from adjacent particles, i.e., for low-volume fraction and not-too-long echo times. Then, from Eq. [11] *r′/r* should be proportional to *(Δω*_{r}*τ*_{CP})^{1/3}. However, this dependence does not limit the increase of r′ with *τ*_{CP}, which is obviously unacceptable: the inner region volume cannot exceed the volume associated with each particle. The ratio *r′/r* must thus approach a constant value (depending on volume fraction) for very long echo times. We will therefore set:

- (14)

where *x* = *Δω*_{r}*τ*_{CP}. Equation [14] accounts for both requirements mentioned above: *r′/r* is proportional to *x*^{1/3} for small *x*, and it approaches *b*^{-1/3}, i.e., a constant, when *bx* becomes larger than *α*. At the limit of very large *x* (very long echo times), we expect (4/3)*πr′*^{3} to go to the volume associated with one particle, i.e., (4/3)*πr*^{3}/v, from which we deduce that the limit of (*r′/r*)^{3}, i.e., *b*^{−1}, is 1/*v.* We will thus write *b* = *βv,* so that Eq. [14] may be written as:

- (14a)

with an expected value of 1 for the adjustable parameter *β*.

The best fit of Eqs. [13]–[14a] to the data of Figs. 1–3, using only data for *τ*_{D} ≥ 0.01 ms, yielded *α* = 1.34 and *β* = 0.99, and produced very good agreement with the data. Both values are easily interpreted. We showed above that the condition for the SDR to be effectively reached is given by Eq. [5]. For *x* ≤ 1.35, magnetization will be restored by the pulses in the entire space, which implies that Eq. [13] may be applied with *r′* = *r*. In other words, Eq. [5] shows that *r′* should be equal to *r* for *x* = 1.35, almost identical to the fitted value *α* = 1.34. The value of 0.99 obtained for *β* is almost equal to the expected value (i.e., 1).

For each echo time it is possible to estimate the smallest radius for which the echoes will be at least partially effective (which is also the largest radius for which the rate is approximately given by the SDR theory): starting from large radii, to which our model may be applied, the limit corresponds to the radius for which the rate given by Eq. [13], which increases when *r* decreases, will reach the SDR rate (Eq. [2]), which is the maximum possible. The rates corresponding to the two regimes intersect when *τ*_{D}*= τ*_{l}, with:

- (15)

#### Inner Region — Fast Decay Component

Protons sufficiently close to the particles will experience gradients so strong that they will be rapidly dephased. These protons contribute a fast signal decay that is unobservable with MRI techniques, but that is visible with computer simulations (see Fig. 4). This was verified by simulations in which the spins were not allowed to get closer to the particles than a distance *r″*. For parameters *r* = 0.2 μm, *τ*_{CP} = 0.2 ms, *Δω*_{r} = 2.36 × 10^{7} rad/s, *v* = 5 × 10^{−6}, and *D* = 2.5 × 10^{−5} cm^{2}/s, the recorded relaxation rate was 17.3 s^{−1}. As *r″* was varied over the range *r* ≤ *r″* ≤ 10*r*, the rate remained equal to 17.5 ± 0.3 s^{−1}, decreasing only to 11.9 s^{−1} for *r″* = 15*r*.

These results clearly show that in this case eliminating an inner region does not affect the relaxation rate unless the excluded region is more than 1000 times the particle volume. Applying Eq. [14a] with the fitted parameters to the data of Fig. 4 (*x* = 4720) leads to *r′* = 15.1*r*: the fact that the rate starts decreasing for a radius slightly smaller than the model boundary arises from the roughness of our modeling, which does not account for the decay due to escaping spins.

Figures 4 and 5 show the decay of the echoes for *r″* = *r* (this corresponds to one of the points recorded in Fig. 1, without forbidden region) and for *r″* = 10 *r* (i.e., with a relatively large forbidden region): the only difference is the vanishing of the fast decaying component in Fig. 5.

#### Application of the PRM Model to Hardy and Henkelman Simulations

We also consider simulations by Hardy and Henkelman (9) of water diffusing among magnetic dipolar points (instead of spheres). Thus, there is no particle radius, which is an important parameter of modeling. The consequence of this is the complete vanishing of the motional-averaging regime, as well as the breakdown of the slow diffusion condition. Indeed, if we attribute an arbitrary small radius *r* to the dipoles and apply the condition of validity of the motional-averaging regime (*τ*_{D}*Δω*_{r} < 1), with *Δω*_{r} = *γμ/r*^{3}, where *μ* is the nominal magnetic moment of the dipolar points, we obtain *γμ/D* < *r*, which cannot be satisfied for an arbitrary small *r*. Any comparison of the simulated results with outer sphere theory would therefore be meaningless.

A first consequence of this is that*T* is given by the SDR rate, whatever the parameters of the problem, as noted in the original article of Brown (4). Rewriting Eq. [2] for dipolar points, with *v* = (4/3)π*r*^{3}*C* and *C* the dipole concentration, we obtain:

- (16)

with all factors depending on *r* canceling each other.

Our Eqs. [13], [14] can also be rewritten without an explicit *r* factor. If we assign an arbitrary radius *r* to the dipoles, Eqs. [13], [14] then become:

- (17)

and all factors depending on *r* cancel. This independence was shown to have physical meaning earlier, when the relaxation rate was shown to be unaffected when an inner region is excluded from the diffusion space.

The reference parameters used in the simulation were magnetic moment 1.57 × 10^{−9} emu, concentration 10^{6}/cm^{3}, and *τ*_{CP} = 10 ms. This concentration corresponds to an interparticle distance *a* = 100 μm and an interparticle diffusion time τ_{p} = 4 s (*D* = 2.5 × 10^{−5} cm^{2}/s). The parameters were then varied and, for the ranges considered, 1/*T*_{2} was found to increase as the 0.9 power of *D* and as the 0.6 power of *τ*_{CP} and *μ*.

The *D* dependence recorded by the authors is not very different from the linear dependence in Eq. [17], which also exhibits a common dependence on *μ* and *τ*_{CP}.

Fits of Eq. [17] to the data are shown in Figs. 6–8. The fitted parameters are *α* = 0.83 and *β* = 0.13, which remain reasonable values, although different from those given earlier. We do not see any fundamental justification for this discrepancy, except the fact that the huge dephasing that could result from a random jump leading a diffusing magnetic moment very near a point dipole could influence the recorded rate.

The agreement is very good for the echo time and concentration (reproducing, namely, the “superlinear” dependence) and for large values of the dipolar moment *μ*. The disagreement for low values of *μ* is striking, but corresponds to values of *μ* out of the domain of validity of the model, i.e., too close to the SDR. Indeed, the “outer region” modeling requires *τ*_{D} > *τ*_{l}, where *τ*_{l} is defined by Eq. [15]. For dipolar points, this condition becomes:

- (18)

The limit defined by Eq. [18] is reached for *μ* = 7 × 10^{−13} emu, a value not far from the smallest values shown in Fig. 7, corresponding approximately to the reference moment (1.57 × 10^{−9} emu) divided by 2000.

### DISCUSSION

- Top of page
- Abstract
- PREVIOUSLY PUBLISHED THEORIES
- MATERIALS AND METHODS
- RESULTS
- A NEW MODEL
- DISCUSSION
- CONCLUSION
- REFERENCES

The simulated relaxation rates induced by small and very strongly magnetized impenetrable spheres show a functional dependence on diffusion times and particle magnetization incompatible with the slow motion theories (6, 7). This feature, although not surprising (the slow diffusion condition is not satisfied), is a clear indication that the broadening of the validity domain of these theories, suggested by their authors and confirmed by the reanalysis of previously published data (10), does not extend to the whole range of parameters characteristic of the problem (concentration, particle size, magnetization, echo time).

However, our modeling provides another way of understanding the limitations imposed to the slow diffusion regime. Equation [7] is indeed the asymptotic limit of Eqs. [13]–[14a] when *τ*_{CP}*Δω*_{r} > 1/*v*, *α* being then negligible against *βvτ*_{CP}*Δω*_{r}. The two equations are strictly identical for *β* = 0.73. Equations [13]–[14a] thus contain Eq. [7] as a limit for long echo times rather than a limit for long diffusion times. Now, Eqs. [13]–[14a] are submitted to the condition *τ*_{D} > *τ*_{l}, where *τ*_{l} is given by Eq. [15], as a consequence of the ceiling imposed to the relaxation rate by the static dephasing regime. Remarkably, and omitting constants *α* and *β* (both on the order of 1), the condition *τ*_{D} > *τ*_{l} reduces to the strict slow diffusion condition (*τ*_{D} > *τ*_{CP}) when *τ*_{CP}*Δω*_{r} is on the order of 1, *α* being then much larger than *βvτ*_{CP}*Δω*_{r}, that is, at the lower limit of the strong magnetization domain. The same condition becomes *τ*_{D} > *v*^{5/3}*Δω*_{r}*τ*_{CP}^{2} or:

- (19)

if the “very long” echo time condition (*τ*_{CP}*Δω*_{r} > 1/*v*) is met. Inequality, Eq. [19], is exactly the weakened slow motion condition.

The most unexpected feature arising from our analysis is the good quality of the fit using Eqs. [13]–[14a], which completely rests on the contribution from the only outer region. For increasing echo times, and thus for increasing inner volume, the contribution from the inner region will become larger and larger, up to the vanishing of the slow component observed in our simulations. The magnetization decay is then due to relaxation into the inner region, governed by the static dephasing regime, which is indeed the limit the relaxation rate has to reach, i.e., 1/*T*. The model thus contains an intrinsic mechanism leading from *T*_{2} to*T*.

### CONCLUSION

- Top of page
- Abstract
- PREVIOUSLY PUBLISHED THEORIES
- MATERIALS AND METHODS
- RESULTS
- A NEW MODEL
- DISCUSSION
- CONCLUSION
- REFERENCES

Computer simulations made possible an analysis of susceptibility-induced *T*_{2}-shortening by strongly magnetized particles. This analysis first shows that motional averaging remains valid for particles sufficiently small, regardless of the relative magnitudes of *τ*_{CP} and (*Δω*_{r})^{−1}, provided that the motional averaging condition (*τ*_{D} < 1/*Δω*_{r}) is satisfied. In this regime the relaxation rate increases linearly with *τ*_{D} and is independent of echo time. While this increase continues until *τ*_{D} ≅ *τ*_{CP} for weakly magnetized particles, for strongly magnetized particles it is limited by *Δω*_{r}—that is, the static dephasing regime appears as a relaxation ceiling (4, 5).

For very small highly magnetized particles, the predictions of the slow diffusion theories are shown to disagree with simulation results obtained. CPMG sequences appear to operate a kind of spatial demixing of the different aforementioned regimes; the recorded rate is the slowest one (we called it the MGDT result), and a model of partial refocusing was shown to reproduce quite satisfactorily our new simulation results, as well as older ones of Hardy and Henkelman (9). The model is based on a spatial separation between two regions, an inner region where the gradients are too strong to allow refocusing, and an outer one where refocusing occurs, with a boundary depending on echo time, magnetization, and volume fraction. The theoretical result by Jensen and Chandra (7) appears as the limit of our “outer” rate for very long echo times (or for very large magnetizations).

Our conclusions are summarized through the definition of three regimes, according to the magnitude of *τ*_{D}:

- 1
for

*τ*_{D}*< τ*_{SDR} = π

/(4

*Δω*_{r}) = 3.04/

*Δω*_{r}, motional averaging remains valid and the relaxation rate is given by outer sphere theory (Eq. [

1]), with

*T*_{2} =

*T*;

- 2
for

*τ*_{SDR}*< τ*_{D} <

*τ*_{l} (see Eq. [

15] for the definition of

*τ*_{l}), the rate reaches and keeps its maximum value as given by Eq. [

2] (SDR): 1/

*T* = π

*vΔω*_{r}/9 = 1.35

*vΔω*_{r} (= 1/

*T*_{2});

- 3
when

*τ*_{D} >

*τ*_{l}, the partial refocusing model (Eqs. [

13]–[14a]) applies and the rate decreases like 1/

*τ*_{D}—the slow diffusion regime (Eq. [

7]) appears here as the very long echo time limit of the model (

*τ*_{CP} > 1/(v

*Δω*_{r})).

*T*, which is no more equal to

*T*_{2}, is still given by Eq. [

2].

The condition *τ*_{D} > *τ*_{l} must be satisfied for reaching the regime where the rate is inversely proportional to *τ*_{D} (Eqs. [13]–[14a]). This condition is equivalent to the strict slow diffusion condition at the limit *Δω*_{r}*τ*_{CP} ≈ 1. For longer echo times, this condition is progressively relaxed towards its weakened version, an evolution simultaneous to that of Eqs. [13]–[14a] towards eq. 7 of Jensen and Chandra (7).