The nuclear magnetic resonance (NMR) water signal obtained from a biological tissue may be strongly influenced by magnetic field inhomogeneities (MFIs) generated by spatial variations in magnetic susceptibility. These variations can be due to macroscopic structures, such as air cavities, bone, or large veins. More interestingly, there can also be microscopic MFIs, with length scales ranging roughly from 1 to 100 μm, which are produced, for example, by capillaries, iron rich cells, or the presence of an extracellular paramagnetic contrast agent. Information about these types of microscopic magnetic tissue structures is therefore encoded in the NMR signal.
The conventional MR method for exploiting the effect of MFIs has been to measure the relaxation rates R2 ≡ 1/T2 or R ≡ 1/T. This approach has, in particular, been applied to study the distribution of iron in the brain (1) and to quantify iron overload in the liver (2, 3) and heart (4, 5). A possible disadvantage of using these relaxation rates to assess MFIs is that they are also influenced by molecular relaxation mechanisms, such as dipole-dipole interactions. Thus R2 and R are not specific indicators of MFIs.
Some studies have also utilized the derived relaxation rate R ≡ R – R2 (6–8). Since molecular relaxation mechanisms typically increase the R2 and R by about the same amount, R can provide a better characterization of MFIs than either R2 or R separately. However, the precise physical meaning of R is not always apparent because of its complicated theoretical relationship to MFIs (9–15). Moreover, R may sometimes be poorly defined, as NMR signal decay in the presence of MFIs may deviate significantly from a monoexponential form (9–18). Measured R values will then be dependent on the experimental details.
Here we describe a new MRI method for determining a quantitative measure of MFIs, the magnetic field correlation (MFC). The MFC has a much more direct relationship to the MFIs than does R, allowing for a clearer physical interpretation. In addition, MFC imaging is based on a detailed theoretical model of NMR relaxation in the presence of MFIs that includes nonmonoexponential decay effects. As a consequence, MFC imaging data may be less sensitive than R to variations in experimental technique. MFC imaging was originally proposed by Jensen et al (19, 20) and Helpern et al (21), but a detailed presentation of the method has not been previously published.
The main purpose of this article is to explain the theoretical basis of MFC imaging. In addition, we give results from phantom experiments to support the theory's validity. Finally, we show some human brain MFC images, primarily to demonstrate the method's feasibility for in vivo studies. This preliminary data suggests that MFC imaging may be useful for brain iron quantification, which is of interest for a variety of neuropathologies including Alzheimer's and Parkinson's diseases (22).
Definition of Magnetic Field Correlation
The total magnetic field experienced by an individual water molecule at time t can be decomposed as
where B0 is the uniform average field and B(t) reflects the MFIs. The time dependence of B(t) is due to water diffusion. If there are significant microscopic MFIs with a length scale L, then B(t) will vary on a time scale of T ≈ L2/6D, where D is the water diffusion coefficient. For L = 10 μm and D = 1 μm2/ms, which is a typical diffusion coefficient value for biological tissues, this yields a time scale of about 17 ms.
Now consider the temporal correlation function
where the angle brackets indicate an averaging over all the water molecules within a voxel. This depends only on the time difference t – t′ because of time translation invariance. With increasing time, K(t) will normally decrease due to the effect of diffusion. Its initial value, K(0), is simply the variance for the MFIs. It should be noted that, by definition, 〈B(t)〉=0.
In biological tissues, K(0) is generally less than about 10–12 T2, at clinical MRI field levels of 1.0 to 3.0 T. For this reason, it is convenient to define the MFC at a time t as γ2K(t), where γ = 2.675 × 108 s–1T–1 is the proton gyromagnetic ratio. MFC values will then usually be in the range of 0 to 50,000 s–2.
If the components of a tissue are all either diamagnetic or paramagnetic, then the MFIs will depend linearly on the applied field strength. The MFC will then grow quadratically with the applied field, as follows from Eq. . However, this quadratic scaling of the MFC may not always be exact, since ferritin iron, which is a common nonheme form of iron in the human body, can be weakly superparamagnetic (23).
Measuring the MFC with MRI
The central idea of this work is that MRI can be used to estimate the MFC in biological tissues in a simple, direct manner (19). The method is based on the weak field theory of Jensen and Chandra (13). Within this approximation, the magnitude of the NMR signal at a time t is given by
where S0 is the initial signal intensity, σ(ta) is the spin flip function, and T2 is the transverse relaxation rate in the absence of the MFIs. The spin flip function has a magnitude of unity and changes sign at the time of any 180° refocusing pulses, as described in detail in Ref.13.
For the case of a single refocusing pulse at a time tp, the spin flip function is simply
where it is assumed that t > tp. Note that tp = t/2 corresponds to a symmetric spin echo (SSE) at the time t and that tp ≠ t/2 corresponds to an asymmetric spin echo (ASE).
As shown in the Appendix, one may deduce from Eq.  that
Physically, Eq.  states that the ratio of the signal intensity of an SSE acquired at a time t to an ASE acquired at the same time depends only on the MFC for times ranging from –ts + t/2 to ts + t/2. If the pulse time shift ts is not too large, then it is reasonable to utilize the Taylor expansion K(s + t/2) = K(t/2) + sK′(t/2) + O(s2) in the integral of Eq. , which gives
Hence for small ts, the SSE to ASE signal intensity ratio can be used to determine the MFC at the spin echo refocusing pulse time of t/2.
so that the ASE signal intensity is approximately a Gaussian function in the refocusing pulse time shift ts. In practice, it is by fitting MRI data to Eq.  that MFC estimates are obtained. Note that Eq.  does not depend on the T2 relaxation time.
Following a similar derivation, the single spin echo formula of Eq.  can be generalized to multiple spin echo sequences as
where 2Δt is the interecho time and Sn(ts) is the signal intensity of the nth echo. The nth echo is acquired at the time 2nΔt, and ts indicates the shift of the refocusing pulses from their usual times of (2n – 1)t. All the refocusing pulses must be shifted by the same amount. Adopting the convention S0 (ts) ≡ S0, Eq.  reduces to Eq.  for n = 1. In analogy with Eq. , Eq.  implies the approximation
The anomalous sign in the exponent for even n, reflects an enhanced refocusing for even order echoes as compared to odd order echoes. By fitting Eq.  to multiple spin echo data, the MFC at several different times can be acquired in parallel. A disadvantage, however, of a multiple spin echo approach is the possibility of confounding effects due to refocusing pulse imperfections.
In deriving the basic result of Eq.  (or Eq. ), essentially two key assumptions have been made: Condition 1, that the weak field theory is valid; and Condition 2, that ts is short enough to justify the Taylor expansion used to obtain Eq. . Condition 2 can always be satisfied, since the pulse time shift can be chosen to be arbitrarily small. However, MFC estimates made with too small ts values may have large statistical uncertainties due to noise. If we assume that the MFIs scale with the applied field (which is generally the case), then Condition 1 can also be guaranteed, in principle, by reducing the field level.
Although our derivation of Eq.  has made use of the weak field theory, its validity is not restricted to the weak field regime. In particular, in the absence of diffusion, Eq.  can be shown to be exact (in the limit of small ts). To see this, we first note, following the formalism of Jensen and Chandra (13), that without diffusion
By taking the spin flip function for a single refocusing pulse of Eq. , Eq.  reduces to
provided t > tp. The logarithm of the SSE to ASE signal intensity ratio is then seen to be
Expanding this in powers of ts yields
This is identical to Eq.  when diffusion effects are neglected, which shows that Eq.  holds in this case. The main criterion for the validity of the expansion of Eq.  is that
It is plausible, if not rigorously justified, to take the natural generalization of Eq. ,
as a basic condition for the applicability of Eq. . Equation , as well as the generalization of Eq. , may be readily extended to the multiple spin echo formula of Eq. .
While Eq.  provides a rough guideline for the validity of the MFC imaging method, it can be difficult to determine a priori whether Eq.  holds for a particular experimental arrangement. However, numerical evidence based on Monte Carlo simulations indicates this may well be the case for biological tissues at clinical field levels (19). In practice, it is convenient to take the viewpoint that a fit to Eq.  defines an “apparent” MFC much as a monoexponential fit to the signal intensity versus b-value data is used to define an apparent diffusion coefficient in diffusion-weighted MRI. The apparent MFC can then be regarded as an approximation to the true MFC, with the understanding that the nature of this approximation may at times be subtle.
An implicit assumption we have made is that the T2 relaxation rate, in the absence of the MFIs, is the same throughout the voxel. This implies that partial volume effects could alter measured MFC values. In addition, the presence of fat is a potential confounding factor, since the significant resonant frequency difference between fat and water can strongly affect ASE signal intensities. Indeed, one of the original applications of ASEs is to separate fat and water signals (24).
Of course, any MRI acquisition requires the imposition of imaging gradients. These are assumed to have a negligible effect on measured MFC values. Another practical consideration is that the receiver bandwidth be large enough to afford a sufficient temporal resolution. More specifically, the bandwidth/pixel should be much greater than (MFC)1/2.
To appreciate the physical significance of MFC imaging data, it is instructive to cite some established approximations for K(t), such as those that have been discussed in detail by Jensen and Chandra (13). Perhaps the most commonly used approximation is
where τ is a characteristic decay time (25). Alternatively, for unrestricted water diffusion, Jensen and Chandra (13) have argued that a more accurate functional form is given by
where rc is a characteristic length for the MFIs. This implies that the MFC will decrease to half its initial value in a time t ≈ rc2/(6.81 · D), which is nearly the same as the mean time for a water molecule to diffuse a distance rc.
For the particular model of MFIs generated by randomly distributed, identical penetrable spheres of radius R, one has the exact expression (13)
where ρ is the number of spheres per unit volume and χ is the magnetic susceptibility difference between the spheres and the surrounding media. The function U is given by
where Φ(x) is the probability integral. In this case, K(t) = K(0)/2 for t ≈ R2/(10 · D).
Microscopic magnetic susceptibility variations within a voxel are often the main source of its MFIs. However, macroscopic susceptibility variations, produced for example by air cavities or large veins, can generate field gradients that affect many nearby voxels. To understand how such macroscopic gradients alter MFC values, it is helpful to formally separate the macroscopic and microscopic contributions to the MFC.
The gradient field, G, within a voxel is determined by the condition
with r0 = 〈r(t)〉 representing the position of the voxel's center. Observe that G is independent of t due to time translation invariance (a small effect of water molecules leaving and entering the voxel is neglected). If we assume the voxel is a rectangular parallelepiped with edges of lengths Lx, Ly, and Lz oriented parallel to the x, y, and z Cartesian coordinate axes, then the components of G are
where we have used the conventional component notations defined by r = (x, y, z) and r0 = (x0, y0, z0).
From Eq. , one may show that the MFIs can be decomposed as
with the microscopic contribution satisfying the conditions
The last term on the right hand side of Eq.  has the order of magnitude D|t – t′| · |G|2, which is much smaller than the second term provided the diffusion length, (6D|t – t′|)1/2, is small compared to the voxel dimensions. This will almost always be the case in practice, since MRI experiments probe diffusion on time scales of no more than about 200 ms, corresponding to a diffusion length of about 35 μm.
The third term on the right hand side of Eq.  represents the cross correlation of the microscopic MFIs with the macroscopic gradients. On physical grounds, it is reasonable to suppose that this term is small, since the sources of the macroscopic gradients are largely from outside the voxel and typically independent of the microscopic MFIs which are due primarily to intravoxel magnetic susceptibility variations. More rigorously, we have the inequality
which is demonstrated in the Appendix. Thus the cross correlation can, at worst, become significant only on time scales for which the MFC has decreased substantially below its initial value.
The above considerations support the approximation
where Kmicro(|t−t′|)=〈Bmicro(t)Bmicro(t′)〉. Hence the MFC can be regarded as the sum of microscopic and macroscopic contributions, which differ in important respects. The microscopic contribution depends on time, but is independent (or weakly dependent) on the voxel size. In contrast, the macroscopic contribution is time independent, but depends quadratically on the voxel dimensions.
Since it is the microscopic component of the MFC that is usually of interest, it is advantageous to image with the smallest voxel size that gives an adequate signal-to-noise ratio (SNR), in order minimize the effect of any macroscopic gradients. In addition, isotropic voxels are preferable both for minimizing the macroscopic gradient contribution for a given SNR and for minimizing the dependence of the MFC on the voxel's spatial orientation.
Equation  implies that the difference between the MFC obtained at two different times is independent of the macroscopic gradients. Such a differential MFC value can be deduced from MFC estimates based on two separate single spin echo experiments. Alternatively, one can use a dual spin echo sequence to derive an MFC difference more directly. From Eq. , one can show that
Therefore, the use of dual spin echoes may be advantageous under circumstances with large macroscopic gradients. More generally, one may show that the signal intensity of all the even order echoes described by Eq.  are independent of the macroscopic gradients.
Connection to Carr-Purcell-Meiboom-Gill R2 Values
Microscopic MFIs in both the brain and blood have been previously investigated by measuring the dependence of R2, as measured with a Carr-Purcell-Meiboom-Gill (CPMG) sequence, on the interecho time (13, 26–30). Because of the combined effects of MFIs and water diffusion, R2 is observed to increase as the interecho time is lengthened.
Within the weak field approximation, the relationship between the MFC and R2 as measured with a CPMG sequence is (13)
where R2,0 is the transverse relaxation rate in the absence of the MFIs and 2Δt is the interecho time. Thus the interecho time dependence of R2 also provides information about the MFC. However, as Eq.  implies, the connection between R2 and the MFC is mathematically complex, and it would be difficult to obtain model independent values for the MFC at a specific time using only R2 data. Our MFC imaging procedure, in contrast, yields MFC estimates in a straightforward manner.
A phantom was constructed from six 60-ml plastic bottles. One bottle was filled with a mixture of water and Sephadex (G-25 Superfine; Amersham Biosciences, Uppsala, Sweden). Sephadex consists of particles of cross-linked dextran with a wet bead size range of 17 to 70 μm for the grade employed. Since water and dextran have different magnetic susceptibilities, microscopic MFIs could then be expected in the Sephadex bottle when placed in a strong magnetic field. A second bottle contained water doped with a small amount of gadopentetate dimeglumine (Gd-DTPA) in order to shorten T1. The remaining bottles were filled with corn syrup and were included to mechanically stabilize the other two. Corn syrup was used because of its extremely short T2, which effectively eliminates the signal from these bottles in T2-weighted images and thereby reduces imaging artifacts. The six bottles were attached with a rubber band, placed in a plastic container, and surrounded with a corn syrup bath. The corn syrup bath served to reduce macroscopic MFIs within the bottles.
The phantom was first imaged at room temperature with a Siemens 3-T Trio MRI scanner. A segmented echo planar imaging (EPI) ASE sequence was used with an EPI factor of 37. EPI was chosen to be consistent with the human study, for which EPI is critical in reducing both motion artifacts and the acquisition time. The field of view (FOV) was 256 × 256 mm2, the acquisition matrix size was 128 × 128, and the slice thickness was 2 mm, resulting in isotropic 2-mm3 voxels. The bandwidth was 1345 Hz/pixel, and the repetition time (TR) was 2 s. Images for a single slice were obtained at echo times (TE) of 40, 50, 60, 80, and 100 ms. For each TE, images were acquired with refocusing pulse time shifts of ts = 0, –4, –8, –12, and –15 ms, where the negative sign indicates a reduction of the interval between the initial 90° excitation pulse and 180° refocusing pulse from the SSE value of TE/2. (Note: TE is the acquisition time for the images and should note be confused with the spin echo refocusing time of TE + 2ts.) For each set of imaging parameters, 20 images were obtained and averaged to increase the SNR. In order to test the field dependence of the MFC, the phantom was subsequently imaged with a Siemens 1.5-T Avanto MRI scanner using an identical imaging protocol.
In addition to the ASE images needed for the determination of the MFC, the phantom was scanned at 3 T with a proton-density-weighted sequence, in order to estimate the water content of the Sephadex bottle, and with a diffusion-weighted sequence, in order to estimate the diffusion coefficient. Both the proton-density-weighted and diffusion-weighted sequences had the same FOV and resolution as the ASE sequence. Other imaging parameters for the proton-density-weighted sequence were TE = 5 ms, TR = 1.5 s, and flip angle = 10°. For the diffusion-weighted sequence, the other imaging parameters were TE = 91 ms, TR = 2 s, and b values of 0 and 1000 s/mm2. The diffusion sensitizing gradients were applied in three orthogonal directions so that the trace of the diffusion tensor could be calculated.
The mean signal intensities in the water and Sephadex bottles were estimated for each image by averaging 12 × 12 pixel square regions of interest (ROIs). The signal intensities for the ASE images were fit to the equation
with a separate fit being performed for each TE. The parameter η reflects the background noise level and was estimated from the signal intensity in air. The two fitting parameters, a1 and a2, were determined by χ2 minimization using the Levenberg-Marquardt method (31).
In the absence of background noise (η = 0), fitting to Eq.  is equivalent to fitting to Eq. . The modification of Eq.  is a conventional method for approximately including the effects of noise for MRI magnitude images (32). By comparing Eqs.  and , it is seen that the parameter a2 provides an estimate for the MFC. MFC maps for the phantom were constructed in an analogous manner by fitting the images on a pixel by pixel basis to Eq. .
To estimate the water content of the Sephadex bottle, the signal intensity in the proton-density-weighted images for this bottle was divided by that of the water bottle. The diffusion coefficient in the Sephadex bottle was determined from the trace of the diffusion tensor (i.e., the average of the apparent diffusion coefficient in the three orthogonal directions). This is not to imply that the diffusion was anisotropic; in fact the measured apparent diffusion coefficient was nearly the same in each direction.
Our human study was performed on the same 3-T scanner and with the same segmented EPI ASE sequence as for the phantom study and was approved by the New York University School of Medicine institutional review board. The EPI factor and the TR were the same as for the phantom study, but the refocusing pulse time shifts were ts = 0, –4, –8, –12, and –16 ms. In addition, the FOV was 220 × 220 mm2, the slice thickness was 1.1 mm, and the bandwidth was 1185 Hz/pixel. The slices were oriented axially with a slight tilt to avoid coming too close to the sinus cavity, which is a potential source of macroscopic MFIs. Fat saturation pulses were used to suppress the fat signal.
In a first scanning session, images where obtained with TE values of 42 ms and 70 ms and with an acquisition matrix size of 192 × 192. The entire protocol was run twice, as a test of reproducibility. The data for these two trials were analyzed separately. In a second scanning session, images where obtained with a TE of 42 ms and acquisition matrices of 192 × 192, 128 × 128, and 64 × 64, to test the effect of resolution on the MFC values. As indicated by Eq. , a component of the MFC due to macroscopic gradients oriented in the image plane would be expected to increase quadratically with the voxel dimensions.
The acquisition of each set of imaging parameters was preformed 10 times. Image repetitions were coregistered and averaged to improve the SNR. The averaged images for a given slice but different refocusing pulse time shifts were also coregistered, prior to the MFC analysis. This coregistration procedure was employed to minimize motion artifacts in the MFC maps.
The ROI analysis and the MFC maps were performed in the same way as for the phantom study, except that the ROIs were chosen in the putamen. The putamen was selected because it is a large structure with a high iron concentration. The adjacent globus pallidus also has high iron content, but was less suitable for this study because the SNR is significantly lower in this region. From the pair of SSE images (i.e., ts = 0) with TE = 42 ms and 70 ms, R2 maps were computed from the formula
where TE1 = 42 ms and TE2 = 70 ms.
The phantom arrangement is demonstrated in the proton-density-weighted image of Fig. 1. Bottles 1–4 contain corn syrup and are hypointense both because of their low water content and their extremely short T2 relaxation times. The ratio of the signal intensities of the water bottle (Fig. 1, A) to the Sephadex bottle (Fig. 1, B) indicates a water fraction of 0.820 ± 0.003 in bottle B. The 3-T SSE image of Fig. 1 was obtained from the ASE sequence with TE = 40 ms and ts = 0. Only bottles A and B are visible because of the rapid transverse relaxation of the signal in bottles 1–4 and in the surrounding corn syrup bath.
The 3-T MFC map for t = TE/2 = 20 ms is also shown in Fig. 1. The MFC of the Sephadex bottle is much larger than that of the water bottle, which is consistent with the presence of microscopic MFIs in the Sephadex bottle. A small MFC is also observed in the water bottle, possibly reflecting a macroscopic gradient. The MFC maps for longer times look similar, except for a marked decrease of the MFC in the Sephadex bottle with increasing time.
Fits of the 3-T signal intensity vs. the refocusing pulse shift are illustrated in Fig. 2 for TE = 40 ms and TE = 50 ms. A good fit to Eq.  is seen in both cases, and fits of similar quality were obtained for the other TE values. The goodness-of-fit was quantified by the confidence level for the χ2. This confidence level, Q, is given by
where ν is the number of degrees of freedom and Γ(a, x) is the incomplete gamma function. The number of degrees of freedom ν is the number of data points minus the number of fitting parameters. If all the ts values are used, then ν = 5 – 2 = 3, for our case. A confidence level of 0.50 or 50% means that there is a 50% chance of obtaining a χ2 of this magnitude or larger, assuming the theoretical model is accurate (31). Confidence levels exceeding 5% are usually regarded as indicating an acceptable fit. This threshold is well surpassed for each value of TE, supporting the validity of Eq. . For every TE, the highest confidence level was obtained by using all five ts values.
Both the 1.5-T and 3-T MFC values vs. time for the Sephadex and water bottles are shown in Fig. 3. In the Sephadex bottle, the MFC is seen to decrease monotonically with time, which is the theoretically expected behavior when microscopic MFIs are present. At 1.5 T, the MFC in the water bottle is zero within the experimental uncertainties. A small, time-independent MFC is observed in the water bottle at 3 T, consistent with the presence of a macroscopic gradient. The microscopic MFIs should, of course, be nearly zero in the water bottle. The macroscopic gradients within the ROIs could potentially have been affected by both the slice location and phantom placement, and thus the macroscopic contributions to the MFC for the 1.5 T and 3 T scans may not be well correlated.
The time dependence of the microscopic MFC in the Sephadex bottle can reasonably be expected to approximately follow Eq. . As suggested by Eq. , any macroscopic gradients contribute a constant offset. Therefore the total MFC should be given by
where C is the macroscopic gradient offset. Because Sephadex is diamagnetic, the ratio γ2Kmicro(0)/B is independent of the applied field strength. The lines in Fig. 3 are global fits of Eq.  to the measured MFC data. The fitting parameters γ2K(0)/B and 4Dt/rc2 were constrained to be the same for both the 1.5-T and 3-T data sets, but the macroscopic gradient offset was allowed to differ for the two field levels. In carrying out the fits, the precise applied field strengths of B0 = 1.4944 T and 2.8936 T were used. The resulting fitting parameters are
If these estimated C values are subtracted from the measured MFC values, then the ratios of the microscopic components of the MFC for 3 T and 1.5 T are found to be 3.76, 3.71, 3.44, 4.03, and 2.81 for t = 20, 25, 30, 40, and 50 ms, respectively. These ratios are comparable the exact square field ratio of (2.8936/1.4944)2 = 3.75, in agreement with theory.
The water diffusion coefficient in the Sephadex bottle was measured to be 1.29 ± 0.01 μm2/ms. This combined with Eq.  then gives rc = 6.5 ± 1.4 μm for the characteristic length of the MFIs, which is comparable to the size (17 μm) of the smaller of the Sephadex particles. We note that for the random sphere model considered by Jensen and Chandra (13), rc is about half the sphere diameter.
Accepting Eq.  as a rough approximation for the microscopic component of the MFC, we can also estimate the susceptibility difference χ between Sephadex and water. Since U(0) = 1, Eq.  gives
Equating the measured water fraction of 0.820 ± 0.003 with 1 – 4πρR3/3 then yields χ ≈ 5.4 × 10–8. This is not far from the range of susceptibility difference values, 0.7 to 1.1 × 10–8, measured by Kuznetsov et al (33) for starch grains, which are chemically similar to Sephadex.
The anatomy of the analyzed slice for the first scanning session is shown in the TE = 42 ms SSE images appearing in Fig. 4. The ROI indicated in the Trial 1 image indicates the putamen, which was selected for the quantitative analysis; a similar ROI was identified in the Trial 2 image. Fits of the signal intensity versus the refocusing pulse time shift are given in Fig. 5. The confidence levels are maximized when the data points for |ts| = 16 ms are excluded, in which case the confidence levels are consistent with a good fit. This suggests that the form of Eq.  is appropriate up to |ts| ≤ 12 ms, but may become less accurate for larger shifts.
The MFC maps shown in Fig. 4 were calculated using just the refocusing pulse shifts with |ts| ≤ 12 ms, as these gave the best fit in the putamen. The large hyperintense regions near the center of the image correspond to the putamen and globus pallidus. It is reasonable that these would have a large MFC due to the high iron concentration in these structures (34). Smaller focal bright features, presumably blood veins, are also apparent. Blood veins can generate high MFC values due to the large paramagnetic susceptibility of deoxyhemoglobin. The elevated MFC values seen in the scalp and near the brain perimeter are likely due to macroscopic gradients or EPI-related artifacts. The slight differences between the first and second trials may be attributed to motion and noise. The corresponding R2 maps, appearing in Fig. 4, are also hyperintense in the putamen and globus pallidus. However, the contrast between these two basal ganglia structures and the surrounding tissue is not as great as for the MFC maps. For example, the contrast-to-noise ratio between the putamen and frontal white matter in the MFC maps is about 2.4, while it is 1.1 for the R2 maps.
The MFC values for the two trials for both t = TE/2 = 21 ms and t = TE/2 = 35 ms are shown in Fig. 6. The statistical error in the MFC estimates is about 10%. The MFC drop from 21 to 35 ms is also about 10%. This is consistent with the theoretical expectations, but is not statistically significant.
SSE images and MFC maps obtained in the second scanning session for three different resolutions are shown in Fig. 7. The data for the three resolutions were obtained serially, and the resolutions shown in Fig. 7 correspond to the acquisition matrix of the raw data. The slice position is approximately the same for each resolution, but slight differences due to movement are visible. Also shown in Fig. 7 are histograms for the MFC maps. The histograms include only values that are >0 and <1000 s–2 in order to exclude the background and values associated with the scalp and large vessels. The similarity of the three histograms suggests that there is little contribution to the MFC values from macroscopic gradients parallel to the image plane. As follows from Eq. , an effect of macroscopic gradients would result in a quadratic increase of the MFC with the voxel dimensions.
The central idea of this article is that the MFC can be estimated by fitting ASE data to the Gaussian form of Eq. , for single-spin echoes, or Eq.  for multiple-spin echoes. Of course, ASEs have long been used in MRI (24), and the use of Gaussian fits has been previously introduced for both empirical (16, 35, 36) and theoretical reasons (37, 38). Here, however, we have given the first detailed, rigorous derivation of a relationship between MFIs and the MFC that includes the effect of diffusion through microscopic MFIs.
A related but distinct imaging method has been previously proposed by Ma and Wehrli (7). This method, termed gradient-echo sampling of free induction decay and echo (GESFIDE), is also based on ASE acquisitions, but differs in three important aspects. First, the position of the 180° refocusing pulse is fixed, but the image acquisition time is varied. Second, the data is fit to monoexponential signal decay forms. Third, the method yields estimates for R2, R, and R, rather than for the MFC. A possible limitation of this method is that significant deviations from monoexponential decay have been observed in both blood (16) and bone (18).
In prior work, we have reported numerical evidence obtained with Monte Carlo simulations (19) and experimental evidence for yeast cell suspensions (20) to support the validity and feasibility of MFC imaging, and an initial human MFC image was presented in Ref.21. The phantom results given here confirm and extend the conclusions obtained with yeast cell suspensions. The close agreement between the theoretical form of Eq.  and the data shown in Fig. 2 attests to the accuracy, in this case, of a Gaussian fit. The predicted decrease of the MFC with time is clearly demonstrated by the data of Fig. 3. The fit of this data to Eq. , with reasonable values for the fitting parameters, also supports the underlying theory of MFC imaging. In particular, the phantom data confirms that the measured MFC values scale with the applied field as predicted theoretically.
Our human results are consistent with those of the phantom study, with a good fit to Eq.  observed in the putamen, but with some indication of a deviation from Gaussian behavior for shifts |ts| ≥ 16 ms. While a decrease of the MFC with increasing time is observed, it is not statistically significant. A high MFC is found in both the putamen and the globus pallidus, which is likely due to the known high iron concentration of these regions (34). A high MFC is also seen in other brain areas, such as near large blood veins. The MFC maps of Fig. 4 are distinguished from the corresponding R2 maps because they have a much larger contrast between the basal ganglia structures and the surrounding tissue. This suggests the possibility that MFC imaging may provide a more sensitive method of brain iron quantification than the use of R2 maps.
An important issue is the length scale of the MFIs that contribute to the MFC values observed in the brain parenchyma. The results shown in Fig. 7 indicate at most a small contribution from macroscopic gradients parallel to the image plane, although this result may depend on the slice selection and orientation. The decrease in the MFC with time in the putamen points to a significant contribution from microscopic MFIs with a length scale on the order of micrometers. However, additional investigation of the origin of the MFIs is warranted. It should nonetheless be emphasized that any contribution of macroscopic gradients can be strongly suppressed by reducing the voxel dimensions.
The MFC maps of Figs. 4 and 7 show a significant amount intraregional MFC heterogeneity within the putamen and other brain regions. This may reflect spatial inhomogeneities in nonheme brain iron, and indeed Haacke et al. (39) have reported qualitatively similar results in the basal ganglia with susceptibility-weighted imaging. Alternatively, blood veins, motion artifacts, and noise could be contributing factors. It may be noted that some spatial inhomogeneity is also apparent in the phantom MFC map of Fig. 1, which is likely due, at least in part, to imaging artifacts rather than true MFC variations.
As discussed, the MFC is affected by water diffusion, as well as the MFIs. Therefore, it can be helpful to acquire diffusion-weighted MRI data in conjunction with MFC measurements in order to better interpret MFC differences. This may be particularly important for tumors and tissues with edema. In normal brain, however, diffusion coefficients are fairly uniform within the brain parenchyma, and so regional variations in water diffusion are unlikely to alter MFC contrast significantly. As demonstrated with our phantom study, the effects of water diffusion and the MFIs can potentially be separated by measuring the MFC at several different times and fitting to a model, such as the one given by Eq. .
Further experimental validation and development of MFC imaging is needed to demonstrate practical advantages over conventional relaxations rates for the quantification of tissue properties, such as iron concentration. Most importantly, the connection between MFC and brain iron should be investigated in greater detail both with animal studies, which would allow for a direct comparison to histological iron measures, and with more human studies. In addition, the effect of imaging artifacts, such as EPI-related ghosting, motion, and vascular pulsation, should be explored. Finally, the confounding effect of macroscopic MFIs should be evaluated more thoroughly; the explicit decomposition of the MFC into microscopic and macroscopic components given by Eq.  hints at a refined MFC imaging method that would allow these to be measured independently.
Because the MFC grows approximately with the square of the applied field strength, high field scanners are especially suitable for MFC imaging. It should nevertheless be remembered that the main theoretical basis for MFC imaging relies on a weak field approximation, and Eqs.  and  may become inaccurate if the field level is too high (19).
The quadratic dependence of the MFC on the applied field also suggests that the “rescaled MFC” ≡ MFC/B may be regarded an intrinsic tissue property. However, the previously noted weak nonlinearities in magnetic characteristics of ferritin iron (23) imply that this rescaled MFC can, in fact, have a residual field dependence. An additional dependence on the relative orientation of a tissue to the direction of the applied field can occur both because of macroscopic gradients, which may change with body alignment, and anisotropic tissue properties (as in white matter).
Because iron compounds, such as ferritin and hemosiderin, are the most commonly occurring paramagnetic substances within the human body, iron quantification is a potential application of MFC imaging. Additional tissue properties could be studied by administering a paramagnetic contrast agent, for example Gd-DTPA. Since these agents remain in the extracellular spaces, they can generate significant microscopic MFIs. It is then at least conceivable that MFC imaging may be able to yield information about the cellular morphology of pathological tissues, including tumors.
We have shown how a new physical quantity, the MFC, can, in principle, be measured in vivo by means of MRI. The key to MFC imaging is the fitting of ASE data to Eq.  (or Eq. ). Since MFC images are prone to motion artifacts, an EPI acquisition and image coregistration were utilized for the preliminary MFC maps shown here. In addition, voxels with small dimensions were employed to reduce the effects of macroscopic gradients. While the data presented supports the feasibility of MFC imaging, considerable work remains in the validation and optimization of this technique.
This work was supported in part by grants from the National Institutes of Health (NIH/NIBIB R21/R33 EB003305 to J.H.J) and from the Institute for the Study of Aging and the Werner Dannheisser Testamentary Trust (to J.A.H.).
Derivation of Eq. 
From Eq. , it is straightforward to demonstrate
where ts is the shift of tp from the SSE value of t/2.
A rearrangement of the integrals in Eq. [A1] results in
With the change of variables ta → t – tb and tb → t – ta in the second integral on the right hand side of Eq. [A2], we find
A second change of variables in the second integral, ta → ta – ts and tb → tb – ts, gives
By combining the two right hand side terms and making the transformation ta → ta + t/2, one then sees that
A final variable change of ta – tb → s and (ta + tb)/2 → u allows the integral over u to be performed explicitly, leading directly to the result of Eq. .
Derivation of Eq. 
Note that Schwarz's inequality gives
From time translation invariance, it can be deduced that
where it has been assumed that Kmicro(t) ≥ 0. From Eqs. [A6] and [A7], one then sees that
As previously discussed in the Theory section, for a practical MRI experiment
Applying this result together with the algebraic inequality x2 + y2 ≥ 2xy to Eq. [A8] then yields