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Quantifying magnetic susceptibility in the brain from the phase of the MR signal
+
= Introduction =
provides a non-invasive means for measuring the accumulation
 
of iron believed to occur with aging and neurodegenerative disease.
 
Phase observations from local susceptibility distributions,
 
however, are corrupted by external biasfields, which
 
may be identical to the sources of interest. Furthermore,
 
limited observations of the phase makes the inversion ill-posed. We
 
describe a variational approach to susceptibility estimation that
 
incorporates a tissue-air atlas to resolve ambiguity in the forward model, while eliminating additional biasfields
 
through application of the Laplacian. Results show qualitative improvement
 
over two methods commonly used to infer underlying
 
susceptibility values, and quantitative susceptibility estimates
 
show better correlation with postmortem iron concentrations than
 
competing methods.
 
 
 
= Description =
 
  
 
There is increasing evidence that excessive iron deposition in specific regions
 
There is increasing evidence that excessive iron deposition in specific regions
Line 24: Line 9:
 
In magnetic resonance imaging (MRI) experiments, differences
 
In magnetic resonance imaging (MRI) experiments, differences
 
in magnetic susceptibility cause perturbations in the local magnetic field, which
 
in magnetic susceptibility cause perturbations in the local magnetic field, which
can be computed from the phase of the MR signal.
+
can be computed from the phase of the MR signal (in a gradient echo sequence, the observed field is proportional to the MR phase).
 +
 
 +
= Description =
 +
 
 +
In MRI, magnetic susceptibility differences cause measurable perturbations in the local magnetic field that can be modeled as the convolution of a dipole-like kernel with the spatial susceptibility distribution. In the Fourier domain, the kernel exhibits zeros at the magic angle, preventing direct inversion of the field map; also, limited observations make the problem ill-posed. The observed data is also corrupted by confounding fields (ie. those from tissue/air interfaces, mis-set shims, and other non-local sources). Previous work has shown that MR images can be successfully reconstructed from under-sampled observations by exploiting the sparsity of in-vivo data under various transforms using methods from compressed sensing [2]. In susceptibility estimation, the forward model results in under-sampling of the data in the Fourier domain, but accurate estimates can be obtained using  the Laplacian and L1 norm, which promote sparse solutions while removing external field artifacts. Our variational method for susceptibility estimation is described in Figs. 1-2.
 +
 
 +
{|
 +
|[[File:Namic wiki fig1.png|thumb|400px|Fig 1. Relevant notation]]
 +
|}
 +
 
 +
{|
 +
|[[File:Latex pdf zoomed to paint equations.PNG|thumb|400px|Fig 2. Applying the Laplacian to the forward model in [1] eliminates non-local phase artifacts to give [2]. The first term in [3] provides regularization, penalizing large differences in spatial frequency relative to Magnitude data, while the second penalizes departures from [2], enforcing agreement of high frequency phase effects.]]
 +
|}
  
 +
== Forward Model (Eq. 1) ==
  
The field perturbations caused by magnetic susceptibility differences can be
+
The forward model relates the perturbing field to the unknown susceptibility through a local term and convolution of the second z-derivative of the Green’s function of the Laplacian with the unknown susceptibility map [3].
modeled as the convolution of a dipole-like kernel with the spatial susceptibility
 
distribution. In the Fourier domain, the kernel exhibits zeros at the magic angle,
 
preventing direct inversion of the fieldmap [3]. Critically, limited observations of
 
the field make the problem ill-posed. The observed data is also corrupted by
 
confounding biasfields (ie. those from tissue-air interfaces, mis-set shims, and other
 
non-local sources). Eliminating these fields is critical for accurate susceptibility
 
estimation since they corrupt the phase contributions from local susceptibility
 
sources.
 
  
 +
== Bias Field Elimination (Eq. 2) ==
  
In general, methods that rely heavily on agreement between observed and
+
Applying the Laplacian removes non-local phase effects such as shim fields, which are a solution to the Laplace equation.  
predicted field values computed using kernel-based forward models [2, 3, 6] are
 
inherently limited since they cannot distinguish between low frequency biasfields
 
and susceptibility distributions that are eigenfunctions of the model. Examples
 
of such distributions include constant, linear, and quadratic functions of
 
susceptibility along the main field (ie. 'z') direction. Applying the forward model to
 
these distributions results in predicted fields that are proportional to the local
 
susceptibility sources, but also identical in form to non-local biasfields (ie. those
 
produced by a z-shim). Therefore, removing all low frequency fields prior to
 
susceptibility estimation will eliminate the biasfield as well as fields due to the
 
sources of interest, potentially preventing accurate calculation of the underlying
 
susceptibility values. In contrast, inadequate removal of the biasfield may result
 
in the estimation of artifactual susceptibility eigenfunctions in areas where the
 
biasfield is strong, such as regions adjacent to tissue-air interfaces. This suggests
 
that additional information such as boundary conditions or priors may be necessary to
 
regularize an incomplete forward model and prevent the mis-estimation
 
of low frequency biasfields.
 
  
 +
== Objective Function (Eq. 3) ==
  
We present a variational approach for Atlas-based Susceptibility Mapping
+
The first term provides regularization, penalizing solutions with large differences in spatial frequency structure relative to the magnitude image.
(ASM) that performs simultaneous susceptibility estimation and biasfield
+
The second term penalizes departures from Eq. 2, by enforcing agreement of high frequency phase effects while eliminating low order bias fields.
removal using the Laplacian operator and a tissue-air susceptibility atlas. In [7,
 
8, 6] it was shown that applying the Laplacian to the observed field eliminates
 
non-local biasfields due to mis-set shims and remote susceptibility
 
distributions (ie. the neck/chest).  
 
In this method, large deviations from the susceptibility atlas are penalized,
 
discouraging the estimation of artifactual susceptibility eigenfunctions in regions near
 
tissue-air boundaries where the Laplacian may not be sufficient to eliminate the
 
contribution of non-local sources and substantial signal loss corrupts the observed field.
 
Agreement of predicted and observed fields
 
within the brain is also enforced, but deviations in estimated susceptibility values outside the
 
brain are not penalized, allowing values at the boundary to vary from
 
the atlas-based prior to account for unmodeled external field sources (ie. shims).
 
  
= Results =
+
== Data Acquisition ==
  
The method is evaluated by comparison of susceptibility maps estimated using ASM to results from Susceptibility Weighted Imaging (SWI) and Field Dependent Relaxation Imaging (FDRI). In SWI a filtered phase map is obtained by applying a high-pass filter to the phase data, and the resulting SWI map is commonly used as a proxy for susceptibility.  
+
Cylindrical and rectangular phantoms were made using Magnevist (gadopentetate dimeglumine) solutions of 0.5, 1.0, 2.0, and 3.0 mM corresponding to susceptibility values of 0.15, 0.31, 0.62, and 0.94 ppm [4,5]. Field maps were obtained using a 3D multi-echo GRE sequence on a 3T Siemens Trio MRI.
While SWI has shown some correlation with magnetic susceptibility differences
 
due to iron and other sources, the phase maps it yields are only an indirect
 
measure of susceptibility due to the non-local effects of the convolution kernel.
 
In addition, the filtering process may remove some low frequency fields due to
 
sources inside the brain. In FDRI, R2 maps are acquired at two different field strengths (ie. 1.5
 
and 3 Tesla) and the difference in R2 divided by the difference in field strength
 
gives the FDRI. The mean FDRI in several regions of interest was previously compared
 
to the mean iron concentration obtained from postmortem analysis and showed
 
stronger correlation with iron content than the SWI maps computed for the same
 
subjects. Obtaining FDRI measurements would be impractical for most studies,
 
however, since it requires images to be collected on two separate scanners.
 
Quantitative results were obtained by comparison of mean susceptibility values in the thalamus (TH), caudate (CD), putamen (PT)
 
and globus pallidus (GP) to corresponding results from SWI, FDRI and postmortem data.
 
  
 +
= Results =
  
ASM results for a young subject are shown in Fig. 1. Column 1 shows the T1
+
Application of the Laplacian removes substantial inhomogeniety in the field map in both phantoms as shown in Fig. 3 (Rectangular phantom) and Fig. 4 (Cylindrical phantom). Rectangular phantom: mean estimated susceptibility values for water and Magnevist were -9.049 and 0.6273 ppm, with true values of -9.050 and 0.6270 ppm. Cylindrical phantom: the estimated susceptibility map allowed different concentrations of Magnevist to be clearly identified and reasonable estimates were obtained in the presence of significant noise and bias due to external field effects.
structural (row 1) and acquired fieldmap (row 2). Application of the Laplacian
 
to the field map (row 2, column 2) removes substantial B0 inhomogeneities that
 
bias the observed field. The susceptibility atlas is shown in row 1, column 2
 
and estimated external sources are shown in row 1, column 3. The estimated
 
susceptibility map (row2, column 3) shares high frequency structure with the
 
Laplacian of the observed field, while low frequency structure is preserved by
 
enforcing agreement with additional information provided by the atlas-based
 
prior and observed field.
 
  
 +
{|
 +
|[[File:Box mag.jpg|thumb|300|Fig 3a. Magnitude Image]]
 +
|[[File:Box fmap.png|thumb|300|Fig 3b. Field map]]
 +
|[[File:Box fmap lp.png|thumb|300|Fig 3c. Laplacian of the Field]]
 +
|[[File:Box susc.png|thumb|300|Fig 3d. Estimated Susceptibility (ppm)]]
 +
|}
  
 
{|
 
{|
|[[File:Fig1 compound lighter v2.png|400px|thumb|Fig. 1: ASM Results.  
+
|[[File:Cyl mag.png|thumb|300|Fig 4a. Magnitude Image]]
The first column shows the T1 structural image (row 1) and field
+
|[[File:Cyl fmap.png|thumb|300|Fig 4b. Field map]]
map (row 2) with substantial inhomogeneity that was obtained from a young subject.
+
|[[File:Cyl fmap lp.png|thumb|300|Fig 4c. Laplacian of the Field]]
Column 2 shows the susceptibility atlas (row 1), in which voxels take continuous values
+
|[[File:Cyl susc.png|thumb|300|Fig 4d. Estimated Susceptibility (ppm)]]
between [0,1] corresponding to susceptibility values between air and tissue. Taking
+
|[[File:Susc plot2.png|thumb|100|Fig 4e. Estimated vs. True mean susceptibility values for each tube (ppm)]]
the Laplacian of the fieldmap successfully eliminates biasfields (row 2, column 2).  
 
Estimates of external sources are shown in row 1, column 3. The estimated susceptibility
 
map (row 2, column 3) shares similar high frequency structure with the Laplacian of
 
the observed field while low frequency structure is preserved by enforcing agreement
 
with the atlas and observed field.]]
 
 
|}
 
|}
  
Fig. 2 shows results from FDRI (row 1, column 2), SWI (row 1,column 3),
+
= Future Directions =
ASM for a young subject (row 2, column 3), and ASM results for 2 elderly
 
subjects (row 2, columns 1,2). The FDRI shows strong constrast between the
 
ROIs and surrounding tissue, but less high frequency structure than the SWI.
 
The SWI retains high frequency phase effects, but indiscriminately removes low
 
order fields from both internal and external sources, resulting in artifactual low
 
frequency structure. The ASM method accurately preserves the high frequency
 
phase effects seen in SWI while showing improved estimation of low order susceptibility
 
distributions. In addition, ASM provides direct estimates of susceptibility
 
values rather than filtered phase proxies for susceptibility.
 
  
 +
Future work will focus on quantifying magnetic susceptibility and iron content in the brain. Further development of the method described above has generated the preliminary results shown below in Fig 5.
  
 
{|
 
{|
|[[File:Fig2 compound lighter.png|400px|thumb|Fig. 2: Comparison of Results.
+
|[[File:Miccai fig1 crop.png|thumb|600px|Fig 5. The field map (left), laplacian of the field (center) and estimated susceptibility map (right) for a young healthy subject is shown. Taking the Laplacian of the fieldmap successfully eliminates the substantial biasfields in the observed field. The estimated susceptibility map shares similar high frequency structure with the Laplacian of the observed field while low frequency structure is preserved by additional modeling constraints.]]
Row 1 shows the T1 structural image (column 1), FDRI
 
(column 2) and SWI (column 3) results for a young subject. ASM results are shown in
 
row 2 for young (column 3) and elderly (columns 1,2) subjects. The FDRI shows strong
 
constrast between ROIs and adjacent tissue, but less high frequency structure than
 
the SWI. The SWI retains high frequency phase effects, but indiscriminately removes
 
low order fields from both internal and external sources, resulting in artifactual low
 
frequency structure. ASM accurately preserves the high frequency structure seen in
 
SWI while showing improved estimation of low order susceptibility distributions.]]
 
 
|}
 
|}
  
 +
= References =
 +
 +
1. Zecca L, et al. Nat Rev Neurosci, 5:863{73, Nov 2004.
  
Quantitative results from ASM and previously reported results from FDRI
+
2. Lustig M,et al. MRM. 2007. 58(6):1182.  
and SWI for the same elderly subjects are shown in Fig. 3. The mean
 
susceptibility values (relative to tissue susceptibility) in each ROI from all elderly subjects are plotted
 
against the corresponding iron concentrations from postmortem analysis (only
 
the mean and SD in each ROI was reported in [17]). ASM shows a high
 
correlation with postmortem values, which is comparable to that seen in FDRI and
 
substantially better than the correlation between phase and iron concentration
 
obtained with SWI. In addition, for the structures that we analyzed, ASM results
 
compare favorably to the correlation between postmortem iron and
 
susceptibility estimates in corresponding ROIs computed from multi-angle acquisitions [6].
 
  
 +
3. Jenkinson M, et al. MRM. 2004. 52(3):471.
  
{|
+
4. de Rochefort L, et al. MRM.2010. 63(1):194.  
|[[File:Fig3 elderly compound.png|800px|thumb|Fig. 3: Quantitative Results. The Mean +/- SD iron concentration
 
(mg/100g fresh weight)
 
in each ROI determined from postmortem analysis [17] is plotted on the x-axis. The y-
 
axes show the Mean +/- SD FDRI (s^{-1}/Tesla), Mean +/- SD SWI (radians), and Mean +/- SD
 
ASM susceptibility (ppm). Mean susceptibility values from ASM show a high
 
correlation with the postmortem data, which agrees well with previous results from FDRI
 
and shows improvement over SWI values reported for the same data [5].]]
 
|}
 
  
 +
5. Weisskoff RM, et al. MRM. 1992. 24(2):375.
  
 
= Key Investigators =
 
= Key Investigators =
  
* MIT: Clare Poynton, Polina Golland, Elfar Adalsteinsson
+
* MIT: Clare Poynton, Elfar Adalsteinsson
* BWH/Harvard: William (Sandy) Wells
+
* Harvard/BWH: William Wells
 
* Stanford: Adolf Pfefferbaum, Edith Sullivan
 
* Stanford: Adolf Pfefferbaum, Edith Sullivan

Latest revision as of 19:47, 28 November 2012

Home < Projects:QuantitativeSusceptibilityMapping

Introduction

There is increasing evidence that excessive iron deposition in specific regions of the brain is associated with neurodegenerative disorders such as Alzheimer's and Parkinson's disease [1]. The role of iron in the pathogenesis of these diseases remains unknown and is difficult to determine without a non-invasive method to quantify its concentration in-vivo. Since iron is a ferromagnetic substance, changes in iron concentration result in local changes in the magnetic susceptibility of tissue. In magnetic resonance imaging (MRI) experiments, differences in magnetic susceptibility cause perturbations in the local magnetic field, which can be computed from the phase of the MR signal (in a gradient echo sequence, the observed field is proportional to the MR phase).

Description

In MRI, magnetic susceptibility differences cause measurable perturbations in the local magnetic field that can be modeled as the convolution of a dipole-like kernel with the spatial susceptibility distribution. In the Fourier domain, the kernel exhibits zeros at the magic angle, preventing direct inversion of the field map; also, limited observations make the problem ill-posed. The observed data is also corrupted by confounding fields (ie. those from tissue/air interfaces, mis-set shims, and other non-local sources). Previous work has shown that MR images can be successfully reconstructed from under-sampled observations by exploiting the sparsity of in-vivo data under various transforms using methods from compressed sensing [2]. In susceptibility estimation, the forward model results in under-sampling of the data in the Fourier domain, but accurate estimates can be obtained using the Laplacian and L1 norm, which promote sparse solutions while removing external field artifacts. Our variational method for susceptibility estimation is described in Figs. 1-2.

Fig 1. Relevant notation
Fig 2. Applying the Laplacian to the forward model in [1] eliminates non-local phase artifacts to give [2]. The first term in [3] provides regularization, penalizing large differences in spatial frequency relative to Magnitude data, while the second penalizes departures from [2], enforcing agreement of high frequency phase effects.

Forward Model (Eq. 1)

The forward model relates the perturbing field to the unknown susceptibility through a local term and convolution of the second z-derivative of the Green’s function of the Laplacian with the unknown susceptibility map [3].

Bias Field Elimination (Eq. 2)

Applying the Laplacian removes non-local phase effects such as shim fields, which are a solution to the Laplace equation.

Objective Function (Eq. 3)

The first term provides regularization, penalizing solutions with large differences in spatial frequency structure relative to the magnitude image. The second term penalizes departures from Eq. 2, by enforcing agreement of high frequency phase effects while eliminating low order bias fields.

Data Acquisition

Cylindrical and rectangular phantoms were made using Magnevist (gadopentetate dimeglumine) solutions of 0.5, 1.0, 2.0, and 3.0 mM corresponding to susceptibility values of 0.15, 0.31, 0.62, and 0.94 ppm [4,5]. Field maps were obtained using a 3D multi-echo GRE sequence on a 3T Siemens Trio MRI.

Results

Application of the Laplacian removes substantial inhomogeniety in the field map in both phantoms as shown in Fig. 3 (Rectangular phantom) and Fig. 4 (Cylindrical phantom). Rectangular phantom: mean estimated susceptibility values for water and Magnevist were -9.049 and 0.6273 ppm, with true values of -9.050 and 0.6270 ppm. Cylindrical phantom: the estimated susceptibility map allowed different concentrations of Magnevist to be clearly identified and reasonable estimates were obtained in the presence of significant noise and bias due to external field effects.

Fig 3a. Magnitude Image
Fig 3b. Field map
Fig 3c. Laplacian of the Field
Fig 3d. Estimated Susceptibility (ppm)
Fig 4a. Magnitude Image
Fig 4b. Field map
Fig 4c. Laplacian of the Field
Fig 4d. Estimated Susceptibility (ppm)
Fig 4e. Estimated vs. True mean susceptibility values for each tube (ppm)

Future Directions

Future work will focus on quantifying magnetic susceptibility and iron content in the brain. Further development of the method described above has generated the preliminary results shown below in Fig 5.

Fig 5. The field map (left), laplacian of the field (center) and estimated susceptibility map (right) for a young healthy subject is shown. Taking the Laplacian of the fieldmap successfully eliminates the substantial biasfields in the observed field. The estimated susceptibility map shares similar high frequency structure with the Laplacian of the observed field while low frequency structure is preserved by additional modeling constraints.

References

1. Zecca L, et al. Nat Rev Neurosci, 5:863{73, Nov 2004.

2. Lustig M,et al. MRM. 2007. 58(6):1182.

3. Jenkinson M, et al. MRM. 2004. 52(3):471.

4. de Rochefort L, et al. MRM.2010. 63(1):194.

5. Weisskoff RM, et al. MRM. 1992. 24(2):375.

Key Investigators

  • MIT: Clare Poynton, Elfar Adalsteinsson
  • Harvard/BWH: William Wells
  • Stanford: Adolf Pfefferbaum, Edith Sullivan