Quantitative susceptibility mapping
Encyclopedia
Quantitative Susceptibility
Mapping (QSM) provides a novel contrast
mechanism in Magnetic Resonance Imaging (MRI) different from traditional Susceptibility Weighted Imaging
. The voxel intensity in QSM is linearly proportional to the underlying tissue apparent magnetic susceptibility
, which is useful for chemical identification and quantification of specific biomarkers including iron, calcium, gadolinium
, and super paramagnetic iron oxide (SPIO) nano-particles. QSM utilizes phase images, solves the magnetic field
to susceptibility
source inverse problem, and generates a three dimensional susceptibility
distribution. Due to its quantitative nature and sensitivity to certain kinds of material, potential QSM applications include standardized quantitative stratification of cerebral microbleeds and neurodegenerative disease, accurate gadolinium
quantification in contrast enhanced MRI, and direct monitoring of targeted theranostic drug biodistribution in nanomedicine
.
, the local field
induced by non-ferromagnetic biomaterial susceptibility
along the main polarization B₀ field is the convolution
of the volume susceptibility distribution
with the dipole
kernel
: 
. This spatial convolution
can be expressed as a point-wise multiplication in Fourier domain
:
. This Fourier
expression provides an efficient way to predict the field perturbation when the susceptibility distribution is known. However, the field to source inverse problem involves division by zero at a pair of cone surfaces at the magic angle
with respect to B₀ in the Fourier domain
. Consequently, susceptibility
is underdetermined at the spatial frequencies on the cone surface, which often leads to severe streaking artifacts in the reconstructed QSM.
gradient echo sequence can be used for data acquisition. In practice, high resolution imaging with a moderately long echo time is preferred to obtain sufficient susceptibility
effects, although the optimal imaging parameters depend on the specific applications and the field strength. A multi-echo acquisition is beneficial for accurate B₀ field measurement without the contribution from B1 inhomogeneity. Flow compensation may further improve the accuracy of susceptibility
measurement in venous blood, but there are certain technical difficulties to devise a fully flow compensated multi-echo sequence.
quantitative susceptibility
mapping, only the local susceptibility
sources inside the brain are of interest. However, the magnetic field
induced by the local sources is inevitably contaminated by the field induced by the air-tissue interface, whose susceptibility
difference is orders of magnitudes stronger than that of the local sources. Therefore, the background field generated by the air-tissue interface needs to be removed for clear visualization on phase images and precise quantification on QSM.
Traditional heuristic methods, including high-pass filter
ing, are useful for the background field removal, but they also tamper with the local field and degrade the quantitative accuracy. Two recent methods which are based on physical principles, Projection onto Dipole Fields (PDF) and Sophisticated Harmonic Artifact Reduction on Phase data (SHARP), demonstrated improved contrast and higher precision on the estimated local field. Both methods model the background field as a magnetic field
generated by an unknown background susceptibility
distribution, and differentiate it from the local field using either the approximate orthogonality or the harmonic property.
from multiple orientations. COSMOS utilizes the fact that the zero cone surface in the Fourier domain
is fixed at the magic angle
with respect to the B₀ field. Therefore, if an object is rotated with respect to the B₀ field, then in the object's frame, the B₀ field is rotated and thus the cone. Consequently, data that cannot be calculated due to the cone becomes available at the new orientations.
COSMOS assumes a model-free susceptibility
distribution and keeps full fidelity to the measured data. This method has been validated extensively in in vitro
, ex vivo
and phantom experiments. Quantitative susceptibility
maps obtained from in vivo
human brain imaging
also showed high degree of agreement with previous knowledge about brain anatomy. It should be noted that 3 orientations are generally required for COSMOS, limiting the practicality for clinical applications. However, it may serve as a reference standard when available for calibrating other techniques.
. This observation is translated into mathematics in MEDI, where edges in a QSM which do not exist in the corresponding magnitude image are sparsified by solving a weighted
norm minimization problem.
MEDI has also been validated extensively in phantom, in vitro
and ex vivo
experiments. In in vivo
human brain
, MEDI calculated QSM showed similar results compared to COSMOS without statistically significant difference. MEDI only requires a single angle acquisition, so it is a more practical solution to QSM.
is only at the location of the cone and its immediate vicinity. For this region in k-space
, spatial-frequencies of the dipole kernel are set to a predetermined non-zero value for the division. Investigation of more advanced strategies for recovering data in this k-space
region is also a topic of ongoing research.
Truncated k-space
division only requires a single angle acquisition, and benefits from the ease of implementation as well as the fast calculation speed. However, streaking artifacts are frequently present in the QSM and the susceptibility
value is underestimated compared to COSMOS calculated QSM.
and phantom experiments that cortical bones, whose major composition is calcification, are diamagnetic
compared to water. Therefore, it is possible to use this diamagnetism
to differentiate calcifications from iron deposits that usually demonstrate strong paramagnetism
. This may allow QSM to serve as a problem solving tool for the diagnosis of confounding hypointense findings on T2* weighted images.
quantification of gadolinium
or SPIO
concentrations.
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
Mapping (QSM) provides a novel contrast
Contrast (vision)
Contrast is the difference in visual properties that makes an object distinguishable from other objects and the background. In visual perception of the real world, contrast is determined by the difference in the color and brightness of the object and other objects within the same field of view...
mechanism in Magnetic Resonance Imaging (MRI) different from traditional Susceptibility Weighted Imaging
Susceptibility weighted imaging
Susceptibility weighted imaging , originally called BOLD venographic imaging, uses a type of contrast in magnetic resonance imaging different from traditional spin density, T1, or T2 imaging. SWI uses a fully flow compensated, long echo, gradient echo scan to acquire images...
. The voxel intensity in QSM is linearly proportional to the underlying tissue apparent magnetic susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
, which is useful for chemical identification and quantification of specific biomarkers including iron, calcium, gadolinium
MRI contrast agent
MRI contrast agents are a group of contrast media used to improve the visibility of internal body structures in magnetic resonance imaging . The most commonly used compounds for contrast enhancement are gadolinium-based. MRI contrast agents alter the relaxation times of tissues and body cavities...
, and super paramagnetic iron oxide (SPIO) nano-particles. QSM utilizes phase images, solves the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
to susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
source inverse problem, and generates a three dimensional susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
distribution. Due to its quantitative nature and sensitivity to certain kinds of material, potential QSM applications include standardized quantitative stratification of cerebral microbleeds and neurodegenerative disease, accurate gadolinium
MRI contrast agent
MRI contrast agents are a group of contrast media used to improve the visibility of internal body structures in magnetic resonance imaging . The most commonly used compounds for contrast enhancement are gadolinium-based. MRI contrast agents alter the relaxation times of tissues and body cavities...
quantification in contrast enhanced MRI, and direct monitoring of targeted theranostic drug biodistribution in nanomedicine
Nanomedicine
Nanomedicine is the medical application of nanotechnology. Nanomedicine ranges from the medical applications of nanomaterials, to nanoelectronic biosensors, and even possible future applications of molecular nanotechnology. Current problems for nanomedicine involve understanding the issues related...
.
Background
In MRIMagnetic resonance imaging
Magnetic resonance imaging , nuclear magnetic resonance imaging , or magnetic resonance tomography is a medical imaging technique used in radiology to visualize detailed internal structures...
, the local field
induced by non-ferromagnetic biomaterial susceptibilityMagnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
along the main polarization B₀ field is the convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of the volume susceptibility distribution
with the dipoleDipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...
kernel
: . This spatial convolutionConvolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
can be expressed as a point-wise multiplication in Fourier domain
K-space (MRI)
k-space is a formalism widely used in magnetic resonance imaging independently introduced in 1983 by Ljunggren and Twieg.In MRI physics, k-space is the 2D or 3D Fourier transform of the MR image measured. Its complex values are sampled during an MR measurement, in a premeditated scheme controlled...
:
. This FourierK-space (MRI)
k-space is a formalism widely used in magnetic resonance imaging independently introduced in 1983 by Ljunggren and Twieg.In MRI physics, k-space is the 2D or 3D Fourier transform of the MR image measured. Its complex values are sampled during an MR measurement, in a premeditated scheme controlled...
expression provides an efficient way to predict the field perturbation when the susceptibility distribution is known. However, the field to source inverse problem involves division by zero at a pair of cone surfaces at the magic angle
Magic angle
The magic angle is a precisely defined angle, the value of which is approximately 54.7356°. The magic angle is a root of a second-order Legendre polynomial, P_2=0 \,, and so any interaction which depends on this second-order Legendre polynomial vanishes at the magic angle...
with respect to B₀ in the Fourier domain
K-space (MRI)
k-space is a formalism widely used in magnetic resonance imaging independently introduced in 1983 by Ljunggren and Twieg.In MRI physics, k-space is the 2D or 3D Fourier transform of the MR image measured. Its complex values are sampled during an MR measurement, in a premeditated scheme controlled...
. Consequently, susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
is underdetermined at the spatial frequencies on the cone surface, which often leads to severe streaking artifacts in the reconstructed QSM.
Data acquisition
In principle, any 3DThree-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
gradient echo sequence can be used for data acquisition. In practice, high resolution imaging with a moderately long echo time is preferred to obtain sufficient susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
effects, although the optimal imaging parameters depend on the specific applications and the field strength. A multi-echo acquisition is beneficial for accurate B₀ field measurement without the contribution from B1 inhomogeneity. Flow compensation may further improve the accuracy of susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
measurement in venous blood, but there are certain technical difficulties to devise a fully flow compensated multi-echo sequence.
Background field removal
In human brainBrain
The brain is the center of the nervous system in all vertebrate and most invertebrate animals—only a few primitive invertebrates such as sponges, jellyfish, sea squirts and starfishes do not have one. It is located in the head, usually close to primary sensory apparatus such as vision, hearing,...
quantitative susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
mapping, only the local susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
sources inside the brain are of interest. However, the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
induced by the local sources is inevitably contaminated by the field induced by the air-tissue interface, whose susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
difference is orders of magnitudes stronger than that of the local sources. Therefore, the background field generated by the air-tissue interface needs to be removed for clear visualization on phase images and precise quantification on QSM.
Traditional heuristic methods, including high-pass filter
High-pass filter
A high-pass filter is a device that passes high frequencies and attenuates frequencies lower than its cutoff frequency. A high-pass filter is usually modeled as a linear time-invariant system...
ing, are useful for the background field removal, but they also tamper with the local field and degrade the quantitative accuracy. Two recent methods which are based on physical principles, Projection onto Dipole Fields (PDF) and Sophisticated Harmonic Artifact Reduction on Phase data (SHARP), demonstrated improved contrast and higher precision on the estimated local field. Both methods model the background field as a magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
generated by an unknown background susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
distribution, and differentiate it from the local field using either the approximate orthogonality or the harmonic property.
Field-to-source inversion
The field-to-source inverse problem can be solved by several methods with various associated advantages and limitations.Calculation Of Susceptibility through Multiple Orientation Sampling (COSMOS)
COSMOS solves the inverse problem by oversamplingOversampling
In signal processing, oversampling is the process of sampling a signal with a sampling frequency significantly higher than twice the bandwidth or highest frequency of the signal being sampled...
from multiple orientations. COSMOS utilizes the fact that the zero cone surface in the Fourier domain
K-space (MRI)
k-space is a formalism widely used in magnetic resonance imaging independently introduced in 1983 by Ljunggren and Twieg.In MRI physics, k-space is the 2D or 3D Fourier transform of the MR image measured. Its complex values are sampled during an MR measurement, in a premeditated scheme controlled...
is fixed at the magic angle
Magic angle
The magic angle is a precisely defined angle, the value of which is approximately 54.7356°. The magic angle is a root of a second-order Legendre polynomial, P_2=0 \,, and so any interaction which depends on this second-order Legendre polynomial vanishes at the magic angle...
with respect to the B₀ field. Therefore, if an object is rotated with respect to the B₀ field, then in the object's frame, the B₀ field is rotated and thus the cone. Consequently, data that cannot be calculated due to the cone becomes available at the new orientations.
COSMOS assumes a model-free susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
distribution and keeps full fidelity to the measured data. This method has been validated extensively in in vitro
In vitro
In vitro refers to studies in experimental biology that are conducted using components of an organism that have been isolated from their usual biological context in order to permit a more detailed or more convenient analysis than can be done with whole organisms. Colloquially, these experiments...
, ex vivo
Ex vivo
Ex vivo means that which takes place outside an organism. In science, ex vivo refers to experimentation or measurements done in or on tissue in an artificial environment outside the organism with the minimum alteration of natural conditions...
and phantom experiments. Quantitative susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
maps obtained from in vivo
In vivo
In vivo is experimentation using a whole, living organism as opposed to a partial or dead organism, or an in vitro controlled environment. Animal testing and clinical trials are two forms of in vivo research...
human brain imaging
Neuroimaging
Neuroimaging includes the use of various techniques to either directly or indirectly image the structure, function/pharmacology of the brain...
also showed high degree of agreement with previous knowledge about brain anatomy. It should be noted that 3 orientations are generally required for COSMOS, limiting the practicality for clinical applications. However, it may serve as a reference standard when available for calibrating other techniques.
Morphology Enabled Dipole Inversion (MEDI)
A unique advantage of MRI is that it provides not only the phase image but also the magnitude image. In principle, the contrast change, or equivalently the edge, on a magnitude image arises from the underlying change of tissue type, which is the same cause for the change of susceptibilityMagnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
. This observation is translated into mathematics in MEDI, where edges in a QSM which do not exist in the corresponding magnitude image are sparsified by solving a weighted
norm minimization problem.MEDI has also been validated extensively in phantom, in vitro
In vitro
In vitro refers to studies in experimental biology that are conducted using components of an organism that have been isolated from their usual biological context in order to permit a more detailed or more convenient analysis than can be done with whole organisms. Colloquially, these experiments...
and ex vivo
Ex vivo
Ex vivo means that which takes place outside an organism. In science, ex vivo refers to experimentation or measurements done in or on tissue in an artificial environment outside the organism with the minimum alteration of natural conditions...
experiments. In in vivo
In vivo
In vivo is experimentation using a whole, living organism as opposed to a partial or dead organism, or an in vitro controlled environment. Animal testing and clinical trials are two forms of in vivo research...
human brain
Brain
The brain is the center of the nervous system in all vertebrate and most invertebrate animals—only a few primitive invertebrates such as sponges, jellyfish, sea squirts and starfishes do not have one. It is located in the head, usually close to primary sensory apparatus such as vision, hearing,...
, MEDI calculated QSM showed similar results compared to COSMOS without statistically significant difference. MEDI only requires a single angle acquisition, so it is a more practical solution to QSM.
Truncated K-space Division (TKD)
The underdetermined data in Fourier domainK-space (MRI)
k-space is a formalism widely used in magnetic resonance imaging independently introduced in 1983 by Ljunggren and Twieg.In MRI physics, k-space is the 2D or 3D Fourier transform of the MR image measured. Its complex values are sampled during an MR measurement, in a premeditated scheme controlled...
is only at the location of the cone and its immediate vicinity. For this region in k-space
K-space (MRI)
k-space is a formalism widely used in magnetic resonance imaging independently introduced in 1983 by Ljunggren and Twieg.In MRI physics, k-space is the 2D or 3D Fourier transform of the MR image measured. Its complex values are sampled during an MR measurement, in a premeditated scheme controlled...
, spatial-frequencies of the dipole kernel are set to a predetermined non-zero value for the division. Investigation of more advanced strategies for recovering data in this k-space
K-space (MRI)
k-space is a formalism widely used in magnetic resonance imaging independently introduced in 1983 by Ljunggren and Twieg.In MRI physics, k-space is the 2D or 3D Fourier transform of the MR image measured. Its complex values are sampled during an MR measurement, in a premeditated scheme controlled...
region is also a topic of ongoing research.
Truncated k-space
K-space (MRI)
k-space is a formalism widely used in magnetic resonance imaging independently introduced in 1983 by Ljunggren and Twieg.In MRI physics, k-space is the 2D or 3D Fourier transform of the MR image measured. Its complex values are sampled during an MR measurement, in a premeditated scheme controlled...
division only requires a single angle acquisition, and benefits from the ease of implementation as well as the fast calculation speed. However, streaking artifacts are frequently present in the QSM and the susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...
value is underestimated compared to COSMOS calculated QSM.
Differentiating calcification from iron
It has been confirmed in in vivoIn vivo
In vivo is experimentation using a whole, living organism as opposed to a partial or dead organism, or an in vitro controlled environment. Animal testing and clinical trials are two forms of in vivo research...
and phantom experiments that cortical bones, whose major composition is calcification, are diamagnetic
Diamagnetism
Diamagnetism is the property of an object which causes it to create a magnetic field in opposition to an externally applied magnetic field, thus causing a repulsive effect. Specifically, an external magnetic field alters the orbital velocity of electrons around their nuclei, thus changing the...
compared to water. Therefore, it is possible to use this diamagnetism
Diamagnetism
Diamagnetism is the property of an object which causes it to create a magnetic field in opposition to an externally applied magnetic field, thus causing a repulsive effect. Specifically, an external magnetic field alters the orbital velocity of electrons around their nuclei, thus changing the...
to differentiate calcifications from iron deposits that usually demonstrate strong paramagnetism
Paramagnetism
Paramagnetism is a form of magnetism whereby the paramagnetic material is only attracted when in the presence of an externally applied magnetic field. In contrast with this, diamagnetic materials are repulsive when placed in a magnetic field...
. This may allow QSM to serve as a problem solving tool for the diagnosis of confounding hypointense findings on T2* weighted images.
Quantification of contrast agent
For exogenous susceptibility sources, the susceptibility value is theoretically linearly proportional to the concentration of the contrast agent. This provides a new way for in vivoIn vivo
In vivo is experimentation using a whole, living organism as opposed to a partial or dead organism, or an in vitro controlled environment. Animal testing and clinical trials are two forms of in vivo research...
quantification of gadolinium
MRI contrast agent
MRI contrast agents are a group of contrast media used to improve the visibility of internal body structures in magnetic resonance imaging . The most commonly used compounds for contrast enhancement are gadolinium-based. MRI contrast agents alter the relaxation times of tissues and body cavities...
or SPIO
MRI contrast agent
MRI contrast agents are a group of contrast media used to improve the visibility of internal body structures in magnetic resonance imaging . The most commonly used compounds for contrast enhancement are gadolinium-based. MRI contrast agents alter the relaxation times of tissues and body cavities...
concentrations.