HighResolution Magnetocardiogram Restoration for Cardiac Electric Current Localization

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First Claim
1. A magnetocardiogram (MCG) system comprising:
 a sensor unit including M×
M electromagnetic sensors producing a sparse measurement output of M×
M data units, said sparse measurement output constituting a first MCG image;
a linear model defining a second MCG image of substantially higher resolution than said first MCG image, said second MCG image having a P×
P resolution where P>
M, said linear model establishing interpolation patterns between characteristics of the linear model and any data point of said M×
M measurement output; and
a high resolution MCG image synthesizer for producing a third MCG image by projecting said first MCG image onto the subspace of the linear model, and establishing coefficients for said third MCG image in accordance with the linear model and said M×
M data units.
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Abstract
Magnetocardiogram (MCG) provides temporal and spatial measurements of cardiac electric activities, which permits current localization. An MCG device usually consists of a small number of magnetic sensors in a planar array. Each sensor provides a highly lowresolution 2D MCG map. Such a lowres map is insufficient for cardiac electric current localization. To create a high resolution MCG image from the sparse measurements, an algorithm based on model learning is used. The model is constructed using a large number of randomly generated high resolution MCG images based on the BiotSavart Law. By fitting the model with the sparse measurements, high resolution MCG image are created. Next, the 2D position of the electric current is localized by finding the peak in the tangential components of the high resolution MCG images. Finally, the 2D current localization is refined by a nonlinear optimization algorithm, which simultaneously recovers the depth of the electric current from the sensor and its magnitude and orientation.
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20 Claims
 1. A magnetocardiogram (MCG) system comprising:
a sensor unit including M×
M electromagnetic sensors producing a sparse measurement output of M×
M data units, said sparse measurement output constituting a first MCG image;a linear model defining a second MCG image of substantially higher resolution than said first MCG image, said second MCG image having a P×
P resolution where P>
M, said linear model establishing interpolation patterns between characteristics of the linear model and any data point of said M×
M measurement output; anda high resolution MCG image synthesizer for producing a third MCG image by projecting said first MCG image onto the subspace of the linear model, and establishing coefficients for said third MCG image in accordance with the linear model and said M×
M data units. View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
 13. A method of creating a magnetocardiogram (MCG) image from a sparse measurement output provided by a sensor unit including a plurality of electromagnetic sensors, each electromagnetic sensor contributing its output data to said sparse measurement output, said method comprising;
defining high resolution to mean a resolution substantially higher than the resolution provided by said sparse measurement output; creating a plurality of synthesized high resolution magnetocardiogram images based on simulated electrical impulses within a threedimensional spatial heart volume, as it would be perceived in an expected magnetocardiogram system; creating a linear model of the synthesized high resolution magnetocardiogram images to establish interpolation patterns between characteristics of the linear model and any sparse measurement output; and creating a representative high resolution MCG image by projecting said sparse measurement output onto the subspace of the linear model, and establishing coefficients.  View Dependent Claims (14, 15, 16, 17, 18, 19, 20)
1 Specification
1. Field of Invention
The present invention relates the field of magnetocardiogram (MCG) imaging. More specifically, it relates to the generation of highresolution MCG images from sparse data input from an electromagnetic sensor unit.
2. Description of Related Art
Cardiac electric currents are generated by electrophysiological processes in the heart. Localizing abnormal electric currents is very important for diagnosing ischemic diseases such as myocardial infarction, angina cordis, etc. It also benefits patients in the catheter lab for both treatment and followup, as is explained in “Forty Years of Magnetocardiology”, by F. Stroink, in Int. Conf. on Biomagnetism Advances in Biomagnetism, 28:18, 2010.
Traditionally, cardiac electric activities such as arrhythmia are diagnosed by means of an electrocardiogram (ECG). However, since an ECG only provides temporal information, it cannot localize abnormal currents in the heart directly, even if the ischemic disease has been detected. However, by using a large number of electrodes (leads), Body Surface Potential Mapping (BSPM) is able to reconstruct a body surface potential map, as is explained in “Noninvasive volumetric imaging of cardiac electrophysiology”, by Wang et al., in CVPR, pages 21762183, 2009. Nonetheless, the accuracy of electric current localization is still limited because the signals are often distorted due to the poor conductivity of body tissue.
The advent of the magnetocardiogram, or magnetocardiography, (MCG) provides more accurate measurements of cardiac electric currents, both spatially and temporally. With reference to
Each sensor 13 is a capture point, and hereinafter may be referenced as a capture 13. Each capture 13 measures a onedimensional (i.e. 1D) magnetic waveform in a direction perpendicular to the sensor planar array (i.e. the zdirection) emanating from the patient'"'"'s chest 21 (i.e. human torso). The MCG sensor unit 11 is usually placed five to ten centimeters above the patient'"'"'s chest 21, and measures the patient'"'"'s heart magnetic field in a noninvasive way. At each capture 13 a low resolution (hereinafter, lowres), twodimensional (2D) MCG map of electromagnetic activity is measured.
Compared to ECG, MCG has a few advantages. First, the magnetic field generated by the heart'"'"'s electrical impulses (i.e. currents) is not distorted in the direction perpendicular to the body surface (i.e., z direction), due to the magnetic property of body tissue. Thus MCG is more accurate and sensitive to weak electrical activities in the early stage of heart disorders. Second, the MCG sensor array can localize the position of electrical currents in the heart. Finally, MCG measurements are noninvasive. After forty years of research in MCG, cardiac electric current localization and high resolution visualization for MCG measurements are attracting more and more interest from both research and clinical areas.
However, there are a number of difficulties associated with MCG, which so far has prevented MCG from becoming a mainstream medical diagnostic tool in cardiology. One difficulty is that the lowres 2D MCG maps are not sufficient for localizing electric currents in the heart. For example, a 64 channel Hitachi™ MCG system with a 25 mm sensor interval (as described in “Newly Developed Magnetocardiographic System for Diagnosing Heart Disease”, by Tsukada et al., in Hitachi Review, 50(1):1317, 2001) only measures an 8×8 MCG map (i.e. an 8×8 array of 64 measurement points).
The resolution of an MCG map is limited due to the size of the sensors 13, which limit the number of sensors 13 that an MCG sensor unit 11 may have.
Thus, a necessary step in MCG, is creating a high resolution (hereinafter Highres) MCG image, or map, from a lowres 2D MCG map. Two image examples 23 and 25 of such highres images are shown in
Most current MCG systems use curve fitting interpolation methods to reconstruct highres MCG images from lowres 2D MCG maps, as is shown in “Magnetocardiographic Localization of Arrhythmia Substrates: A Methodology Study With Accessory PathWay Ablation as Reference”, by B. A. S. et al., in Ann Noninvasive Electrocardiol, 10(2):152160, 2005, and shown in “Evaluation of an Infarction Vector by Magnetocardiogram: Detection of Electromotive Forces that Cannot be Deduced from an Electrocardiogram”, by Nomura et al, in Int. Congress Series, 1300:512515, 2007. Unfortunately, the accuracy of curve fitting methods is usually limited.
Another method for improving the accuracy of highres MCG images first reconstructs a threedimensional (3D) position, magnitude and orientation of electric currents, given the lowres MCG measurements. This method is generally called the inverse problem, and is more fully explained in “Magnetocardiographic Localization of Arrhythmia Substrates: A Methodology Study with Accessory Pathway Ablation as Reference”, by R. J. et al., in Europace, 11(2):169177, 2009, and explained in “Conversion of Magnetocardiographic Recordings Between Two Different Multichannel Squid Devices”, by M. B. et al., in IEEE Trans. on Biomedical Engineering, 47(7):869875, 2000. This method generally computes a highres MCG image based on current reconstructed by the BiotSavart law.
As it is known in the art, however, according to the Helmboltz reciprocity principal, the inverse problem for MCG is an ill posed problem unless the number of electric currents is known. But even when the current number is known, it requires solving a large scale nonlinear optimization problem which is often computationally expensive and may lead to undesired local minimum.
R. J. et al. therefore propose a simplified solution by assuming a single electric current located at the world origin and far from the sensor array. As it is known in the art, linear solutions may be presented for special cases where the current positions are fixed at uniform grids in the heart. The presented linear solutions can be overconstrained or underconstrained. These methods, however, make another assumption that the sensor array and heart are perfectly aligned. In practice, these assumptions often can be difficult to satisfy.
Therefore, highres MCG image restoration based on these types of methods can often be unreliable.
Recently machine learning techniques have been applied to highres MCG image restoration. An example of this approach applies learned nonlinear interpolation functions using neural networks.
An object of the present invention is a method of creating more accurate highres MCG images.
Another object of the present invention is to provide a less computingintensive approached toward generating accurate highres MCG images form lowres 2D maps.
A further object of the present invention is to eliminate the need for assumption regarding the alignment of an MCG system and a patient'"'"'s torso.
These objects are achieved by considering the highres MCG image restoration as an example based superresolution problem. Typically, one would require a library of true examples from which to learn characteristic of such true examples. However, since it is infeasible to measure dense magnetic fields, and thus not feasible to obtain such true example from direct measures, the presently preferred embodiment uses a model learning algorithm based on synthetic highres MCG images. The sample images are randomly generated based on the BiotSavart Law. From these samples the algorithm constructs a linear model by principal component analysis (PCA). By projecting the sparse measurements into the subspace of the linear model, the model coefficients are estimated and the highres MCG image can be restored as a model instance.
As is explained above, an MCG image typically provides lowres, 2D MCG maps, and the inverse problem (i.e. reconstruction of the position and moment of the electric current) would typically be applied to the lowres, 2D MCG maps. It is presently preferred, however, that the inverse problem be applied to the restored highres MCG image.
Given the highres MCG image, the 2D position of the electric current can be localized as the maximal point of the tangential components of the highres MCG image. To improve the 2D localization accuracy, a nonlinear optimization algorithm is developed to solve the inverse problem. At the same time, the depth, magnitude and orientation of the electric current are also recovered. More specifically, the preferred algorithm alternates two steps iteratively. The first step estimates the 3D current position, and the second step reconstructs its magnitude and orientation. The 2D current location estimated from the model based restoration is used as the initialization. The present method is efficient, accurate and reliable without the need of special assumptions.
The above objects are met in a magnetocardiogram (MCG) system comprising: a sensor unit including M×M electromagnetic sensors producing a sparse measurement output of M×M data units, said sparse measurement output constituting a first MCG image; a linear model defining a second MCG image of substantially higher resolution than said first MCG image, said second MCG image having a P×P resolution where P>M, said linear model establishing interpolation patterns between characteristics of the linear model and any data point of said M×M measurement output; and a high resolution MCG image synthesizer for producing a third MCG image by projecting said first MCG image onto the subspace of the linear model, and establishing coefficients for said third MCG image in accordance with the linear model and said M×M data units.
In this MCG system, the third MCG image has a P×P resolution.
Preferably, the MCG system further has an electric current localizer for determining a position and momentum of an electric current in accord with said third MCG image, said electric current localizer evaluating the electromagnetic output data from each electromagnetic sensor in an xy orientation (Bxy) assuming single dipole, computing dense Bxy from dense Bz, finding the image maximum in said third MCG image, and using this determined position information as a starting point in an iterative process for identifying a threedimensional position vector {right arrow over (p)} and momentum vector {right arrow over (J)} for said electric current.
In this approach, the identifying a threedimensional position vector {right arrow over (p)} and momentum vector i for said electric current further includes: Given said third MCG image B_{z}(i,j)(i=1, 2, . . . ,N; j=1, 2, . . . ,N), the maximal point of the tangential components B′_{xy}(i,j) of B_{z}(i,j) refers to the 2D position (x_{p}, y_{p}) of the electric current, and the tangential components of B_{z}(i,j) is computed as B_{xy}(i, j)=√{square root over ((∂B_{z}(i,j)/∂x)^{2}+(∂B_{z}(i,j)/∂y)^{2})}{square root over ((∂B_{z}(i,j)/∂x)^{2}+(∂B_{z}(i,j)/∂y)^{2})}; and said iterative process for identifying position vector {right arrow over (p)} and momentum vector {right arrow over (J)} for said electric current includes: (a) defining the BiotSarvart Law as {right arrow over (B)}^{m}={right arrow over (J)}×{right arrow over (R)}_{m}=−{right arrow over (R)}_{m}×{right arrow over (J)}, where {right arrow over (B)}^{m}={right arrow over (B)}({right arrow over (r)}_{m}), {right arrow over (J)}={right arrow over (J)}({right arrow over (p)}) and
(b) expanding this definition of the BiotSarvart Law to a matrix form by using a skewsymmetric matrix:
where the z component of the magnetic field is computed as:
B_{z}^{m}=[R_{m}^{2},−R_{m}^{1}]·[J^{1},J^{2}]′
where R_{m}^{1},R_{m}^{2 }are x,y components of {right arrow over (R)}_{m}, and for said M×M electromagnetic sensors one has a linear system:
where B is a measured M×1 vector, R is a M×2 position matrix that is computed from given {right arrow over (p)} and {right arrow over (δ)}_{m}, and a lease square solution for J provides an estimation of J defined as J=(R^{T}R)^{−1}R^{T}B; (c) defining the BiotSarvart Law
letting α=4π/μ_{0 }and {right arrow over (ε)}_{0}={right arrow over (r)}_{0}−{right arrow over (p)}, identifying {right arrow over (δ)}_{m }as known for each sensor to redefining the BiotSarvart Law as
letting {right arrow over (τ)}_{m}={right arrow over (J)}×{right arrow over (δ)}_{m }and {right arrow over (ε)}=(x_{ε},y_{ε},z_{ε})^{T }and computing {right arrow over (τ)}_{m }from {right arrow over (J)}, for each sensor m=1:M, defining a nonlinear equation in terms of (x_{ε},y_{ε},z_{ε}) as
letting F=(f^{1}; f^{2}; . . . ; f^{M})=0, and solving a least square solution of the nonlinear system F for {right arrow over (ε)}_{0}; (d) using {right arrow over (ε)}_{0 }from step (c) to update the position matrix R and recompute J as in step (b), and iteratively repeat steps steps (b) and (c) until converges is achieved; and (e) defining the {right arrow over (p)}={right arrow over (r)}_{0}−{right arrow over (ε)}_{0}, and defining the initial depth z and magnitude ∥{right arrow over (J)}∥ of the electric current as
where d is the distance between two magnetic poles in the third MCG image.
Further preferably, in the present MCG system, the linear model is defined as creating a plurality of synthesized magnetocardiogram images having the same resolution as said second MCG image, said synthesized magnetocardiogram images being based on simulated electrical impulses within a threedimensional spatial heart volume, as it would be perceived in an expected magnetocardiogram system.
In this case, the plurality of synthesized magnetocardiogram images includes at least one thousand synthesized images simulating perceived electrical impulses per predefined depth level within said heart volume.
Additionally, the synthesized MCG images are synthesized using the BiotSavart Law.
Also, the synthesized MCG images are based on randomly generated currents within said heart volume.
Furthermore, the linear model is created using by principal component analysis (PCA).
In the presently preferred MCG system, the interpolation patterns are established by the following steps: (A) defining the following notation: N×N dense Bz magnetic field map to form a vector; M×M sparse measurement to form a vector; K randomly generated single current dipoles Q; (B) for each randomly generated current Q, compute N×N magnetic field map using BiotSavart equation and stack the image to a vector f_{10}; (C) repeating step (B) to obtain K samples and get a data matrix A=└f_{1}, f_{2}, . . . f_{K}┘; and (D) training a PCA model given input data A, to obtain the eigenmatrix Σ_{f}.
In an embodiment of the present invention, the third MCG image is created by: given a new dipole and M×M sparse measurements g_{j }finding the corresponding rows in the eigenmatrix, and denoting a resultant submatrix as Σ_{g}; projecting the sparse measurement to the PCA subspace and computing the coefficients as c_{g}=Σ_{g}^{+}(g_{j}−g_{mean}), where Σ_{g}^{+ }is an estimation of the inverse of Σ_{g}; and using the computed coefficients and original PCA space to reconstruct the dense magnetic field map Bz, as f_{j}=Σ_{f}c_{g}+f_{mean}.
Also in an embodiment of the present invention, the producing of said third MCG image includes: defining the sparse measurement output as a vector g; defining the linear model as Σ; extracting from Σ the row corresponding to sparse measurement output to form a subeigenmatrix Σ_{g}; projecting g onto Σ_{g}; defining the establishment of coefficients as c_{g}=Σ_{g}^{+}(g_{i}=μ_{g}), where Σ_{g}^{+} is the pseudo inverse of Σ_{g}, μ_{g }are extracted coefficients from a mean vector μ of linear model Σ; and defining the high resolution MCG image vector h as h=Σ·c_{g}+μ
The above objects also are met in a method of creating a magnetocardiogram (MCG) image from a sparse measurement output provided by a sensor unit including a plurality of electromagnetic sensors, each electromagnetic sensor contributing its output data to said sparse measurement output, said method comprising; defining high resolution to mean a resolution substantially higher than the resolution provided by said sparse measurement output; creating a plurality of synthesized high resolution magnetocardiogram images based on simulated electrical impulses within a threedimensional spatial heart volume, as it would be perceived in an expected magnetocardiogram system; creating a linear model of the synthesized high resolution magnetocardiogram images to establish interpolation patterns between characteristics of the linear model and any sparse measurement output; and creating a representative high resolution MCG image by projecting said sparse measurement output onto the subspace of the linear model, and establishing coefficients.
Preferably in this method, the plurality of synthesized high resolution magnetocardiogram images includes more than one thousands images simulating perceived electrical impulses at different depths within said heart volume.
Further preferably, the synthesized high resolution MCG images are synthesized using the BiotSavart Law.
If desired, the synthesized high resolution MCG images may be randomly generated.
In a preferred embodiment, the linear model is created using by principal component analysis (PCA).
Also in a preferred embodiment, the interpolation patterns are established by the following steps: (A) defining the following notation: N×N dense Bz magnetic field map to form a vector; M×M sparse measurement to form a vector; K randomly generated single current dipoles Q; (B) for each randomly generated current Q, compute N×N magnetic field map using BiotSavart equation and stack the image to a vector f_{1}; (C) repeating step (B) to obtain K samples and get a data matrix A=└f_{1}, f_{2}, . . . f_{K}┘; and (D) train a PCA model given input data A, to obtain the eigenmatrix Σ_{f}.
In this approach, the representative high resolution MCG image is created by: given a new dipole and M×M sparse measurements g_{j}, finding the corresponding rows in the eigenmatrix, and denoting a resultant submatrix as Σ_{g}; projecting the sparse measurement to the PCA subspace and computing the coefficients as c_{g}=Σ_{g}^{+}(g_{j}−g_{mean}), where Σ_{g}^{+ }is an estimation of the inverse of Σ_{g}; and using the computed coefficients and original PCA space to reconstruct the dense magnetic field map Bz, as f_{j}=Σ_{f}c_{g}+f_{mean}.
More specifically, the step of creating a representative high resolution MCG image includes: defining the sparse measurement output as a vector g; defining the linear model as Σ; extracting from Σ the row corresponding to sparse measurement output to form a subeigenmatrix Σ_{g}; projecting g onto Σ_{g}; defining the establishment of coefficients as c_{g}=Σ_{g}^{+}(g_{i}−μ_{g}), where E_{g}^{+ }is the pseudo inverse of Σ_{g}, μ_{g }are extracted coefficients from a mean vector μ of linear model Σ; and defining the high resolution MCG image vector h as h=Σ·c_{g}+μ.
Other objects and attainments together with a fuller understanding of the invention will become apparent and appreciated by referring to the following description and claims taken in conjunction with the accompanying drawings.
In the drawings wherein like reference symbols refer to like parts.
The presently preferred embodiment considers the highres MCG image restoration as an example based superresolution problem. Typically, one would require a library of true examples from which to learn characteristics of such true examples. However, since it is impractical to measure dense magnetic fields, and thus not feasible to obtain such true examples from direct measures, the presently preferred embodiment uses a model learning algorithm based on synthetic highres MCG images.
The sample images are preferably randomly generated based on the BiotSavart Law. From these sample images, a linear model is constructed a by use of principal component analysis (PCA). Sparse measurements from an MCG sensor unit are then projected into the subspace of the linear model to estimate model coefficients and restore a highres MCG image as a model instance.
With a highres MCG image thus reconstructed, it can then be analyzed to identify the location, depth, magnitude and orientation of an electric current.
As is explained above, an MCG image typically provides lowres, 2D MCG maps that do not provide enough information for directly recovering specific information of electrical currents. However, once the highres MCG image is reconstructed, the 2D position of the electric current can be localized as the maximal point of the tangential components of the highres MCG image. To improve the 2D localization accuracy, a nonlinear optimization algorithm is developed to solve the inverse problem. At the same time, the depth, magnitude and orientation of the electric current are also recovered. More specifically, the preferred algorithm alternates two steps iteratively. The first step estimates the 3D current position, and the second step reconstructs its magnitude and orientation. The 2D current location estimated from the model based restoration is used as the initialization. The present method is efficient, accurate and reliable without the need of special assumptions. For the sake of simplicity, the presently preferred system/method is illustrated as applied to a single electric current case only. It is to be understood, however, that extension of the present system/method to multiple currents is straightforward.
The present embodiment utilizes various computing devices (or data processing devices) to learn (i.e. create) a linear model from a set of highres MCG images generated by random electric currents. Sparse data (i.e. a low resolution image) received from an MCG sensor unit is then projected onto the linear model, and a high resolution image representation of the low resolution image is created there from. An example of this approach is illustrated in
With reference to
It is to be understood that the number of virtual sensors, and thus the value of P is design choice. A later experiment described below, for example, incorporates a higher number of virtual sensors to produce an even higher resolution MCG image.
Given a single electric current, a magnetic field at each sensor 13 can be computed based on the BiotSarvart Law, equation Eq. 1, where {right arrow over (J)}({right arrow over (p)})is the moment of the current including its magnitude and orientation. In this case, {right arrow over (p)} is the 3dimensional (i.e. 3D) position vector of the current. Note that this representation of electric current is an approximation by assuming the size (or magnitude) of the current is zero. One can consider that the volume (size, or density) information is included in the moment vector {right arrow over (J)}. {right arrow over (B)}({right arrow over (r)}_{m}) is the magnetic vector measured by the m_{th }sensor at position {right arrow over (r)}_{m}={right arrow over (r)}_{o}+{right arrow over (δ)}_{m}, where r_{o }is the reference point of the sensor plane and δm indicates the offset of the m_{th }sensor with respect to r_{o}. As it is known in the art, μ_{o }is the magnetic constant.
In typical MCG systems, only the z component of {right arrow over (B)} is measured.
From Eq. 1 one may compute B_{z }(the z component of {right arrow over (B)}) by means of equation Eq. 2, where J^{1},J^{2},J^{3 }represent the three components of the current moment vector {right arrow over (J)}; x_{p},y_{p},z_{p }represent the three components of the current position vector {right arrow over (p)}; and r_{m}^{1},r_{m}^{2},r_{m}^{3 }represent the three components of the sensor position vector {right arrow over (r)}_{m}.
In a training step, a set of highres P×P MCG images (where P>>M) are generated. To generate each P×P MCG image, a single electric current with both random moment and 3D position is created. The highres P×P MCG image is computed based on Eq. 2.
Some examples of training images are shown in
A received sparse M×M measurement from an MCG sensor unit defines a vector g. To restore (i.e. create) a highres MCG image given a sparse M×M measurement (vector g), one first extracts the corresponding rows from the eigenmatrix Σ to form a subeigenmatrix Σg. Similarly, vector g'"'"'s corresponding elements from mean vector μ form a submean vector μ_{g}. Vector g is then projected to subeigenmatrix Σg, and model coefficients c_{g }are calculated as c_{g}=Σ_{g}^{+}(g_{j}−ρ_{g}), where Σ_{g}^{+} is the pseudo inverse of Σg. Finally the original eigenmatrix Σ along with estimated coefficients c_{g }are used to reconstruct the highres MCG image vector h, as h=Σ·c_{g}+μ, where h is a P×P vector.
An illustration of these depth layers 41 is shown in
In the present approach, the sensor positions {right arrow over (r)}_{m}, the 2D position (x_{p}, y_{p}), and the moment J of the electric current are fixed. B_{z }is only affected by the depth z of the current. Thus, Eq. 2 can be simplified to Eq. 3, where a_{m }and b_{m }are constants but unknowns, c=20 cm is the depth of the sensor, and z is the depth of the current, which varies between 0 to 10 cm within the heart volume bounding box. Preferably, a_{m }lies in a range from −7.5 to 7.5 cm, and b_{m }lies in a range from 0 to 112.5 cm.
By applying Taylor expansion to Eq. 3, one obtains Eq. 4. By ignoring O(Δz^{3}), one only needs to prove that
is close to zero for any possible z and any sensor.
A graph of
versus depth, z, is shown in
in 64 trials with random a_{m }and b_{m }in each trial. As shown,
demonstrates a very small value (close to zero) when z varies from 0 to 10 cm. Therefore, a set of depth layers was sampled within this depth range, as is illustrated in
In the present experiments, 1000 samples were generated in each of 10 evenly distributed depth layers. The presently preferred method of creating a restored highres MCG image was then compared with the bicubic interpolation method, as well as with the actual, ground truth images.
With reference to
As is mentioned above, a 2D estimate of the electric current location can be obtained by analyzing the highres MCG image. A presently preferred method for improving the localization accuracy is to solve a nonlinear optimization that reconstructs both 3D position and moment of the electric current, i.e. the inverse problem. An accurate highres MCG image restored by the linear model provides a good initialization for the inverse problem and helps it converge on the global optimum more quickly. The preferred method for generating a 2D estimate from a highres MCG image is as follows.
Given a highres MCG image B_{z}(i,j)(i=1, 2, . . . ,N; j=1, 2, . . . ,N), the maximal point of the tangential components B′_{xy}(i,j) of B_{z}(i,j) refers to the 2D position (x_{p},y_{p}) of the electric current. This may be seen in the second row images of
The inverse problem is to solve both 3D position {right arrow over (p)} and moment {right arrow over (j)} of the electric current. This approach may be better understood with reference to
Eq. 6 is then expanded to a matrix form by using a skew symmetric matrix, which results in Eq. 7. In this case, the z component of the magnetic field can be computed as shown in Eq. 8, where R_{m}^{1},R_{m}^{2 }are x,y components of {right arrow over (R)}. Given M sensors, a linear system is defined as illustrated in equation Eq. 9, where B is a measured M×1 vector, R is a M×2 position matrix that is computed from given {right arrow over (p)} and {right arrow over (δ)}_{m}. In the present case, J is a 2×1 unknown vector. When rank(R)≧2 (this holds for the single electric current case with 64 sensors), one can solve a least square solution for J, as illustrated in equation Eq. 10.
Note that by only measuring B_{Z }it is impossible to recover J^{3}. In fact, the magnetic field generated by the z component of the current only propagates along the horizontal direction and never reaches outside of the body. For the following computation, one sets J^{3}=0. Given an estimated current moment {right arrow over (J)}=[J,0], one can update the current position {right arrow over (p)}.
Eq. 1 is rewritten as equation Eq. 11. One may then let α=4π/μ_{0 }and {right arrow over (ε)}_{0}={right arrow over (r)}_{0}−{right arrow over (p)}. {right arrow over (δ)}_{m }is known for each sensor. One may then apply equation Eq. 12 to obtain α{right arrow over (B)}^{m}. In Eq. 12, let {right arrow over (τ)}_{m}={right arrow over (J)}×{right arrow over (δ)}_{m }and {right arrow over (ε)}_{0}=(x_{ε},y_{ε},z_{ε})^{T}. It is noted that {right arrow over (τ)}_{m }can be computed given {right arrow over (J)}. Again, the cross product is removed from Eq. 12 by using a skewsymmetric matrix. Therefore for each sensor m=1:M , one obtains a nonlinear equation in terms of (x_{ε},y_{ε},z_{ε}), as illustrated in Eq. 13. Letting F=(f^{1};f^{2}; . . . ;f^{m})=0 , one then solves a least square solution of the nonlinear system F for {right arrow over (ε)}_{0}.
Once the offset {right arrow over (ε)}_{0 }is obtained, the position matrix R can be updated and J can be recomputed. These iterations are repeated until the algorithm converges. The inverse problem step converges in real time (0.5 seconds on average). Finally {right arrow over (p)}={right arrow over (r)}_{0}−{right arrow over (ε)}_{0}. Since the highres MCG image only provides an estimate for 2D current position (x_{p},y_{p}), the initial depth z and magnitude ∥{right arrow over (J)}∥ of the electric current are given by equation Eq. 14, where d is the distance between two magnetic poles in the highres MCG image.
The present highres MCG image restoration method and electric current localization algorithm was evaluated using both simulations and physical phantom setups. In both scenarios the ground truth of the 3D position {right arrow over (p)}_{g }and moment {right arrow over (J)}_{g }of the electric current are known.
The present simulation setup is similar to the setup showed in
Tables 1 to 4 in
Moreover, since the present method solves the inverse problem, the present method permits the reconstruction of the 3D position of the electric current and its moment. Applicants believe that the the present ability to reconstruct a 3D current is new to the present field.
Table 2 in
Table 3 in
In reality, an electric current is more like a voxel rather than a point. Different sizes of voxel currents were simulated by generating a set of point currents within a small cube by a 0.5 mm interval.
A real phantom experiment is shown in
In this real phantom experiment, the electric current has a physical shape and size. It can be considered as a set of small line segment currents. The present localization algorithm estimates the 2D position of the geometric center of the coil. By synchronizing the fluxgate sensor measurement with the AC generator, one can simulate an 8×8 MCG system. The output of the fluxgate sensor 57 is imported to the spectrum analyzer 59 and converted to measurements in Tesla.
The real phantom setup is totally unshielded thus the measurement noise is big, which is shown in
A couple of parameters can affect the accuracy of the highres MCG image restoration and current localization. Presently, the resolution is decreased by changing N from 351 to 141, i.e. 20 instead of 50 pixels are inserted between adjacent real (or physical) sensors, and the localization error is increased by 150%. On the other hand, when one inserts more than 50 pixels, the accuracy does not change much. The sensor number also affects the accuracy. With the same covering area (17.5×17.5 cm^{2}), the more sensors that are used in the MCG system, the better the accuracy of the present algorithm that one can achieve. For example, with the 5% white Gaussian random noise, the localization error is 0.878 mm for 8×8 sensors; 0.850 mm for 10×10 sensors; 0.837 mm for 12×12 sensors; 0.768 mm for 18×18 sensors; and 0.660 mm for 36×36 sensors. These two parameters are therefore very important for MCG system design.
Hereinabove, only the single electric current localization problem is considered, and a good initialization can be computed from the dense MCG image. In reality there can be more than one electric voxel current. Signal decomposition might be needed for initialization of the multiple current localization. In summary the present method is capable of restoring/creating accurate highres MCG images. The highres MCG images are created in an efficient, accurate and reliable manner for single current 2D localization. In addition the present algorithm can reconstruct the depth and moment of the current. It can also be easily extended to solve for multiple current sources.
While the invention has been described in conjunction with several specific embodiments, it is evident to those skilled in the art that many further alternatives, modifications and variations will be apparent in light of the foregoing description. Thus, the invention described herein is intended to embrace all such alternatives, modifications, applications and variations as may fall within the spirit and scope of the appended claims.