CN101216541B

CN101216541B - Magnetic resonance image-forming system gradient field spherical harmonic coefficient obtaining method

Info

Publication number: CN101216541B
Application number: CN2008100562041A
Authority: CN
Inventors: 代亮; 赵磊; 韦巍
Original assignee: SYMBOW MEDICAL TECHNOLOGY Co Ltd
Current assignee: SYMBOW MEDICAL TECHNOLOGY Co Ltd
Priority date: 2008-01-15
Filing date: 2008-01-15
Publication date: 2012-06-06
Anticipated expiration: 2028-01-15
Also published as: CN101216541A

Abstract

The invention relates to a method for acquiring spherical harmonic coefficient of gradient field of magnetic resonance imaging system, which comprises the following steps of : (a) constructing a water model coordinate system selecting a mark point Pc(u0, v0) closest to the image center as the original point Op, selecting the u axis direction of an MRI image as the c axis direction, selecting the v axis direction in the image as the r axis direction, and selecting the direction forming a right-handed coordinate system with the c and the r axes as the s axis direction; (b) calculating the MRI coordinates of the mark point Pi according to the water model coordinate system by following steps: (1) ignoring the rotation of the water model in the plane thereof and the translation of the original point; (2) considering the rotation by a theta angle in the plane of the water model around the center of Op and a shaft in the direction of a scan layer slice; and (3) considering the translation of the coordination original point of the water model; and (c) obtaining a linear equation group concerning spherical harmonic coefficients a and b if the magnetic field intensity parameters gamma, theta, phi and image coordinates of a certain point are known, and calculating the coefficients a and b by solving the linear equation group.

Description

Method for acquiring gradient field spherical harmonic coefficient of magnetic resonance imaging system

Technical Field

The invention relates to a method for acquiring a coefficient, in particular to a method for acquiring a gradient field spherical harmonic coefficient of a magnetic resonance imaging system.

Background

In nuclear resonance systems, the non-linearity of the gradient field tends to cause geometric distortions of the image. The geometric distortion of the image is different whether on different magnetic resonance imaging apparatuses or within different fields of view (FOV) of the same magnetic resonance imaging apparatus. Therefore, image correction techniques aiming at the problem of nonlinearity of the gradient field are necessary to improve the accuracy of magnetic resonance image navigation surgery and to judge the feasibility of image research among multiple magnetic resonances in practical applications. The basic process of the magnetic resonance image gradient deformation correction is as follows: firstly, the position offset of a limited number of control points in the space of the magnetic resonance imaging equipment is solved by a certain method, then the coordinate of each pixel of the image to be corrected in the space of the magnetic resonance imaging equipment is calculated, and the offset of the pixel space is calculated by an interpolation method according to the coordinate of each pixel; and converting the spatial offset of the pixel into an offset in an image coordinate system according to the conversion relation between the coordinate system of the magnetic resonance imaging equipment and the image coordinate system, and then compensating the offset into the image coordinate of the pixel, namely completing the gradient deformation correction process of the magnetic resonance image.

Methods for calculating the offset of the control point in the magnetic resonance imaging space can be divided into two types: the first one is to use a three-dimensional water model, which provides control points in a three-dimensional space, to analyze the control points in the magnetic resonance image of the three-dimensional water model to obtain the image deformation amount on the control points in the limited space, and to convert the offset in the image coordinate system of the pixel into a space offset according to the conversion relation between the coordinate system of the magnetic resonance imaging device and the image coordinate system; another approach is to use a descriptive function of the magnetic field, the spherical harmonic function, to calculate the offset of a finite number of control points in the imaging space. The first method relies on the three-dimensional water model, the implementation cost is much higher than that of the second method, and the position offset of the control point can be obtained by imaging and analyzing the three-dimensional water model for multiple times, so the implementation efficiency is low. Before applying the second method, i.e. the calculation of spherical harmonics, the relevant correction parameters within the spherical harmonics are first given. The researchers at Harvard university Jorge Jovicich, SilvestercCzanner have their paper, Reliablility in Multi-Site Structural MRI students: effects of Gradient Non-linear Correction on Phantom and Human Data, Neuron Image (in press) (reliability research of multipoint structural magnetic resonance imaging, effect of Gradient Non-linear Correction in water model and Human body Image, translation in neuro Image) provides a method for calculating relevant Correction parameters. The method is to calculate correction parameters for image correction according to the design parameters of the gradient coil, but the design parameters of the gradient coil are only one of the main factors of image deformation, and other factors are not taken into consideration. Therefore, there is still a certain difference between the actual calibration parameters and the design parameters.

Disclosure of Invention

In view of the above problems, an object of the present invention is to provide a method for obtaining spherical harmonic coefficients of a gradient field of a magnetic resonance imaging system, which accurately calculates correction parameters for a specific gradient coil at a limited number of points in an image space, thereby obtaining real magnetic field gradient parameters.

In order to achieve the purpose, the invention adopts the following technical scheme: a method for obtaining the spherical harmonic coefficient of gradient field in magnetic resonance imaging system includes the following steps:

(a) establishing a water model coordinate system, wherein the origin of the MRI image is V (x)₀，y₀，z₀) With the center V (x) of the MRI image closest to the center of the water-mode image_c，y_c，z_c) Is marked point P_c(u₀，v₀) Is the origin O of a water model coordinate system_p(ii) a Taking the u-axis direction in the MRI image as the c-axis direction; taking the v-axis direction in the image as the r-axis direction; the direction of a right-hand coordinate system formed by the c axis and the r axis is taken as the direction of an s axis; arbitrary mark point P_iHas a water model coordinate system coordinate of (c)_i，r_i，s_i) (ii) a According to O_pCalculating the MRI coordinates of the image to obtain

V_{o_{p}} (x, y, z) = v_{c} * u_{0} * p * v_{r} * v_{0} * p + V (x_{0}, y_{0}, z_{0}) - - - (1)

Where p is the distance between adjacent pixels. v. of_c，v_r，v_sThe unit vectors are in the directions of the c, r and s axes, respectively.

(b) Calculating MRI coordinates of the marker point, and comparing the marker point P_i(c_i，r_i，s_i) The MRI coordinate can be divided into the following steps through a water model coordinate system:

(i) irrespective of water mouldRotation in its own plane and translation of origin of coordinates, marking point P_i(c_i，r_i，s_i) The coordinates in the MRI coordinate system are

V_{p_{i}} (x, y, z) = c_{i} * d * v_{c} + r_{i} * d * v_{r} + s_{i} * d * v_{s} - - - (2)

(ii) Considering the water model itself in the plane with O_pA mark point P is rotated by an angle theta by taking the layer direction for scanning as an axis as a circle center_iThe coordinates in the MRI coordinate system need to be multiplied by the rotation matrix T_rot

T_{rot} = [\begin{matrix} t * x * x + c & t * x * y + s * z & t * x * z - s * y & 0 \\ t * x * y - s * z & t * y * y + c & t * y * z + s * x & 0 \\ t * x * z + s * y & t * y * y + c & t * z * z + c & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

c＝cos(θ)，s＝sin(θ)，t＝1-c

Wherein x, y, z are the coordinates of the layer direction vector;

(iii) considering the translation of the origin of the water model coordinates, the point P is marked_iThe coordinates in the MRI coordinate system are

Wherein, V_pi(x ', y ', z ') is the mark point P_iActual coordinates in the MRI coordinate system. The magnetic field strength can be expressed as:

B_r(n，m)(r，θ，_)＝rⁿ[a_v(n，m)cos(m_)+b_v(n，m)sin(m_)]×P_(n，m)(cosθ) (5)

magnetic field gradient function:

<math><mrow> <msub> <mi>G</mi> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>&equiv;</mo> <mfrac> <msub> <mrow> <mo>&PartialD;</mo> <mi>B</mi> </mrow> <mrow> <mi>zv</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> </mfrac> <mo>&equiv;</mo> <mfrac> <mrow> <mo>&PartialD;</mo> <msubsup> <mi>B</mi> <mrow> <mi>zv</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> <mi>L</mi> </msubsup> </mrow> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <msubsup> <mrow> <mo>&PartialD;</mo> <mi>B</mi> </mrow> <mrow> <mi>zv</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> <mi>N</mi> </msubsup> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> </mfrac> <mo>&equiv;</mo> <msubsup> <mi>G</mi> <mi>v</mi> <mi>L</mi> </msubsup> <mo>+</mo> <msubsup> <mi>G</mi> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> <mi>L</mi> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow></math>

the following equations (5), (6) and (7) conclude that:

V_x＝∑rⁿ[a_x(n，m)cos(m_)+b_x(n，m)sin(m_)]×P_(n，m)(cosθ)/a_(1，1)

V_y＝∑rⁿ[a_y(n，m)cos(m_)+b_y(n，m)sin(m_)]×P_(n，m)(cosθ)/b_(1，1) (8)

V_z＝∑rⁿ[a_z(n，m)cos(m_)+b_z(n，m)sin(m_)]×P_(n，m)(cosθ)/a_z(1，0)

wherein a is_v(n，m)、b_v(n，m)Is a constant, and a_v(n，m)、b_v(n，m)Is the coefficient of an n-th order m-order expansion term in the v direction, P_(n，m)(cos θ) is a Legendre polynomial; the left end of the formula (8) is a mark point P_iCoordinate values under the MRI coordinate system;

(c) as described above, if γ, θ, _ and MRI image coordinates of a certain point are known, a linear equation system with respect to spherical harmonic coefficients a, b is obtained, and the coefficients a, b can be found by solving the linear equation system.

And (c) gamma, theta and gamma are coordinates of the marking point in the MRI coordinate system. Can be composed of V_pi(x ', y ', z ') was obtained by the following conversion

The adjacent mark points in the step (a) are arranged at equal intervals, and the interval d of the adjacent mark points is a fixed value.

In the step (b), P is set₁(u₁，v₁) In the c direction and P direction on the water mold_c(u₀，v₀) Adjacent marker points; according to P₁(u₁，v₁) And P_c(u₀，v₀) The image of (2) is calculated, then the calculation formula of the theta angle

θ＝tg^-1((v₁-v₀)/(u₁-u₀))。

Due to the adoption of the technical scheme, the invention has the following advantages: 1. the invention reversely calculates the magnetic field gradient parameters according to the actual image, and overcomes the defect that the design parameters of the gradient coil are different from the real magnetic field gradient parameters. 2. In the process of reversely solving the gradient parameters of the magnetic field, the influence of the arrangement position of the water model on the accuracy of the algorithm is considered, and the accuracy of the algorithm is improved. 3. The invention adopts the search algorithm of the gradient field spherical harmonic parameters, greatly simplifies the steps of algorithm implementation and improves the algorithm efficiency. The method provided by the invention aims at image deformation correction caused by gradient field nonlinearity in magnetic resonance imaging application, and has important significance for improving the precision of magnetic resonance image navigation operation and feasibility analysis of image research among multiple magnetic resonances.

Drawings

FIG. 1 is a schematic view of the structure of the water mold of the present invention

FIG. 2 is a schematic cross-sectional side view of FIG. 1

FIG. 3 is a schematic diagram of a water model image obtained by scanning and a water model coordinate system established based on the water model image

Detailed Description

The present invention will be described in detail below with reference to the accompanying drawings and examples.

The method for acquiring the spherical harmonic coefficient of the gradient field of the magnetic resonance imaging system provided by the invention is characterized in that limited points are acquired according to the actual image space, magnetic resonance correction parameters, namely the spherical harmonic coefficient, are reversely solved, and only the parameters below 5 th order are calculated due to the comprehensive consideration of the calculation amount and the correction accuracy.

As shown in fig. 1 and 2, the gradient correction water model is a tool for gradient correction, which is used for both searching system parameters and measuring corrected errors. The gradient correction water mold is a square box body 1, columns 2 which are arranged into a square matrix in an equidistant square matrix mode are arranged in the box body 1, and copper sulfate solution is filled in the columns 2 and can form images in magnetic resonance imaging equipment.

The water model is placed in the effective imaging space, the center of the magnetic field is ensured to be in the water model, and the scanning plane is arranged to scan the obtained water model image, as shown in figure 3. Closest to the MRI image center V (x) within the water-mode image_c，y_c，z_c) Is marked point P_cThe water model image coordinate is (u)₀，v₀). MRI images with origin V (x)₀，y₀，z₀) Here, x, y, and z in V (x, y, z) are coordinates in the MRI coordinate system. When a water model coordinate system is established, a mark point P is used_cAs the origin O of a coordinate system_pThe u-axis direction in the MRI image is defined as the c-axis direction, the v-axis direction in the image is defined as the r-axis direction, and the direction constituting a right-hand coordinate system with the c and r-axes is defined as the s-axis direction. The original point O of the water model is obtained according to the placing mode of the water model and the scanning requirement_pDistance V (x) from the center of the image_c，y_c，z_c) Very closely, the image distortion pairs O due to gradient non-linearity can therefore be ignored_pInfluence of position, then directly according to O_pThe MRI coordinates were calculated as follows

V_{o_{p}} (x, y, z) = v_{c} * u_{0} * p + v_{r} * v_{0} * p + V (x_{0}, y_{0}, z_{0}) - - - (1)

Where p is the distance between adjacent pixels, v_c，v_r，v_sThe unit vectors are in the directions of the c, r and s axes, respectively.

And for any one of the marker points P_iSince image distortion due to the non-linearity of the gradient of the position is significant and the MRI coordinates cannot be calculated using the image coordinates any more, P is_iThe MRI coordinates of (1) need to be calculated through a water model coordinate system. Marking point P_iHas a water model coordinate system coordinate of (c)_i，r_i，s_i) And the interval between adjacent mark points is d, and the calculation can be divided into the following steps:

(i) marking point P without considering the rotation of water model in self plane and the translation of coordinate origin_iThe coordinates in the MRI coordinate system are

V_{p_{i}} (x, y, z) = c_{i} * d * v_{c} + r_{i} * d * v_{r} + s_{i} * d * v_{s} - - - (2)

(ii) Considering the water model itself in the plane with O_pAs the center of circle, rotate by theta angle with the slice direction of the scanning layer as the axis, mark point P_iThe coordinates in the MRI coordinate system need to be multiplied by the rotation matrix T_rot，

T_{rot} = [\begin{matrix} t * x * x + c & t * x * y + s * z & t * x * z - s * y & 0 \\ t * x * y + s * z & t * y * y + c & t * y * z + s * x & 0 \\ t * x * z + s * y & t * y * z - s * x & t * z * z + c & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

c＝cos(θ)，s＝sin(θ)，t＝1-c

Wherein x, y and z are coordinates of the slice direction vector; let P₁(u₁，v₁) In the c direction and P direction on the water mold_c(u₀，v₀) Adjacent marker points; according to P₁(u₁，v₁) And P_c(u₀，v₀) The image of (2) is calculated, then the calculation formula of the theta angle

θ＝tg^-1((v₁-v₀)/(u₁-u₀))；

V_pi(x ', y ', z ') is the mark point P_iActual coordinates in the MRI coordinate system.

The formula (5), (6) and (7) are

Magnetic field intensity

B_r(n，m)(r，θ，_)＝rⁿ[a_v(n，m)cos(m_)+b_v(n，m)sin(m_)]×P_(n，m)(cosθ)(5)

Magnetic field gradient function

The following equations (5), (6) and (7) can be derived

V_x＝∑rⁿ[a_x(n，m)cos(m_)+b_z(n，m)sin(m_)]×P_(n，m)(cosθ)/a_x(1，1)

V_y＝∑rⁿ[a_y(n，m)cos(m_)+b_y(n，m)sin(m_)]×P_(n，m)(cosθ)/b_y(1，1) (8)

V_z＝∑rⁿ[a_z(n，m)cos(m_)+b_z(n，m)sin(m_)]×P_(n，m)(cosθ)/a_z(1，0)

The left end of the formula (8) is a coordinate value of a certain imaging point after deformation in the MRI coordinate system, and can be obtained by calculating the image coordinate of the imaging point. Wherein a is_v(n，m)、b_v(n，m)Is a constant number a_v(n，m)、b_v(n，m)Is the coefficient of the n-order m-order expansion term in the v direction and is the inherent characteristic of the nonlinear gradient of the magnetic field. P_(n，m)(cos θ) is a Legendre polynomial. γ, θ, _ at the right end are coordinates of the point in the MRI coordinate system (spherical coordinate system). Can be composed of V_pi(x ', y ', z ') is obtained by conversion of the formula (9).

As described above, when γ, θ, _ and image coordinates of a certain point are known, a linear equation system for a and b is obtained, and coefficients a and b can be obtained by solving the linear equation system. Can then pass through the mark point P_iCalculates the MRI coordinate V after the point deformation_pi(x，y，z)。

V_{p_{i}} (x, y, z) = u * v_{c} * p + v * v_{r} * p + V (x_{0}, y_{0}, z_{0}) - - - (10)

For the comprehensive consideration of the calculation amount and the correction accuracy, only the parameter calculation below 5 th order is performed. In spherical harmonics, the parameter a_v(n，m)，b_v(n，m)Below order 5, only some of the values are non-zero, zero terms may be disregarded in the calculation, and the non-zero terms are listed in table 1:

TABLE 1

X	a	b
			(n＝1，m＝1)	1	0
(n＝3，m＝1)	10^-4	0
			(n＝5，m＝1)	10^-7	0
Y	a	b
			(n＝1，m＝1)	0	1
(n＝3，m＝1)	0	10^-4
			(n＝5，m＝1)	0	10^-7
Z	a	b
			(n＝1，m＝0)	1	0
(n＝3，m＝0)	10^-4	0
			(n＝5，m＝0)	10^-7	0

Note: wherein, the non-zero values of a and b are estimated values.

To calculate a_x、b_xFor example, equation (8) can be simplified as:

(11)

let lagrange's function

Equation (11) can be written in the form of a matrix multiplication:

gamma, theta, gamma, theta, gamma_piSubstituting (x, y, z) into equation (12) can obtain a linear equation system,

by solving the system of linear equations, the following can be calculated:

[\begin{matrix} a_{x (1,1)} \\ a_{x (3,1)} \\ a_{x (5,1)} \end{matrix}] .

since the y and z parameters are calculated in a similar manner, the detailed description is omitted here.

Claims

1. A method for acquiring a gradient field spherical harmonic coefficient of a magnetic resonance imaging system comprises the following steps:

(a) placing a gradient correction water model in an effective imaging space of a magnetic resonance imaging device, ensuring that the center of a magnetic field is in the gradient correction water model, and setting a scanning plane to scan to obtain a water model image;

(b) establishing a water model coordinate system, wherein the origin of the MRI image is V (x)₀，y₀，z₀) With the center V (x) of the MRI image closest to the center of the water-mode image_c，y_c，z_c) Is marked point P_c(u₀，v₀) Is the origin O of a water model coordinate system_p(ii) a Taking the u-axis direction in the MRI image as the c-axis direction; taking the v-axis direction in the image as the r-axis direction; the direction of a right-hand coordinate system formed by the c axis and the r axis is taken as the direction of an s axis; arbitrary mark point P_iHas a water model coordinate system coordinate of (c)_i，r_i，s_i) (ii) a According to O_pCalculating the MRI coordinates of the image to obtain

V_{o_{p}} (x, y, z) = v_{c} * u_{0} * p + v_{r} * v_{0} * p + V (x_{0}, y_{0}, z_{0}) - - - (1)

Where p is the distance between adjacent pixels, v_c，v_r，v_sUnit vectors in the directions of the c, r and s axes, u₀、v₀Is a mark point P_cThe coordinates of (a);

(c) calculating MRI coordinates of the marker point, and comparing the marker point P_i(c_i，r_i，s_i) The MRI coordinate can be divided into the following steps through a water model coordinate system:

(i) irrespective of whether the water mould is in placeRotation in the body plane and translation of the origin of coordinates, the index point P_i(c_i，r_i，s_i) The coordinates in the MRI coordinate system are

V_{p_{i}} (x, y, z) = c_{i} * d * v_{c} + r_{i} * d * v_{r} + s_{i} * d * v_{s} - - - (2)

Where p is the distance between adjacent pixels, v_c，v_r，v_sUnit vectors in the directions of c, r and s axes, d is the interval between adjacent mark points, c_i、r_i、s_iIs a mark point P_iThe coordinates of the water model coordinate system;

T_{rot} = [\begin{matrix} t * x * x + c & t * x * y + s * z & t * x * z - s * y & 0 \\ t * x * y - s * z & t * y * y + c & t * y * z + s * x & 0 \\ t * x * z + s * y & t * y * z - s * x & t * z * z + c & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

c＝cos(θ)，s＝sin(θ)，t＝1-c

Wherein x, y, z are the coordinates of the layer direction vector;

Wherein,

is the mark point P_iActual coordinates in the MRI coordinate system;

magnetic field intensity

Magnetic field gradient function

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The following equations (5), (6) and (7) can be derived

(d) if the magnetic field intensity parameters gamma, theta and theta at a certain point are known,

And MRI image coordinates, and substituting them into equation (8) can obtain a linear equation set regarding spherical harmonic coefficients a, b, and the coefficients a, b can be obtained by solving the linear equation set.

2. The method for acquiring the spherical harmonic coefficients of the gradient field of the magnetic resonance imaging system as set forth in claim 1, wherein: gamma, theta, or a mixture thereof in the step (c),

Is the coordinate of the marking point in the MRI coordinate system, which can be expressed by

Obtained by the following conversion

3. A method of acquiring spherical harmonic coefficients of a gradient field of a magnetic resonance imaging system as claimed in claim 1 or 2, characterized by: the adjacent mark points in the step (a) are arranged at equal intervals, and the interval d of the adjacent mark points is a fixed value.

4. A method of acquiring spherical harmonic coefficients of a gradient field of a magnetic resonance imaging system as claimed in claim 1 or 2, wherein: in the step (b), P is set₁(u₁，v₁) In the c direction and P direction on the water mold_c(u₀，v₀) Adjacent marker points; according to P₁(u₁，v₁) And P_c(u₀，v₀) The image of (2) is calculated, then the calculation formula of the theta angle

θ＝tg^-1((v₁-v₀)/(u₁-u₀))。

5. A method of acquiring spherical harmonic coefficients of a gradient field of a magnetic resonance imaging system as defined in claim 3, wherein: in the step (b), P is set₁(u₁，v₁) In the c direction and P direction on the water mold_c(u₀，v₀) Adjacent marker points; according to P₁(u₁，v₁) And P_c(u₀，v₀) The image of (2) is calculated, then the calculation formula of the theta angle

θ＝tg^-1((v₁-v₀)/(u₁-u₀))。