CN114708350A - Conductivity visualization method for electrical impedance tomography of brain - Google Patents

Conductivity visualization method for electrical impedance tomography of brain Download PDF

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CN114708350A
CN114708350A CN202210384135.7A CN202210384135A CN114708350A CN 114708350 A CN114708350 A CN 114708350A CN 202210384135 A CN202210384135 A CN 202210384135A CN 114708350 A CN114708350 A CN 114708350A
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施艳艳
娄亚君
王萌
张宇航
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Abstract

The invention discloses a conductivity visualization method for brain electrical impedance tomography, which combines a Tikhonov regularization method with a denoising algorithm to realize brain imaging. And obtaining a relative boundary voltage measured value and a sensitivity matrix required by reconstruction according to the field to be measured, solving by using a Tikhonov regularization method to obtain an initial conductivity variation, and further combining a denoising algorithm to obtain an optimized conductivity variation. The specific process is as follows: designing a minimization objective function; setting initialization parameters; updating the value of the solution with an alternating minimization method; judging whether the parameters meet the limiting conditions; judging whether the iteration is finished; and performing image reconstruction according to the finally obtained conductivity variation and the coordinate position information. The method provided by the invention can effectively improve the spatial resolution of the reconstructed image, further improve the imaging quality, has wide applicability and has great application potential in brain electrical impedance tomography.

Description

Conductivity visualization method for electrical impedance tomography of brain
Technical Field
The invention belongs to the technical field of electrical tomography, and particularly relates to a conductivity visualization method for electrical impedance tomography of a brain.
Background
Medical imaging is a key component of clinical procedures and offers the possibility of diagnosing, monitoring and treating human diseases. Currently, many medical imaging techniques have been developed, among which Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) are commonly used for clinical diagnosis. However, CT is radioactive and MRI is expensive. In recent years, Electrical Impedance Tomography (EIT) has attracted considerable research interest to scholars. Compared with CT and MRI, EIT has the advantages of portability, no wound and low cost, and is more suitable for bedside monitoring of patients. Due to its many advantages, EIT has shown great potential in medical applications such as breast cancer detection, breast tomography and brain imaging.
Intracranial hemorrhage, a devastating disease of the brain, is known for its high mortality and poor prognosis. Accurate imaging examinations are of critical importance in clinical treatment in order to reduce patient mortality and improve prognostic outcome. It is well known that changes in pathological tissue lead to changes in the conductivity distribution, which provides the possibility of monitoring cerebral hemorrhage using EIT techniques. In brain imaging based on EIT techniques, a safe current is injected into a pair of opposing electrodes attached to the scalp, and the boundary voltage between the remaining pair of electrodes is measured to restore the conductivity distribution within the examination area. However, the inverse problem of electrical impedance tomography is severely ill-conditioned, leading to poor reconstruction quality. Especially in brain imaging, reconstructing the conductivity distribution becomes more difficult due to the low conductivity of the skull. To solve this problem, the regularization method has proven to be an effective method to stabilize the solution of the inverse problem by adding regularization terms to the objective function. For example, Haoding Li et al, 2017, published in Physiological measurements (Physiological Measurement) volume 38, 1776, 1790, entitled "Using dynamic brain EIT to reveal the development of intracranial lesions" -evaluation of the current reconstruction algorithms "(unseiling the depth of intracranial input using dynamic brain EIT: an evaluation of current reconstruction algorithms). Among various regularization-based reconstruction methods, the Tikhonov method is simple, stable and rapid, and is widely applied. However, the image reconstructed by the Tikhonov method has low definition, which may result in inaccurate information of the reconstructed target object. Some methods are currently combined with Tikhonov methods to improve the image reconstruction quality, such as f.margotti, published in 2016 in "Inverse Problem" (Inverse distribution) volume 32, document No. 125012, entitled Mixed gradient-Tikhonov method for solving nonlinear ill-posed problems in the Banach space (Mixed gradient-Tikhonov methods for solving nonlinear Problem-placed schemes in the Banach spaces), and so on. However, the above-mentioned Tikhonov combination method mainly reconstructs the conductivity distribution around the field having a simple structure and regular boundaries, and a target region of such a complex structure of the head is less studied.
In order to improve the spatial resolution of a reconstructed image, particularly the quality of brain imaging, the invention provides a conductivity visualization method for electrical impedance tomography of a brain.
Disclosure of Invention
The invention provides a conductivity visualization method for electrical impedance tomography of a brain, which improves the spatial resolution of a reconstructed image by combining a Tikhonov regularization method and a denoising algorithm, thereby improving the imaging quality.
The invention adopts the following technical scheme for solving the technical problems, and the electrical conductivity visualization method for the electrical impedance tomography of the brain is characterized by comprising the following specific steps of:
step S1, completing standard model construction on a computer according to the shape information of the brain target field and the conductivity information in the field;
step S2, obtaining a relative boundary voltage measurement value b and a sensitivity matrix a required by respective reconstruction for the four typical models, respectively, the specific process is as follows:
step S201, uniformly distributing 16 electrodes at equal intervals outside a measured field, sequentially collecting boundary voltages under cyclic excitation cyclic measurement by adopting a mode that relative current excites adjacent voltage measurement and excitation electrodes do not measure, and obtaining 192 measurement values in total, wherein a relative boundary voltage measurement value b is the difference between a blank field boundary voltage measurement value b0 without inclusion and a boundary voltage measurement value b1 with an object field containing inclusion, namely the relative boundary voltage measurement value b is b1-b 0;
step S202, a sensitivity matrix A is obtained by combining sensitivity theory calculation according to the measured value of the void field boundary voltage without inclusion, and the calculation formula is as follows:
Figure BDA0003593022670000021
wherein A isijFor the sensitivity coefficient of the jth electrode pair to the ith electrode pair, phii、φjI and j electrode pairs respectively at excitation currenti、IjTime-field domain potential distribution; sNRepresenting a measurement field;
Figure BDA0003593022670000025
is a gradient operator;
step S3, regarding the electrical tomography as a linear ill-posed problem b ≈ A · g, constructing a Tikhonov regularization algorithm target function:
Figure BDA0003593022670000022
the solution is as follows: g ═ aTA+λI)-1ATb, wherein lambda is a regularization parameter, g is a conductivity variation, and I is an identity matrix;
in step S4, the minimization objective function is designed as:
Figure BDA0003593022670000023
wherein xi iseIs an auxiliary variable, S is a sick matrix,
Figure BDA0003593022670000024
the change in conductivity, g, as determined by the Tikhonov regularization algorithm in step S3*In order to optimize the amount of change in conductivity,
Figure BDA0003593022670000031
as fidelity term, h (g)*) Is a prior term;
step S5, introducing a pseudo inverse matrix into the minimization objective function
Figure BDA00035930226700000318
The transformation is:
Figure BDA0003593022670000032
wherein the content of the first and second substances,
Figure BDA0003593022670000033
is a half norm;
step S6, converting the unconstrained problem in the minimized objective function into a constrained problem minimization form:
Figure BDA0003593022670000034
will restrict the condition
Figure BDA0003593022670000035
Is replaced by
Figure BDA0003593022670000036
And will be
Figure BDA0003593022670000037
Is replaced by
Figure BDA0003593022670000038
Then the minimization of the objective function constraint problem is of the form:
Figure BDA0003593022670000039
wherein δ is a design parameter;
step S7, solving the minimization objective function by an alternative minimization method to obtain:
Figure BDA00035930226700000310
step S701, solving
Figure BDA00035930226700000311
Is subjected to a closed solution to obtain
Figure BDA00035930226700000312
Wherein InIs an identity matrix, and k is the iteration number;
step S702, adopting fast Fourier transform to solve
Figure BDA00035930226700000313
To obtain
Figure BDA00035930226700000314
Wherein
Figure BDA00035930226700000315
Omega is an adjusting factor, and F {. is a fast Fourier transform operator;
in step S8, parameters are selected, and the design parameter δ and the adjustment factor ω should satisfy the following constraint:
Figure BDA00035930226700000316
the fast fourier transform can be adopted to obtain the following conditions:
Figure BDA00035930226700000317
step S9, solving the optimizedChange of conductivity g*The iterative solution process is as follows:
step S901, setting initialization parameters: the initial iteration number k is 0, and the maximum iteration number kmaxInitial amount of change in conductivity
Figure BDA0003593022670000041
Designing a parameter delta, an adjusting factor omega, and an adjusting factor step length delta omega;
step S902, update
Figure BDA0003593022670000042
Step S903, update
Figure BDA0003593022670000043
Step S904, judging whether the delta and the omega meet the limiting conditions in the step S8 and k is larger than 1, if so, carrying out the next operation; if not, restarting the iteration process: k is 0, ω is ω + Δ ω, and the values are updated according to step S902 and step S903, respectively
Figure BDA0003593022670000044
And
Figure BDA0003593022670000045
step S905, judging whether the iteration satisfies an iteration termination condition k not more than kmaxIf yes, the iteration is terminated; if not, setting k to k +1, returning to the step S902, and continuing to iteratively solve;
step S906, updating the conductivity change amount
Figure BDA0003593022670000046
And performing image reconstruction on the updated conductivity variation according to the coordinate position information.
Compared with the prior art, the invention has the following beneficial effects: the invention provides a conductivity visualization method for brain electrical impedance tomography, which comprises the steps of firstly calculating to obtain initial conductivity variable quantity through a Tikhonov regularization method, further obtaining optimized conductivity variable quantity by combining a denoising algorithm, and finally realizing image reconstruction of a brain region. The method can effectively improve the spatial resolution of the reconstructed image, further improve the imaging quality, and has great application potential in brain medical imaging.
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Fig. 1 is a block flow diagram of a conductivity visualization method for electrical impedance tomography of a brain according to the present invention;
FIG. 2 is a diagram of the single section measured field, electrode distribution, excitation current and measured voltage modes of the skull model of the present invention, in which: 1-field to be measured, 2-measuring voltage, 3-electrode, 4-exciting current;
FIG. 3 is a graph showing the image reconstruction results of the Tikhonov method and the method of the present invention under the noise-free condition for four models;
FIG. 4 is a schematic diagram of image reconstruction results of the Tikhonov method and the method of the present invention under the noisy condition for four models;
FIG. 5 is a graph of Blur Radius (BR) for four model reconstruction results, where: (a) is the BR value in the noise-free condition; (b) is the BR value in noisy conditions.
Detailed Description
The method for visualizing the conductivity for electrical impedance tomography of the brain provided by the invention is described in detail with reference to the accompanying drawings and embodiments.
The conductivity visualization method for the electrical impedance tomography of the brain aims at improving the reconstruction quality of the brain imaging, aims at solving the problems of image artifact reconstruction, unclear image background and the like, and improves the spatial resolution of the reconstructed image by combining a Tikhonov regularization method with a denoising algorithm, thereby effectively improving the quality of the reconstructed image.
As shown in fig. 1, the conductivity visualization method for electrical impedance tomography of the brain provided by the invention comprises the following specific steps:
step S1, completing standard model construction on a computer according to the shape information of the brain target field and the conductivity information in the field;
step S2, obtaining a relative boundary voltage measurement value b and a sensitivity matrix a required for respective reconstruction for the four typical models, respectively, the specific process is as follows:
step S201, uniformly distributing 16 electrodes at equal intervals outside a measured field, sequentially collecting boundary voltages under cyclic excitation cyclic measurement by adopting a mode that relative current excites adjacent voltage measurement and excitation electrodes do not measure, and obtaining 192 measurement values in total, wherein a relative boundary voltage measurement value b is the difference between a blank field boundary voltage measurement value b0 without inclusion and a boundary voltage measurement value b1 with an object field containing inclusion, namely the relative boundary voltage measurement value b is b1-b 0;
step S202, a sensitivity matrix A is obtained by combining sensitivity theory calculation according to the measured value of the void field boundary voltage without inclusion, and the calculation formula is as follows:
Figure BDA0003593022670000051
wherein A isijFor the sensitivity coefficient of the jth electrode pair to the ith electrode pair, phii、φjI and j electrode pairs respectively at excitation currenti、IjTime-field domain potential distribution; sNRepresenting a measurement field;
Figure BDA0003593022670000059
is a gradient operator;
step S3, regarding the electrical tomography as a linear ill-posed problem b ≈ A · g, constructing a Tikhonov regularization algorithm target function:
Figure BDA0003593022670000052
the solution is as follows: g ═ aTA+λI)-1ATb, wherein lambda is a regularization parameter, g is a conductivity variation, and I is an identity matrix;
in step S4, the minimization objective function is designed as:
Figure BDA0003593022670000053
wherein ξeIs an auxiliary variable, S is a sick matrix,
Figure BDA0003593022670000054
the change in conductivity, g, as determined by the Tikhonov regularization algorithm in step S3*In order to optimize the amount of change in conductivity,
Figure BDA0003593022670000055
as fidelity term, h (g)*) Is a prior term;
step S5, introducing a pseudo-inverse matrix into the minimized objective function
Figure BDA0003593022670000056
The transformation is:
Figure BDA0003593022670000057
wherein the content of the first and second substances,
Figure BDA0003593022670000058
is a half norm;
step S6, converting the unconstrained problem in the minimized objective function into a constrained problem minimization form:
Figure BDA0003593022670000061
will restrict the condition
Figure BDA0003593022670000062
Is replaced by
Figure BDA0003593022670000063
And will be
Figure BDA0003593022670000064
Is replaced by
Figure BDA0003593022670000065
Then the minimization of the objective function constraint problem is of the form:
Figure BDA0003593022670000066
wherein δ is a design parameter;
step S7, solving the minimization objective function by an alternative minimization method to obtain:
Figure BDA0003593022670000067
step S701, solving
Figure BDA0003593022670000068
Is subjected to a closed solution to obtain
Figure BDA0003593022670000069
Wherein InIs an identity matrix, and k is the iteration number;
step S702, adopting fast Fourier transform to solve
Figure BDA00035930226700000610
To obtain
Figure BDA00035930226700000611
Wherein
Figure BDA00035930226700000612
Omega is a regulating factor, F {. cndot } is a fast Fourier transform operator;
in step S8, the parameters are selected, and the design parameter δ and the adjustment factor ω satisfy the following constraints:
Figure BDA00035930226700000613
the fast fourier transform can be adopted to obtain the following conditions:
Figure BDA00035930226700000614
step S9, solving the optimized conductivity variation g*The iterative solution process is as follows:
step S901, setting initialization parameters: the initial iteration number k is 0, and the maximum iteration number kmaxInitial amount of change in conductivity
Figure BDA00035930226700000615
Designing a parameter delta, an adjustment factor omega, and an adjustment factor step length delta omega;
step S902, update
Figure BDA00035930226700000616
Step S903, update
Figure BDA00035930226700000617
Step S904, determining whether δ, ω satisfy the constraint condition in step S8 and k is greater than 1, if yes, performing the next operation; if not, restarting the iteration process: k is 0, ω is ω + Δ ω, and the values are updated according to step S902 and step S903, respectively
Figure BDA0003593022670000071
And
Figure BDA0003593022670000072
step S905, judging whether the iteration satisfies an iteration termination condition k not more than kmaxIf yes, the iteration is terminated; if not, setting k to k +1, returning to the step S902, and continuing to iteratively solve;
step S906, updating the conductivity change amount
Figure BDA0003593022670000073
And performing image reconstruction on the updated conductivity variation according to the coordinate position information.
As shown in FIG. 2, the single-section measured field 1, the excitation current 4, the mode of the measurement voltage 2 and the distribution of the electrodes 3 of the skull model are shown, and 16 electrodes 3 are uniformly distributed outside the field.
Four typical skull models are selected as an embodiment, the real distribution of inclusion objects in a field is shown in the first row of fig. 3, the embodiment uses COMSOL Multiphysics and MATLAB R2018a to carry out simulation modeling, the simulation parameter setting refers to real human tissue parameters, the brain background conductivity is set to be 0.15S/m of the human brain cerebrospinal fluid conductivity, and the conductivity of the bleeding object inclusion is set to be 0.8S/m. Fig. 3 shows a reconstructed image obtained by the second behavior of the Tikhonov method, and it can be seen that there are many artifacts in the background and the reconstructed inclusions are blurred. Especially for model B, the reconstruction quality is relatively poor, since the sensitivity at the field center is much lower than at the field boundaries. In contrast, the third row of fig. 3 is a reconstructed image obtained by the method of the present invention, and for the four models A, B, C, D, the inclusion can be well reconstructed and there is almost no artifact in the background. The result shows that the method provided by the invention can effectively improve the image quality of brain imaging.
Fig. 4 is a schematic diagram of image reconstruction results of the Tikhonov method and the method of the present invention in the case of the four models in the presence of system noise. Under the influence of noise, inclusions reconstructed using the Tikhonov method are further deformed and produce more pronounced artifacts in the background. Compared with the situation without noise, the reconstructed image of the method provided by the invention is hardly influenced by noise, the impurities can be well reconstructed, and the background of the image is clear. The result shows that the method provided by the invention has good anti-noise performance. Generally speaking, the method provides an alternative method for brain imaging, and can effectively improve the image quality of the brain imaging and improve the spatial resolution of the reconstructed image.
As shown in fig. 5, the fuzzy Radius (BR) of the results of the Tikhonov method and the method of the present invention are reconstructed for the four models. Wherein (a) is a BR value in a noise-free condition, and (b) is a BR value in a noise condition. The expression is shown as the following formula, and the smaller the blur radius value of the reconstructed image is, the better the quality of the reconstructed image is.
Figure BDA0003593022670000074
In the formula, ASIs the area of the target region, A0Is the area of the entire field.
The method provided by the invention has the advantages that the fuzzy radius of the reconstructed image is far smaller than that of the reconstructed image by the Tikhonov algorithm, and the method further proves the superiority of the method and can effectively improve the quality of the reconstructed image. Good performance can be exhibited even under noisy conditions.
The above description is only exemplary of the present invention and should not be taken as limiting the invention, as any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (1)

1. A conductivity visualization method for electrical impedance tomography of a brain is characterized by comprising the following specific steps:
step S1, completing standard model construction on a computer according to the shape information of the brain target field and the conductivity information in the field;
step S2, obtaining a relative boundary voltage measurement value b and a sensitivity matrix a required by respective reconstruction for the four typical models, respectively, the specific process is as follows:
step S201, uniformly distributing 16 electrodes at equal intervals outside a measured field, sequentially collecting boundary voltages under cyclic excitation cyclic measurement by adopting a mode that relative current excites adjacent voltages to measure and exciting the electrodes not to measure, and obtaining 192 measurement values in total, wherein a relative boundary voltage measurement value b is the difference between a blank field boundary voltage measurement value b0 without inclusions and a boundary voltage measurement value b1 with object fields containing inclusions, namely the relative boundary voltage measurement value b is b1-b 0;
step S202, the sensitivity matrix A is calculated according to the voltage measurement value of the void field boundary without inclusion and in combination with the sensitivity theoryThe calculation formula is as follows:
Figure FDA0003593022660000011
wherein A isijFor the sensitivity coefficient of the jth electrode pair to the ith electrode pair, phii、φjI and j electrode pairs respectively at excitation currenti、IjTime-field domain potential distribution; s isNRepresenting a measurement field;
Figure FDA0003593022660000012
is a gradient operator;
step S3, regarding the electrical tomography as a linear ill-posed problem b ≈ A · g, constructing a Tikhonov regularization algorithm target function:
Figure FDA0003593022660000013
the solution is as follows: g ═ aTA+λI)-1ATb, wherein lambda is a regularization parameter, g is a conductivity variation, and I is an identity matrix;
in step S4, the minimization objective function is designed as:
Figure FDA0003593022660000014
wherein ξeIs an auxiliary variable, S is a sick matrix,
Figure FDA0003593022660000015
the change in conductivity, g, as determined by the Tikhonov regularization algorithm in step S3*In order to optimize the amount of change in conductivity,
Figure FDA0003593022660000016
as fidelity term, h (g)*) Is a prior term;
step S5, introducing a pseudo inverse matrix into the minimization objective function
Figure FDA0003593022660000017
The transformation is:
Figure FDA0003593022660000018
wherein the content of the first and second substances,
Figure FDA0003593022660000019
Figure FDA00035930226600000110
is a half norm;
step S6, converting the unconstrained problem in the minimized objective function into a constrained problem minimization form:
Figure FDA00035930226600000111
will restrict the condition
Figure FDA0003593022660000021
Is replaced by
Figure FDA0003593022660000022
And will be
Figure FDA0003593022660000023
Is replaced by
Figure FDA0003593022660000024
Then the minimization of the objective function constraint problem is of the form:
Figure FDA0003593022660000025
wherein δ is a design parameter;
step S7, solving the minimization objective function by an alternative minimization method to obtain:
Figure FDA0003593022660000026
step S701, solving
Figure FDA0003593022660000027
Is subjected to a closed solution to obtain
Figure FDA0003593022660000028
Wherein InIs an identity matrix, and k is the iteration number;
step S702, adopting fast Fourier transform to solve
Figure FDA0003593022660000029
To obtain
Figure FDA00035930226600000210
Wherein
Figure FDA00035930226600000211
Omega is an adjusting factor, and F {. is a fast Fourier transform operator;
in step S8, parameters are selected, and the design parameter δ and the adjustment factor ω should satisfy the following constraint:
Figure FDA00035930226600000212
the fast fourier transform can be adopted to obtain the following conditions:
Figure FDA00035930226600000213
step S9, solving the optimized conductivity variation g*The iterative solution process is as follows:
step S901, setting initialization parameters: first stageThe initial iteration number k is 0, and the maximum iteration number kmaxInitial amount of change in conductivity
Figure FDA00035930226600000214
Designing a parameter delta, an adjusting factor omega, and an adjusting factor step length delta omega;
step S902, update
Figure FDA00035930226600000215
Step S903, update
Figure FDA00035930226600000216
Step S904, judging whether the delta and the omega meet the limiting conditions in the step S8 and k is larger than 1, if so, carrying out the next operation; if not, restarting the iteration process: k is 0, ω is ω + Δ ω, and the values are updated according to step S902 and step S903, respectively
Figure FDA00035930226600000217
And
Figure FDA00035930226600000218
step S905, judging whether iteration satisfies an iteration termination condition k ≤ kmaxIf yes, the iteration is terminated; if not, setting k to k +1, returning to the step S902, and continuing to iteratively solve;
step S906 of updating the conductivity change amount
Figure FDA0003593022660000031
And performing image reconstruction on the updated conductivity variation according to the coordinate position information.
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Publication number Priority date Publication date Assignee Title
CN115436427A (en) * 2022-08-22 2022-12-06 电子科技大学 Eddy current thermal imaging defect reconstruction method based on Tikhonov regularization
CN115444392A (en) * 2022-08-31 2022-12-09 河南师范大学 Nonlinear stroke analysis method based on electrical impedance tomography

Cited By (4)

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Publication number Priority date Publication date Assignee Title
CN115436427A (en) * 2022-08-22 2022-12-06 电子科技大学 Eddy current thermal imaging defect reconstruction method based on Tikhonov regularization
CN115436427B (en) * 2022-08-22 2023-10-17 电子科技大学 Eddy current thermal imaging defect reconstruction method based on Tikhonov regularization
CN115444392A (en) * 2022-08-31 2022-12-09 河南师范大学 Nonlinear stroke analysis method based on electrical impedance tomography
CN115444392B (en) * 2022-08-31 2024-05-14 河南师范大学 Nonlinear cerebral apoplexy analysis method based on electrical impedance tomography

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