Electrical impedance tomography (EIT) is a promising visualization technique with low hardware cost, which reconstructs the conductivity change inside a domain from the voltage change on the boundary . EIT has the potential to evaluate the effect of electrical muscle stimulation (EMS) on human muscles [2,3], where EMS is being expected to replace physical exercises of human body in the future. However, the EIT image has low accuracy , which restricts the accurate reconstruction of muscle compartments and the quantitative evaluation of EMS effect [5,6]. To increase the reliability of quantitative evaluation of EMS, the reconstruction accuracy of muscle compartments needs to be improved.
EIT reconstructs the image by matching the measured voltage change ΔU with an approximated voltage change u(Δσ) resulting from the conductivity change Δσ, in which the approximation error e between ΔU and u(Δσ) causes the low accuracy of image reconstruction. Generally, u(Δσ) is formulated as the product of a sensitivity matrix and the conductivity change based on a linear model . With conductivity σ changing from the background-field σb to the object-field σo = σb+Δσ, the voltage U changes from Ub to Uo = Ub+ΔU accordingly, and the nonlinear function U = f(σ) mapping σ to U changes from fb(σb) to fo(σo). Replacing operator fo with fb and using the Taylor formula on fb(σo) the result u(Δσ) is formulated as JbΔσ , where Jb is the Jacobian matrix of fb(σb). For simplicity, Jb is replaced by the sensitivity matrix Sb, which is calculated from a linearized form of fb(σb) . As e between ΔU and u(Δσ) is non-negligible, the reconstructed conductivity change Δσ* has low accuracy, which influences the further quantitative evaluation .
To reduce e between ΔU and u(Δσ), two nonlinear models were modified from the linear model to approximate u(Δσ) by optimizing Sb. The first nonlinear model is the so-called “sensitivity updating model” , in which Sb is updated. In detail, a conductivity σb* as the new background-field is used to calculate the new sensitivity matrix Sb*. The second nonlinear model is the so-called “second-order sensitivity model” , in which the Hessian matrix of fb(σb) is estimated as Sb† to compensate for Sb. In detail, the mth row of Sb† is represented by the pivots of [Sb]mT[Sb]m, where [Sb]m is the mth row of Sb. However, the reduction of e by optimizing Sb in the two nonlinear models is insufficient. Analytically, with σ changing from σb to σo, the operator f changes from fb to fo accordingly. The change of f has an inevitable influence on u(Δσ), in which the influence reflects on σb and Δσ simultaneously. To eliminate e thoroughly, it is necessary to consider the influence of change of f on u(Δσ) completely.
Under these circumstances, a new voltage approximation model for conductivity reconstruction is proposed, which is called the “object-oriented sensitivity matrix estimation model (OO-SME model)”. The OO-SME model is derived by linearizing U = f(σ) as the product of a sensitivity matrix S and the conductivity σ. Thus, two linear equations are formulated simultaneously on σb as Ub = Sbσb and on σo as Uo = Soσo, where Sb and So are the sensitivity matrices corresponding to σb and σo respectively. Therefore, u(Δσ) resulting from Δσ is related to σb, Δσ, Sb, and ΔS in the OO-SME model, where ΔS is the sensitivity matrix change from Sb to So. In the OO-SME model, e between ΔU and u(Δσ) is eliminated, thus, an image with higher accuracy can be reconstructed.
The objectives of this study are (1) to propose the OO-SME model for conductivity reconstruction with high accuracy, (2) to reconstruct the lean meat in meat sample accurately as a mimic reconstruction of muscle compartment, and (3) to evaluate the mass of lean meat quantitatively from the reconstruction.
Conductivity reconstruction with OO-SME model
OO-SME model for voltage change approximation based on object-oriented sensitivity matrix estimation
The approximation error e between approximated voltage change u(Δσ) and measured voltage change ΔU is (1),
where ΔU is the measured voltage change from background-field with conductivity σb to object-field with conductivity σo = σb+Δσ, u(Δσ) is the approximated voltage change resulting from the conductivity change Δσ. In the OO-SME model, u(Δσ) is approximated as (2),
where Sb is the sensitivity matrix calculated from σb, ΔS is the sensitivity matrix change from Sb to So, where So is the sensitivity matrix calculated from σo. Sb and So are calculated by (3a) and (3b),
where e is the element index, Ωe is the pixel of eth element, φm-c(•) and φm-v(•) are the field potentials relevant to mth electrode-combination, m-c and m-v represent the current stimulating with current-stimulation pair and voltage-measurement pair respectively . I is the current amplitude. ΔS is calculated from Sb and So by (4),
and used to optimize u(Δσ) by (2). In practice, an estimated conductivity σo* of the object-field is used to calculate the estimated sensitivity matrix So*, from which an estimated sensitivity matrix change ΔS* is calculated to replace ΔS.
The OO-SME model is derived by linearizing U = f(σ) as a product of the sensitivity matrix S and the conductivity σ as (5) based on the divergence theorem ,
where U is the voltage from the boundary with different electrode-combination, σ is the discretized conductivity in the domain, S is the sensitivity matrix. M is the total number of electrode-combinations, N is the total number of elements. Based on (5), the voltage Ub from the conductivity σb is formulated as (6-a),
Replacing Uo with Ub+ΔU, So with Sb+ΔS, and σo with σb+Δσ in (6-b) and subtracting (6-a) from (6-b) gets the expression of ΔU from Ub to Uo, which is used to formulate u(Δσ) in the OO-SME model in (2).
Compared to the existing linear and nonlinear models, u(Δσ) in the OO-SME model is optimized by considering the influence of ΔS completely. As a result, e between u(Δσ) and ΔU is significantly eliminated. Thus, Δσ is expected to be reconstructed with a higher accuracy.
Conductivity reconstruction with OO-SME model
Fig. 1 shows the flowchart of conductivity reconstruction with the OO-SME model, which includes two steps. In the first step, an initial conductivity change Δσinit* is reconstructed from Sb based on the linear model. In the second step, an updated conductivity change Δσupdt* is reconstructed from So* based on the OO-SME model.
In the 1st step of reconstructing Δσinit*, Sb is calculated from σb by (3a), u(Δσ) is formulated based on the linear model as (7).
Δσinit* is reconstructed by matching ΔU with u(Δσ).
In the 2nd step of reconstructing Δσupdt*, the process includes three parts. Firstly, estimate the conductivity of object-field as σo* = σb + Δσinit*, and calculate the estimated sensitivity matrix So* from σo* by (3b). Secondly, calculate the estimated sensitivity matrix change ΔS* by (4), and formulate u*(Δσ) based on the OO-SME model as (8).
Thirdly, reconstruct Δσupdt* by matching ΔU with u*(Δσ). By using the OO-SME model, Δσupdt* is output with a higher accuracy than Δσinit*.
To stabilize the ill-posed conductivity reconstruction model, the matching between ΔU with u(Δσ) and u*(Δσ) are replaced by minimizing a regularized least square error function as (9a) and (9b) respectively ,
where ‖R(Δσ)‖ is the regularization term, and μ is the regularization factor. Iterative methods such as the steepest descent method (SDM) , the Gauss-Newton method (GN) , and the conjugate gradient method (CG)  have been used to minimize (9a) and (9b). In this study, CG is used because of its lower computational cost and the faster convergence speed.
Conductivity reconstruction by simulation
Voltages and sensitivity matrices of background-field and object-field
Fig. 2 shows a mesh, the conductivity of the background-field, and the object-fields in the simulation. Fig. 2(a) is a 2D mesh with the following parameters: diameter d = 100 mm, nodes number P = 1958, elements number N = 3767, electrodes number L = 16. Fig. 2(b) is a background-field with conductivity σb. Fig. 2(c) – (f) are four object-fields with conductivity σo = σb + Δσ. The positions of objects are defined by parameters a, b, c, d, and e, where a = 0.32d, b = 0.27d, c = 0.34d, and e = 0.18d. The magnitude of σb and σo are 0.021 S/m and 0.267 S/m, respectively, thus, the magnitude of Δσ is 0.246 S/m. In the simulation, a random conductivity noise δσ with a magnitude of 20% of Δσ is added to a partial of 40% of the elements to approximate the inconsistency of conductivity. The electrode-combination for current-stimulation and voltage-measurement is chosen as quasi-adjacent pattern  because of its higher signal-to-noise ratio (SNR) than the adjacent pattern . The amplitude of current stimulation is I = 1 mA.
To obtain the voltages Ub and Uo, and the sensitivity matrices Sb and So of the background-field and object-field in the simulation, an elliptical partial differential equation with Neumann boundary is solved by the finite element method (FEM) to calculate the potential distribution . The governing equation of quasi-static electric field and boundary condition of current stimulation are expressed by (10) and (11) ,
where σ is the conductivity, φ(σ) is the field potential, Ω is the field domain, Γ is the boundary of Ω, dΓ is the outwards area element vector, n is the outwards normal vector on Γ, I is the current amplitude, and El is the boundary of the lth electrode.
The solution of φ(σ) corresponding to the discretized background-field and object-field are represented by φ(σb) and φ(σo). Ub and Uo are extracted from φ(σb) and φ(σo), which are the column vectors with size M×1. Sb and So are calculated from φ(σb) and φ(σo), which are two matrices with size M×N.
Regularization matrix and regularization factor
Regularization is used in (9a) and (9b) to stabilize the conductivity reconstruction , in which regularization matrix ‖R(Δσ)‖ and regularization factor μ are needed.
Different ‖R(Δσ)‖ provides different form of penalty. The Tikhonov  and Noser  regularizations are suggested to penalize the finiteness of conductivity on individual element. The Laplace  and total variation (TV)  regularizations are suggested to consider the continuity of conductivity on adjacent elements. A hybrid regularization is used to balance the finiteness and continuity simultaneously . In this study, a hybrid regularization with Noser and Laplace is used. The regularization term is calculated as (12).
where l, m, n, and e are element indices, lth, mth, and nth elements are adjacent to the eth element.
Regularization factor μ plays crucial role in stabilizing the conductivity reconstruction from noisy voltage ΔU* due to the condition number of the sensitivity matrix is high , where the influence of noise is inhibited by choosing μ properly . In this study, a Gaussian white noise δUb is added to ΔU to generate ΔU* as (15),
where Δσ* is the reconstruction of Δσ, σk is the kth singular value of Sb, uk and vk are the left and right orthonormal vectors of Sb corresponding to σk, and K is the rank of Sb. ΔU is the voltage change from Ub to Uo, δUb is a perturbation of Ub. To ensure stable reconstruction, μ is chosen by guaranteeing the second term is smaller than the first term in (18). Thus, μ for stable reconstruction satisfies (19),
u(Δσ) is different in different conductivity reconstruction models. u(Δσ) in the linear model is formulated by (7). In the sensitivity updating model, u(Δσ) is updated by an updated sensitivity matrix Sb* that is calculated from a new background-field σb*, which is formulated as (20).
In the OO-SME model, u(Δσ) is approximated based on (2), in practice, it is formulated as (8). In the forthcoming subsection, the influence of u(Δσ) among the existing models and the OO-SME model will be compared by Δσ*.
Evaluation of reconstructed conductivity
The relative accuracy (RA) between Δσ* and Δσ is defined to verify the high accuracy of the OO-SME model for conductivity reconstruction. RA is defined as (22),
where Δσ*iand Δσi are the ith component of Δσ* and Δσ respectively as well belong to the object.
Fig. 3 shows the voltage change ΔU* of different objects in the simulation. ΔU* is divided into 16 loops, and in each loop it has 13 measurements, where the value varies periodically with the electrode-combinations. Fig. 3 indicates that the conductivity change in the domain is reflected reliably by the voltage change on the boundary.
Fig. 4 shows the reconstructed conductivity Δσ* based on the linear model, two nonlinear models, and the OO-SME model in the simulation. Fig. 4(a) shows four object-fields with different objects. Fig. 4(b), (c), and (d) show Δσ* based on the linear model, sensitivity updating model, and second-order sensitivity model, respectively. Fig. 4(e) shows Δσ* based on the OO-SME model. Compared to the objects in Fig. 4(a), Δσ* in Fig. 4(b), (c), and (d) displays the position of the object only, in which the quantitative values of Δσ* are not reconstructed accurately. In contrast, besides the accurate position information, the magnitude of Δσ is reliably reconstructed by Δσ* in Fig. 4(e). The comparison in Fig. 4 indicates that the conductivity reconstructed based on the OO-SME model has a higher accuracy than the existing models.
Fig. 5 shows RA of reconstructed conductivity Δσ* in Fig. 4. On average, RA with the linear model is 9.39%. RA with the OO-SME model is 83.98%. RA with the sensitivity updating model is 31.30% and RA with the second-order sensitivity model is 24.61%. Compared to the linear model, RA with the sensitivity updating model and the second-order sensitivity model increased slightly by 21.91% and 15.24% respectively. RA with the OO-SME model increased significantly by 73.59%. The comparisons in Fig. 5 show that the reconstructed conductivity Δσ* with the OO-SME model has a higher accuracy to evaluate the ideal conductivity change Δσ than the existing models.
Conductivity reconstruction by experiment
Fig. 6 shows the experimental setup of an EIT system, which consists of 4 parts, a personal computer (PC), an impedance analyzer, a digital multiplexer, and an EIT sensor. The impedance analyzer is IM3570 made by Hioki. The multiplexer is made based on Arduino, which has 16 channels to switch on and off between different electrode-combinations for current-stimulation and voltage-measurement. The sensor is a polylactic acid-made circular tank printed with a 3D printer. The diameter of the tank is d = 100 mm. The 16 electrodes made of stainless screw are mounted along the circumference of the tank evenly.
As shown in Fig. 6, the PC controls the signals to trigger the impedance analyzer and switch on and off the channels on the multiplexer. The impedance analyzer generates a current signal on two output channels (HC and LC) to stimulate the target and measures the voltage signal from the target via two input channels (HP and LP), from which the impedance of the target is calculated. The multiplexer chooses 4 of 16 channels to stimulate the current and measure the voltage, the electrode-combinations for current-stimulation and voltage-measurement in the experiment are the same as in the simulation.
The experiment is conducted as follow. At first, the impedance from the background-field and the object-field are measured. Then, the voltages of background-field Ub* and object-field Uo* are extracted, and the voltage change ΔU* from Ub* to Uo* is calculated. At last, Δσ* is reconstructed by matching ΔU* with u(Δσ) based on different conductivity reconstruction models.
In the experiment, the meat sample from pig rump was used. The background-field is fat. The object-field is a lean meat mass enclosed by fat. The conductivity of fat and lean meat are σf = 0.021 S/m and σm = 0.267 S/m respectively at f = 100 Hz . The lean meat masses with different shapes and sizes are measured, which have the same dimension as the objects in the simulation. The amplitude of current stimulation is I = 1mA. The current frequency is f = 100 Hz.
Fig. 7 shows the voltage change ΔU* of different lean meat masses in the experiment. ΔU* is divided into 16 loops, and in each loop it has 13 measurements, where the value varies periodically with the changing of electrode-combinations. Fig. 7 indicates that the conductivity change in the meat sample is able to be measured reliably by the voltage change on the boundary.
Fig. 8 shows the reconstructed conductivity Δσ* of lean meat mass based on the linear model, two nonlinear models, and the OO-SME model. Fig. 8(a) shows four object-fields of lean meat masses. Fig. 8(b), (c), and (d) show Δσ* based on the linear model, sensitivity updating model, and second-order sensitivity model, respectively. Fig. 8(e) shows Δσ* based on the OO-SME model. Compared to the lean meat masses in Fig. 8(a), Δσ* in Fig. 8(b), (c) and (d) display the position of lean meat mass only, in which the quantitative value of Δσ* is not accurate. In contrast, reconstructed Δσ* in Fig. 8(e) has a high agreement with ideal Δσ. The comparison in Fig. 8 indicates that the proposed OO-SME model reconstructs the lean meat mass in meat sample accurately. Furthermore, the reconstruction can be used to evaluate the mass of lean meat quantitatively.
Fig. 9 shows RA of reconstructed conductivity Δσ* in Fig. 8. On average, RA with the linear model is 7.74%. RA with the OO-SME model is 54.60%. RA with the sensitivity updating model is 34.45% and RA with the second-order sensitivity model is 24.62%. Compared to the linear model, RA with the sensitivity updating model and the second-order sensitivity model increased slightly by 26.71% and 16.88% respectively. RA with the OO-SME model increased significantly by 46.86%. In conclusion, images in EIT reconstructed based on the OO-SME model have a higher accuracy to quantify the conductivity change in lean meat mass.
Approximation error of the OO-SME model
Due to the approximation error e between ΔU* and u(Δσ), reconstructed conductivity Δσ* has low accuracy. Fig. 10 shows the comparison between ΔU* and u(Δσ) from different conductivity reconstruction models in the simulation. Fig. 10(a), (b), (c) and (d) are corresponding to four different objects, respectively. The comparison shows that e from the linear model and the second-order sensitivity model are obvious. The relative ratio of e to ΔU* reaches up to 622% in the linear model and 477% in the second-order sensitivity model on average. Comparing to the linear model and second-order sensitivity model, even though e is significantly reduced in the sensitivity updating model, the relative ratio of e is still around 50% on average. In contrast, e from the OO-SME model is sufficiently eliminated, the relative ratio of e is as low as 2.4% on average. The comparison in Fig. 10 indicates that non-negligible approximation error e in the existing models is significantly reduced in the OO-SME model.
The approximation error e is caused by inaccurate sensitivity matrix estimation in different conductivity reconstruction models. In the linear model for approximation of u(Δσ) as (7), the simplification of omitting the change of f and replacing Jb with Sb leads to non-ignorable e. Consequently, Δσ* of different objects in Fig. 4(b) have similar distributions, and the magnitude of Δσ* does not match with Δσ. In the sensitivity updating model for approximation of u(Δσ) as (20), e is expected to be reduced by replacing Sb with Sb*. Even though Sb* from σb* has a higher accuracy compared to Sb, it is still not accurate enough to reduce e obviously. Therefore, as shown in Fig. 4(c), the accuracy of Δσ* is slightly improved. In the second-order sensitivity model for approximation of u(Δσ) as (21), e is expected to be reduced by considering the estimated second-order sensitivity Sb†. Since Sb† is much smaller than Sb, leaving the contribution of Sb† on u(Δσ) is limited. Thus, e is slightly reduced compared to the linear model. As a result, as shown in Fig. 4(d), the accuracy of Δσ* has not been improved obviously. However, different from the derivation in the linear model based on the Taylor formula, u(Δσ) in the OO-SME model is derived as (2) based on the divergence theorem, in which the influence of ΔS on u(Δσ) is completely considered. Therefore, e is eliminated in the OO-SME model and the accuracy of Δσ* is significantly improved compared to the existing models. As shown in Fig. 4(e), Δσ* based on the OO-SME model obtained the object shape and size accurately.
Contribution of sensitivity change ΔS to u(Δσ)
As expressed by (2), u(Δσ) based on the OO-SME model contains three components, all of which have non-ignorable contributions to u(Δσ). Fig. 11 shows the three components of u(Δσ) from different objects in the simulation. On average, the ratio of SbΔσ to ΔU is around -6.34, the ratio of ΔSσb to ΔU is around 1.18, and the ratio of ΔSΔσ to ΔU is around 6.16. The comparison implies two facts. Firstly, the nonlinear operator f acting on σ changes from fb to fo with σ changes from σb to σo; the contribution of change of f to u(Δσ) should be considered by sensitivity matrix change ΔS. Secondly, the contribution of ΔS to u(Δσ) is expressed as ΔS(σb+Δσ), which consists of two parts; one is ΔSΔσ implying that ΔS induced by Δσ influences the operator f acting on Δσ; another is ΔSσb implying that the influence of ΔS on σb has to be considered to approximate u(Δσ).
Omitting the change of f in the linear model led to a non-negligible e. The reduction of e from the optimized sensitivity matrix Sb* or Sb + Sb† in the two nonlinear models is limited since the contribution of ΔS corresponding to the change of f is not properly considered. In the OO-SME model, e is eliminated by completely considering the contribution of ΔS to u(Δσ), including the components relevant to σb and Δσ simultaneously.
Comparison of sensitivity matrices
Fig. 12 shows the comparison of sensitivity matrix with different conductivity reconstruction models in the simulation, where the sensitivity of each element from all electrode-combinations are collected. Fig. 12(a) shows the conductivity of object-fields. Fig. 12(b), (c) and (d) show Sb in the linear model, Sb* in the sensitivity updating model and Sb + Sb† in the second-order sensitivity model, respectively. Fig. 12(e) shows So* in the OO-SME model. The comparison indicates that the sensitivity matrix change ΔS is necessary to be considered. Compared to the Sb from a homogeneous background-field and the Sb + Sb† that is optimized from Sb directly, Sb* contains information of the object but it is not accurate. In contrast, So* has high accuracy, from which the object information is reliably detected. The comparison in Fig. 12 indicates that the accuracy of the sensitivity matrix in the OO-SME model is significantly improved, from which the estimated sensitivity matrix change ΔS* is calculated to optimize u(Δσ).
Fig. 13 shows the comparison of the sensitivity matrix in the experiment, where the sensitivity of each element from all electrode-combinations are collected. Fig. 13(a) shows the object-field. Fig. 13(b), (c), and (d) show Sb in the linear model, Sb* in the sensitivity updating model and Sb + Sb† in the second-order sensitivity model respectively. Fig. 13(e) shows So* in the OO-SME model. Similar as in Fig. 12, So* in the OO-SME model has a higher accuracy compared to the existing models, from which ΔS* is estimated accurately to optimize u(Δσ).
Quantitative evaluation of measurement object by the OO-SME model
The conductivity reconstruction based on the OO-SME model has high accuracy to evaluate the measurement object quantitatively, which improves the reliability of EIT application in the biomedical field, such as evaluation of effect of EMS on muscle compartments. In this study, the lean meat mass enclosed by fat is accurately reconstructed by the proposed OO-SME model, the relative accuracy RA reaches up 83.98% in the simulation and 54.60% in the experiment, respectively. Compared to the RA of 34.45% in the simulation and 31.30% in the experiment from the sensitivity updating model, the reliability of reconstruction from the OO-SME model is significantly increased. Thus, the OO-SME model can be used for quantitative evaluation of measurement objects in the biomedical fields of EIT applications.
The approximation error in the OO-SME model proposed in this study is eliminated compared to the existing models. The reconstructed conductivity from the OO-SME model has higher accuracy to reflect the shape and size of measurement object.
The lean meat mass in meat sample is accurately reconstructed by the OO-SME model, from which the lean meat mass could be quantitative evaluated.
The relative accuracy of lean meat mass from the reconstructed conductivity based on the OO-SME model reaches up to 83.98% in the simulation and 54.60% in the experiment. The reconstruction has a higher reliability to evaluate the lean meat mass quantitatively.
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