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 [1]. 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 [4], 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 Δ^{b}^{o}^{b}^{b}^{o}^{b}^{b}^{b}^{o}^{o}^{o}^{b}^{b}^{o}^{b}^{b}^{b}^{b}^{b}^{b}^{b}^{b}^{*} has low accuracy, which influences the further quantitative evaluation [4].

To reduce ^{b}^{b}^{b}^{*} as the new background-field is used to calculate the new sensitivity matrix ^{b}^{*}. The second nonlinear model is the so-called “second-order sensitivity model” [11], in which the Hessian matrix of ^{b}^{b}^{b}^{†} to compensate for ^{b}^{th} row of ^{b}^{†} is represented by the pivots of [^{b}_{m}^{T}^{b}_{m}^{b}_{m}^{th} row of ^{b}^{b}^{b}^{o}^{b}^{o}^{b}

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 ^{b}^{b}^{b}^{b}^{o}^{o}^{o}^{o}^{b}^{o}^{b}^{o}^{b}^{b}^{b}^{o}

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.

The approximation error

where Δ^{b}^{o}^{b}

where ^{b}^{b}^{b}^{o}^{o}^{o}^{b}^{o}

where _{e}^{th} element, _{m}_{-c}(•) and _{m}_{-v}(•) are the field potentials relevant to ^{th} electrode-combination, ^{b}^{o}

and used to optimize ^{o}^{*} of the object-field is used to calculate the estimated sensitivity matrix ^{o}^{*}, from which an estimated sensitivity matrix change Δ^{*} is calculated to replace Δ

The OO-SME model is derived by linearizing

where ^{b}^{b}

the voltage ^{o}^{o}

Replacing ^{o}^{b}^{o}^{b}^{o}^{b}^{b}^{o}

Compared to the existing linear and nonlinear models,

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 Δ^{init}^{*} is reconstructed from ^{b}^{updt}^{*} is reconstructed from ^{o}^{*} based on the OO-SME model.

In the 1^{st} step of reconstructing Δ^{init}^{*}, ^{b}^{b}

Δ^{init}^{*} is reconstructed by matching Δ

In the 2^{nd} step of reconstructing Δ^{updt}^{*}, the process includes three parts. Firstly, estimate the conductivity of object-field as ^{o}^{*} = ^{b}^{init}^{*}, and calculate the estimated sensitivity matrix ^{o}^{*} from ^{o}^{*} by (3b). Secondly, calculate the estimated sensitivity matrix change Δ^{*} by (4), and formulate ^{*}(Δ

Thirdly, reconstruct Δ^{updt}^{*} by matching Δ^{*}(Δ^{updt}^{*} is output with a higher accuracy than Δ^{init}^{*}.

To stabilize the ill-posed conductivity reconstruction model, the matching between Δ^{*}(Δ

where ‖

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 ^{b}^{o}^{b}^{b}^{o}

To obtain the voltages ^{b}^{o}^{b}^{o}

where _{l}^{th} electrode.

The solution of ^{b}^{o}^{b}^{o}^{b}^{o}^{b}^{o}^{b}^{o}

Regularization is used in (9a) and (9b) to stabilize the conductivity reconstruction [20], in which regularization matrix ‖

Different ‖

The Noser regularization term of

where _{e}^{th} column of ^{b}_{e}^{th} element of

where ^{th}, ^{th}, and ^{th} elements are adjacent to the ^{th} element.

Regularization factor ^{*} due to the condition number of the sensitivity matrix is high [26], where the influence of noise is inhibited by choosing ^{b}^{*} as (15),

where δ^{b}^{b}

where ^{b}^{*} is the voltage from the experiment, δ^{b}^{*} is the estimated noise from ^{b}^{*} based on reciprocity. The magnitude of δ^{b}

where Δ^{*} is the reconstruction of Δ_{k}^{th} singular value of ^{b}_{k}_{k}^{b}_{k}^{b}^{b}^{o}^{b}^{b}

where _{1} is the maximum singular value of ^{b}

^{b}^{*} that is calculated from a new background-field ^{b}^{*}, which is formulated as (20).

In the second-order sensitivity model, ^{b}^{†}, which is formulated as (21).

In the OO-SME model, ^{*}.

The relative accuracy (^{*} and Δ

where ^{*} _{i}_{i}^{th} component of Δ^{*} and Δ

Fig. 3 shows the voltage change Δ^{*} of different objects in the simulation. Δ^{*} 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 Δ^{*} 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 Δ^{*} based on the linear model, sensitivity updating model, and second-order sensitivity model, respectively. Fig. 4(e) shows ^{*} based on the OO-SME model. Compared to the objects in Fig. 4(a), ^{*} in Fig. 4(b), (c), and (d) displays the position of the object only, in which the quantitative values of ^{*} are not reconstructed accurately. In contrast, besides the accurate position information, the magnitude of ^{*} 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 ^{*} in Fig. 4. On average, ^{*} with the OO-SME model has a higher accuracy to evaluate the ideal conductivity change

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

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 ^{b}^{*} and object-field ^{o}^{*} are extracted, and the voltage change Δ^{*} from ^{b}^{*} to ^{o}^{*} is calculated. At last, Δ^{*} is reconstructed by matching Δ^{*} with

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 ^{f}^{m}

Fig. 7 shows the voltage change Δ^{*} of different lean meat masses in the experiment. Δ^{*} 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 ^{*} 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 ^{*} based on the linear model, sensitivity updating model, and second-order sensitivity model, respectively. Fig. 8(e) shows ^{*} based on the OO-SME model. Compared to the lean meat masses in Fig. 8(a), ^{*} in Fig. 8(b), (c) and (d) display the position of lean meat mass only, in which the quantitative value of ^{*} is not accurate. In contrast, reconstructed ^{*} in Fig. 8(e) has a high agreement with ideal

Fig. 9 shows ^{*} in Fig. 8. On average,

Due to the approximation error ^{*} and ^{*} has low accuracy. Fig. 10 shows the comparison between Δ^{*} and ^{*} 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

The approximation error ^{b}^{b}^{*} of different objects in Fig. 4(b) have similar distributions, and the magnitude of Δ^{*} does not match with Δ^{b}^{b}^{*}. Even though ^{b}^{*} from ^{b}^{*} has a higher accuracy compared to ^{b}^{*} is slightly improved. In the second-order sensitivity model for approximation of ^{b}^{†}. Since ^{b}^{†} is much smaller than ^{b}^{b}^{†} on ^{*} has not been improved obviously. However, different from the derivation in the linear model based on the Taylor formula, ^{*} is significantly improved compared to the existing models. As shown in Fig. 4(e), Δ^{*} based on the OO-SME model obtained the object shape and size accurately.

As expressed by (2), ^{b}^{b}^{b}^{o}^{b}^{o}^{b}^{b}^{b}

Omitting the change of ^{b}^{*} or ^{b}^{b}^{†} in the two nonlinear models is limited since the contribution of Δ^{b}

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 ^{b}^{b}^{*} in the sensitivity updating model and ^{b}^{b}^{†} in the second-order sensitivity model, respectively. Fig. 12(e) shows ^{o}^{*} in the OO-SME model. The comparison indicates that the sensitivity matrix change Δ^{b}^{b}^{b}^{†} that is optimized from ^{b}^{b}^{*} contains information of the object but it is not accurate. In contrast, ^{o}^{*} 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 Δ^{*} is calculated to optimize

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 ^{b}^{b}^{*} in the sensitivity updating model and ^{b}^{b}^{†} in the second-order sensitivity model respectively. Fig. 13(e) shows ^{o}^{*} in the OO-SME model. Similar as in Fig. 12, ^{o}^{*} in the OO-SME model has a higher accuracy compared to the existing models, from which Δ^{*} is estimated accurately to optimize

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

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.