Electrical impedance tomography-based pressure-sensing using conductive membrane
This paper presents a mathematical framework for a flexible pressure-sensor model using electrical impedance tomography (EIT). When pressure is applied to a conductive membrane patch with clamped boundary, the pressure-induced surface deformation results in a change in the conductivity distribution. This change can be detected in the current-voltage data (i.e., EIT data) measured on the boundary of the membrane patch. Hence, the corresponding inverse problem is to reconstruct the pressure distribution from the data. The 2D apparent conductivity (in terms of EIT data) corresponding to the surface deformation is anisotropic. Thus, we consider a constrained inverse problem by restricting the coefficient tensor to the range of the map from pressure to 2D-apparent conductivity. This paper provides theoretical grounds for the mathematical model of the inverse problem. We develop a reconstruction algorithm based on a careful sensitivity analysis. We demonstrate the performance of the reconstruction algorithm through numerical simulations to validate its feasibility for future experimental studies.
Key words. electrical impedance tomography, pressure sensing, conductive membrane, inverse problem, prescribed mean curvature equation.
AMS subject classifications. 35R30, 35J25, 53A10
There is a growing demand for cost-effective flexible pressure sensors. These devices have wide potential applicability, including in smart textiles [5, 21, 22], touch screens , artificial skins , and wearable health monitoring technologies [20, 24]. Electrical measurements have recently been used to measure the pressure-induced surface deformation of conductive membranes. In particular, electrical impedance tomography (EIT) has been used to develop flexible pressure sensors [25, 28, 29], because it allows the electromechanical behavior of an electrically conducting film to be monitored. When a pressure-sensitive conductive sheet is exposed to pressure, the deformation of the surface alters the conductivity distribution, which can be detected by an EIT system. However, rigorous studies employing mathematical modeling and reconstruction methods have not yet been conducted. The purpose of this paper is to provide a systematic mathematical framework for an EIT-based flexible pressure sensor.
Our rigorous mathematical analysis is based on the consideration of a simple model of an EIT-based pressure-sensor using a thin, flexible conductive membrane whose electrical conductance is directly related to pressure-induced deformation. We assume that the conductive membrane is stretched over a fixed frame and has a number of electrodes placed on its boundary as shown in Figure LABEL:membrane. As in a standard EIT system, we use all adjacent pairs of electrodes to inject currents and measure induced boundary voltages between all neighboring pairs of electrodes to get a current-voltage data set, which is a discrete version of a Neumann-to-Dirichlet map. The current-voltage data can probe any external pressure loaded onto the membrane, because the pressure-induced surface deformation results in a change of the current density distribution over the surface, which leads to a change of the current-voltage data. Hence, the change in the current-voltage data can be viewed as a non-linear function of pressure. The inverse problem in this model is to identify the pressure (equivalently the surface deformation) from the boundary current-voltage data.
This paper provides a derivation of an EIT-based pressure-sensing model, which describes the explicit relationship between the measured current-voltage data and the pressure. The mathematical model is associated with an elliptic partial differential equation (PDE) with an anisotropic coefficient, which comes from the pressure-induced surface deformation. To be precise, let be a two-dimensional domain with a smooth boundary occupying the un-deformed membrane in the absence of any pressure. We denote the standard Sobolev space of order as .
Let be the pressure and be the solution of
Under pressure , the current-voltage data are dictated by with being the solution of the elliptic PDE,
Here, is the identity matrix, the superscript denotes the transpose, the unit outward normal vector to , and .
The standard Neumann-to-Dirichlet map is defined by with being the solution of (LABEL:Model). We cannot invert the map with the existing EIT reconstruction methods because of the well-known non-uniqueness result of the inverse problem: there are infinitely many anisotropic coefficients such that . Hence, we must consider the constrained inverse problem of recovering anisotropic coefficient within the set of coefficient tensors associated with pressures. Taking account of the fact that two different pressures and produce the same Neumann-to-Dirichlet map, we need to impose a proper constraint on pressures.
Next, we propose a pressure reconstruction method with the standard -channel EIT system. Owing to the quadratic structure of in , we cannot expect a linearized reconstruction method for , even assuming that pressure is small. Regarding as a piecewise constant function , through the standard discretization of the domain into small elements, , the inverse problem can be approximated by solving a large linear system with a large number of unknowns involving all possible products . (Here, is the indicator function of .) Given that most of the columns of the matrix have relatively small effect on the data, we consider a reduced linear system by eliminating most of the columns. Various numerical simulations verify the feasibility of the reconstruction algorithm.
In section LABEL:sec:framework, we formulate the mathematical model for the EIT-based membrane pressure sensor, and present uniqueness results. In section LABEL:sec:recon_method, we propose a reconstruction method to recover the pressure. In section LABEL:sec:numerical_results, we develop a reconstruction algorithm based on sensitivity analysis, and validate the algorithm by numerical simulation results.
This mathematical study of an EIT-based flexible pressure sensor is in an early stage. The proposed mathematical model requires the assumption of incompressibility, whereas there are many flexible materials that are not incompressible. Constructing a mathematical model that includes compressibility will be a future research topic.
2 Mathematical Framework
2.1 Formulation of the forward problem
Assume that a thin conductive membrane at rest occupies a two-dimensional bounded domain with a smooth boundary . Here, the thickness of the membrane is uniform. Assume that the conductivity of the membrane is homogeneous. Let with . Assume that a pressure lies in the set
where is a positive number. (The assumption is only used to guarantee existence and uniqueness of the prescribed mean curvature equation (LABEL:young-laplace) which will be discussed later.) When the pressure is loaded on , it produces a displacement of the membrane. The displacement at from its rest position is denoted by , and the deformed two-dimensional surface can be expressed as
Here, the boundary of the membrane is fixed so that there is no displacement on the boundary. Because the membrane undergoes deformation to reduce the area change caused by pressure , satisfies the prescribed mean curvature equation,
Let with denoting the duality pair between and . Let be defined by and let be the set of functions in with trace zero on .
To extract EIT-data for pressure-sensing, we inject a current of into the membrane . In the absence of the pressure (), the induced potential due to the injection current of is the solution of the following Neumann problem
In the presence of the pressure (), the induced potential is now defined on the deformed surface , and is governed by
where is the surface gradient, is the outward unit normal vector to the boundary , and is the projection map defined by .
For the derivation of equation (LABEL:NeumannBVP-2), we use the concept of surface conductivity , while regarding the thin membrane as a two-dimensional surface, because the induced current density along the thin membrane can be viewed as a tangent vector field on the surface. If the deformed membrane is uniform in thickness, the resulting potential satisfies the surface Laplace equation, , along the surface with being the Laplace-Beltrami operator. However, under the incompressibility assumption, the thickness of the membrane varies, and so does the surface conductivity. As a small area, , is changed to , the thickness is approximately reduced by a factor of , as is the surface conductivity.
The pair is called current-to-voltage pair. Here, .
Apparently, the voltage difference, , reflects the information of the displacement. Therefore, it is possible to recover from several pairs . The inverse problem is to reconstruct and from the boundary voltage-to-current data .
There are serious difficulties in solving the inverse problem, because the potential in (LABEL:NeumannBVP-2) is defined on the unknown surface in three dimensions, and the measured data, , are given on the boundary of the two-dimensional domain . The relation between the surface, , and the data, , is too complicated to handle the inverse problem. To deal with these difficulties, we introduce the following function defined in the known two-dimensional domain, , as
The following theorem provides a governing equation for , through which the relationship between current and voltage can be understood.
The function in (LABEL:proj) is dictated by the following elliptic equation
where is a symmetric positive definite matrix given by
Proof. Let denote the extension of such that for all and . Then, the surface gradient can be expressed as
where is the unit downward normal vector to the surface and is the three-dimensional gradient. Since , from (LABEL:eq1) a direct computation yields
Here, we used the fact that . The surface divergence of the tangential vector field is written as
It follows from the vector identity, that (LABEL:surf_div) can be expressed as
Replacing with (LABEL:grad1), we obtain
Then, implies , and has the positive eigenvalues and . This completes the proof.
2.2 Unique determination of the pressure support
We have seen that the displacement, , and the current-voltage data, , are involved in (LABEL:proj_con) with the anisotropic coefficient, . In this subsection, we prove that the current-voltage data uniquely determine the pressure support. To do so, we need to investigate the inverse problem of determining from the current-voltage data. An anisotropic coefficient is uniquely determined by the current-voltage data up to a diffeomorphism that fixes the boundary. For any diffeomorphism, with , being the identity map, satisfies
where is a symmetric matrix-valued function given by
where is the Jacobian of and denotes the determinant. This means that two different and produce the same Neumann-to-Dirichlet map. Conversely, the Neumann-to-Dirichlet map can determine the tensor up to the diffeomorphism, , where provided that is approximately constant . In our model, is only involved in the scalar , and it is possible to determine within the set uniquely provided is sufficiently small. Note that two different pressures and produce the same coefficient .
We must consider the constrained inverse problem of recovering the anisotropic coefficient within the set from the current-voltage data. Let us introduce the outer support of , denoted by ; for , there exists an open and connected set such that , , and [8, 18].
For , determines uniquely.
Proof. Let . We assume . We need to prove that . From , it follows that on the boundary . From (LABEL:gamma1), we have
This leads to the following identity with (real) being :
The difference satisfies
where is a matrix given by
and . Since the structure of is the same as in (LABEL:gamma1), is positive-definite and satisfies the elliptic PDE (LABEL:elliptic_A). Hence, by the unique continuation property, it follows that
if on , it follows from the assumption and (LABEL:bound1) that
This leads to
which contradicts (LABEL:runge). This completes the proof.
2.3 Unique determination of the pressure in the monotone case
We now prove the unique determination of the pressure from the current-voltage data in the monotone case.
Let be in . If in and , then either or in .
Proof. To derive a contradiction, we assume that and use exactly the same argument as in the proof of Theorem LABEL:thm_unique. Remember that satisfies the elliptic PDE
with being defined by (LABEL:A). From the strong comparison principle, it follows that
Since on for , we have from Hopf’s lemma
Noting that on , we have
Hence, if in , then we have
However, this is not possible because implies . This concludes that if , then in , which completes the proof.
It is worth emphasizing that two different pressures and having the same support can produce the same displacement near the boundary. More precisely, there exist two different pressures and such that
Let and with being the ball of radius centered at the origin. Consider the following radial symmetric function
where and . A direct computation shows that satisfies
and is the radial derivative. Hence, does not change with : for every , we have
This means that in the non-monotone case there are infinitely many which provide the same displacement near the boundary.
3 Reconstruction method
3.1 Measured data: Discrete Neumann-to-Dirichlet map
We use -channel EIT system in which electrodes are attached on the boundary . Let be the potential in (LABEL:proj_con) with which corresponds to the th injection current using the adjacent pair and . When we inject a current of [mA] along the adjacent electrodes and , the resulting potential satisfies
The th boundary voltage subject to the th injection current is denoted as
Here, . Integration by parts gives the reciprocity principle:
The boundary voltage (LABEL:mea_data) is assumed to be known. We use it as measurement data for recovery of pressure .
3.2 Discrepancy minimization problems
Let be the measured data under an applied pressure . Then it follows from (LABEL:recipro) that the pressure can be obtained by minimizing the discrepancy functional
The inverse problem can be viewed as finding the minimizer of . Unfortunately, it is numerically difficult to compute the minimizer of because is highly non-linear with respect to .
To extract necessary information about from the data , we use the voltage difference data
where is the data in (LABEL:mea_data) with , i.e., the boundary voltage data in the absence of the pressure. With the voltage difference data , the functional in (LABEL:functional) can be rewritten as
The above identity follows from
Due to the high non-linearity of the discrepancy functional in (LABEL:func_J), it is difficult to compute a minimizer directly. To compute minimizers of (LABEL:func_J) effectively, we will make use of various approximations. Assume that is small. We will neglect quantities of fourth order of smallness; for example,
Since we have
From (LABEL:Wp-approx1) and (LABEL:Wp-approx2), we have
Neglecting in (LABEL:ap1), the discrepancy functional in (LABEL:func_J) can be approximated as
Assuming Gaussian measurement noise, we consider the following regularized minimization problem:
and being a regularization parameter.
The displacement can be approximated to with being the solution of Possion’s equation in , because
With this approximation, the minimization problem (LABEL:lin_regul) can be further simplified as follows: find which minimizes the discrepancy functional
where is a positive number. In the next subsection, we minimize the functional defined in (LABEL:F_prop) in order to reconstruct .
3.3 Reconstruction algorithm
Based on the simplified discrepancy functional (LABEL:F_prop), we propose a pressure image reconstruction algorithm. We discretize the domain into triangular elements such that , where is a triangular subregion with side length . For the approximation of the pressure , we assume that is a piecewise constant function contained in the set
We assume that . Then, we can express the pressure by
For each , let be the solution of
Then, can be expressed as
where is the Dirichlet function associated with the domain , that is, the solution to