IOP PUBLISHING Meas. Sci. Technol. 18 (2007) 2847–2858
MEASUREMENT SCIENCE AND TECHNOLOGY
doi:10.1088/0957-0233/18/9/014
Electrical impedance tomography through constrained sequential linear programming: a topology optimization approach C´ıcero R de Lima1, Luis A M Mello1, Raul Gonzalez Lima2 and Em´ılio C N Silva1 1
Department of Mechatronics and Mechanical Systems Engineering, Escola Polit´ecnica da Universidade de S˜ao Paulo, Av. Prof. Mello Moraes, 2231, 05508-900 S˜ao Paulo, SP, Brazil 2 Department of Mechanical Engineering, Escola Polit´ecnica da Universidade de S˜ao Paulo, Av. Prof. Mello Moraes, 2231, 05508-900 S˜ao Paulo, SP, Brazil E-mail:
[email protected],
[email protected],
[email protected] and
[email protected]
Received 28 February 2007, in final form 21 May 2007 Published 20 July 2007 Online at stacks.iop.org/MST/18/2847 Abstract Electrical impedance tomography (EIT) is an imaging method that estimates conductivity distribution inside a body. In EIT, images are obtained by applying a sequence of low intensity electrical currents through electrodes attached to the body. Although in EIT there are serious difficulties to obtain a high-quality conductivity image, for medical applications this technology is safer and cheaper than other tomography techniques. The EIT deals with an inverse problem in which given the measured voltages on electrodes and a finite element (FE) model, it estimates the conductivity distribution, which are parameters of the FE model. In this work, the topology optimization method is applied as a reconstruction algorithm to obtain absolute images in EIT. It is an optimization method that has been applied successfully to structural mechanical applications and consists of systematically finding a conductivity distribution (or material distribution) in the domain that minimizes the difference between measured voltages and voltages calculated by using a computational model. This algorithm combines the finite element method and sequential linear programming (SLP) to solve the inverse problem of EIT. The SLP allows us to easily apply some regularization schemes based on included constraints in the topology optimization problem. Constraints based on image tuning control and weighted distance interpolation (WDI) are proposed, while a material model is applied to ensure the relaxation of the optimization problem. A new formulation to analytically perform the sensitivity analysis is proposed, using Maxwell’s reciprocity theorem. To illustrate, the implemented algorithm is applied to obtain conductivity image distributions of some 2D examples using numerical and experimental data. Keywords: topology optimization, electrical impedance tomography, finite element method, inverse problem, regularization, constraints
an interesting alternative for obtaining images in clinical applications (Holder 1993). In fact, EIT has been applied to Since the beginning of the 1980s, a technique called geophysical sciences (Dines and Lytle 1981), non-destructive electric impedance tomography (EIT) has been studied as testing (Eggleston et al 1989), industrial process tomography 0957-0233/07/092847+12$30.00 © 2007 IOP Publishing Ltd Printed in the UK 2847
1. Introduction
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electrodes
section of body
Figure 1. Electrodes positioned around the body.
(Dickin et al 1992) and most recently, medical applications. EIT is based on an inverse problem, such that, given the voltages measured on electrodes attached to the boundary of body, it searches for the conductivity distribution inside the body which constitutes an image. A sequence of low intensity electrical currents is applied to the body through electrodes attached to the patient’s body, as illustrated in figure 1. Although the EIT technique has a poor resolution, it has potential for clinical applications such as monitoring ventilation of lungs (Holder 1993, Harris et al 1988, Frerichs 2000), breast cancer detection (Zou and Guo 2003), monitoring human brain function (Bagshaw et al 2003), and monitoring heart function and blood flow (Eyuboglu et al 1987). An EIT device is small and portable, which allows its installation for continuous monitoring of bedridden patients, avoiding dangerous patient transportation from ICU (Intensive Care Unit) to the CT-scan room. The technology of EIT is safer than other tomography techniques, because the patient is not exposed to any type of radiation, only to low electrical current levels that do not cause harm to the patient (Cheney et al 1999). A number of deterministic and statistical reconstruction algorithms have been developed for EIT imaging (Lionheart 2004, Kaipio et al 2000). This work presents results obtained from the application of the topology optimization method to EIT absolute image reconstruction, considering regularization constraints. In structural problems, the topology optimization method systematically seeks a material distribution inside a design domain by determining which points of the domain, ideally, should be solid, and which points should be void to minimize an objective function satisfying some specified constraints (Bendsøe and Kikuchi 1988). It is an optimization method that has been successfully applied to structural mechanical applications since the beginning of the 1990s (Suzuki and Kikuchi 1991). Recently, its applications to other scientific investigations can be found (Bendsøe and Sigmund 2003). The topology optimization problem applied to EIT consists of finding a material distribution (absolute values of conductivity or resistivity distribution) of a body that minimizes the difference between electrical potentials obtained from electrode measurements and electrical potentials computed from a model of the body. In this case, the optimization problem is solved by a computational algorithm that combines the finite element method (FEM) (numerical simulation) and an optimizer called sequential linear programming (SLP) (Haftka et al 1996), which allows us to easily include several constraints in the optimization 2848
problem, consequently restricting the solution space, and thus, regularizing the problem. The next section shows the FE model used in this work. In section 3, the basic concept of topology optimization method is described. The formulation of the topology optimization problem applied to EIT and its numerical implementation are presented in sections 4 and 5, respectively. In section 6, the adjoint method for the sensitivity analysis is described, while image reconstruction results from numerical and experimental phantoms are shown in section 7. The phantoms contain high resistivity inclusions to represent a pneumothorax in a clinical scenario. A value of 106 m was assigned to represent air inside the thorax following a previous work (Brown et al 2002). Finally, in section 8 the conclusions are presented.
2. FE model The FE formulation is generated from constitutive equations of a conductive medium (Hayt 1986). For EIT, Maxwell’s equations can be used (Muray and Kagawa 1985). In terms of electric potential (φ), the electric field can be expressed as E = −∇φ,
(1)
where E is the electric field and ∇ is the gradient operator. If we consider the general form of Ohm’s law at a point within a conductive medium, we have J = σ E,
(2)
where J is the electrical current density and σ is the conductivity tensor. When there are no sources within the domain, it can be stated that ∇ · J = 0.
(3)
Combining (1) and (2) and using (3) gives Laplace’s equation ∇ · (σ ∇φ) = 0.
(4)
Laplace’s equation must be constrained using Neumann’s and Dirichlet’s boundary conditions. In the EIT, electrical current is applied through electrodes that are attached to the boundary of a body. In this way, Neumann’s boundary conditions can be written as σ ∂φ = J on the ith electrode i (5) ∂n 0 elsewhere at the boundary, where n is a normal vector of the boundary surface. Dirichlet’s boundary conditions specify the potential at the electrodes. The solution of equation (4), subject to Neumann’s and Dirichlet’s boundary conditions for a given conductivity distribution, is known as a forward problem, whose analytical solution is only possible for simple geometries (Hayt 1986). For most conductivity distributions, the solution of the forward problem must be solved using numerical techniques. In this work, FEM is applied to solve the forward problem of EIT. The FE formulation is derived from constitutive equations of the conductive medium, given by equation (4) and its boundary conditions. Thus, the electrical potential distribution in the discretized domain is obtained by solving the equation of FEM (Bathe 1996) KΦ = I,
(6)
Topology optimization applied to EIT metal electrode electrodes 6 1
I
5 2
II
4 3
0.06
electrode-skin interface electrode element
10
0.05
6 4 3
0.04 2
C(x)
mesh of electrode model section of body
p=1
0.03 0.02 0.01
Figure 2. Electrode model.
where K is the global electric conductivity matrix of the FE model, Φ is a nodal electric potential vector and I is a nodal electric current vector. In this work, four node quadrilateral elements are used and equation (6) is solved using the iterative biconjugate gradient method for a sparse matrix data structure (Press et al 1999). In addition, a FE electrode model, proposed by Hua et al (1993), has been applied to represent the electrical behaviour of the electrode–electrolyte interface layer. Figure 2 shows the electrode elements that are considered in that model. The electric potentials of the outside surface (nodes 4–6) are assumed to be equal, since they represent the metal component of the electrode (Hua et al 1993). The electrical conductivity matrix of the electrode model depends on the width of electrode, on the thickness of the contact interface (electrode–electrolyte) and on the resistivity (inverse of the conductivity) of the contact interface (Hua et al 1993). The product between resistivity and thickness of the contact interface is known as the contact impedance of electrode elements (or electrode parameter ρt). Each electrode model matrix is assembled in the global matrix K, according to its connectivity.
3. Topology optimization concept In the topology optimization method the material at each point of a domain can vary continuously from a material ‘A’ to ‘B’. For instance, material ‘A’ could be air and material ‘B’ could be tissue. The materials are mixed according to a material model, which introduces a relaxation to the topology optimization problem (Bendsøe and Sigmund 1999). In this work, the material model applied is based on SIMP (simple isotropic material with penalization) (Bendsøe and Sigmund 2003), which defines the conductivity properties C(x) at each point of the domain in the following way: C(x) = ρ(x)p CA + (1 − ρ(x)p )CB ,
(7)
where CA and CB are the conductivity properties of base materials of the domain. The value of the variable of the optimization problem (or optimization variable) ρ(x) is defined between 0 (presence of ‘B’ only) and 1 (presence of ‘A’ only). Power p is a coefficient that, together with constraints to be defined shortly, penalizes an excess of intermediate materials (Bendsøe and Sigmund 1999). Figure 3 shows the effect of power p for the material model, equation (7), considering CA < CB . According to this graph, as power p increases and constraints are turned on, higher values of the conductivity properties C(x) at each point of the domain are favoured in the final result.
0 0
0.2
0.4
ρ(x)
0.6
0.8
1
Figure 3. Behaviour of material model equation as p is increased.
In the following section, the topology optimization problem applied to solve the inverse problem of EIT is described. It is a nonlinear optimization problem solved by the SLP algorithm, which requires a sensitivity analysis.
4. Topology optimization problem applied to EIT The EIT image reconstruction can be interpreted as a problem of finding the material distribution inside the domain that reproduces the measured electric potential values at electrodes. In this way, the optimization problem can be defined as ne np 1 (φij − φij 0 )2 (8) Minimize: F = ρ(x),ρt 2 j =1 i=1 such that: electrical conductivity equation 0 ρ(x) 1 ρtmin ρt ρtmax additional constraints, where F is the objective function, φij 0 and φij are the respective measured and computed electrical potential values, and ne and np are the number of applied current patterns and the number of measurement points (electrodes), respectively. The values φij are obtained by the FE analysis of the domain. The optimization problem (8) is a nonlinear and ill-posed problem, i.e., there are different conductivity distributions in the domain that yield the same electrical potential values on electrodes (Sylvester and Uhlmann 1987, Gisser et al 1990). However, as already mentioned, topology optimization method allows us to easily include constraints in the optimization problem. Thus, two constraints are proposed and evaluated in this work. One of them is an image tuning control that forces the integral estimated resistivity values to reach a minimum expected value and, consequently, improving the obtained image quality. This constraint is given by the following equation: ρ(x) d V ∗ , (9)
where is the domain of the model and V ∗ is the material constraint inside the domain. Another constraint is based on weighted distance interpolation (WDI) and is given by 1 ρ(x) d C ∗ , (10) q d(x) 2849
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where d(x)q is the penalized distance between each point and the centre of the domain. C ∗ is a parameter of the optimization process that can be obtained from the average resistivity of the domain and equation (7). In EIT, image sensitivity due to changes of conductivity at the centre of the domain is smaller than the sensitivity due to changes near the boundary of the domain (Bacrie et al 1997). Thus, the constraint, equation (10), makes a balance of conductivity values at each point of the domain by using the inverse of the distance, d(x), from the centre of the domain, increasing the sensitivity of the central region.
Initial data Optimization of electrode parametres (ρtj) Optimization of pseudo-densities (ρk)
No
Convergence?
Yes
5. Numerical implementation 5.1. Continuous approximation of material distribution Considering the material model previously presented in section 3, ρ(x) is interpolated in each element providing a continuum approximation of material distribution (CAMD) in the domain (Matsui and Terada 2004). As seen in section 3, the conductivity properties at each point of the domain are interpolated by equation (7), which in a discretized domain is given by p p (11) Ck = ρ(x)k CA + 1 − ρ(x)k CB , where Ck and ρ(x)k are the conductivity property and ρ(x) within the kth element of the domain, respectively. However, the purpose of CAMD is to provide material distribution variation within the element. Then, ρ(x)k is replaced by a continuous function, n N (x)i ρi , (12) ρ(x)k = i=1
k
where ρi is the ith nodal optimization variable, N (x)i is a FE shape function (Bathe 1996) and n is the number of nodes per element. Therefore, the conductivity is not constant within the element and nodal optimization variables are introduced. Thus, the material model becomes n
n p p N (x)i ρi CA + 1 − N (x)i ρi Ck = CB . i=1
i=1
k
k
(13) In this work, the FE shape function N (x)i is a well-known bi-linear interpolation function of the four node isoparametric quadrilateral element (Bathe 1996), often applied in the FE literature (Bathe 1996); however it can assume any other interpolation function that satisfies the condition 0 ρ(x)k 1 and that it is integrable inside the domain. 5.2. Implementation of topology optimization problem Considering the domain discretized into several elements and using equations (9) and (10) (constraints) as regularization schemes, the numerical implementation of topology optimization problem, equation (8), is given by 1 (φij − φij 0 )2 2 j =1 i=1 ne
minimize: F = ρ,ρt
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np
(14)
Obtained image
Figure 4. A two-phase optimization scheme to estimate electrode parameters and nodal optimization variables.
such that : electrical conductivity equation j = 1, . . . , np ρtmin ρtj ρtmax ; 0 ρi 1; i = 1, . . . , N N Vi ρi V ∗ (tuning) i=1 N N 1 1 ∗ ρ i q q ρ (W DI ), d d i=1 i i=1 i
where Vi is a volume interpolation at each element node and N is the number of nodes of the discretized domain. The optimization variable ρi is a nodal variable and ρtj represents each electrode parameter value which is also obtained by optimization (and, therefore, is also an optimization variable) and is limited by box constraints ρtmin and ρtmax . The parameter ρ ∗ is the average nodal optimization variable computed from the average resistivity of the domain. 5.3. Solution of topology optimization problem The solution of the topology optimization problem shown before is obtained numerically through an iterative optimization algorithm. This optimization algorithm must be able to estimate the unknown conductivity distribution inside the domain and estimate the electrode parameters described in section 2. Since sensitivities of the objective function in relation to the two corresponding sets of optimization variables have different scales, a two-phase method is adopted to estimate electrode parameters and the conductivity distribution for all examples presented in this work. This strategy considers the problem of estimating electrode contact impedances separately from the problem of image estimation (Trigo et al 2004). First, the algorithm estimates the electrode parameters, considering them as optimization variables in the optimization process, keeping the nodal optimization variables that determine the conductivity distribution of the domain fixed. Then, keeping the estimated electrode parameters fixed, the algorithm estimates a new set of nodal optimization variable values. This alternation of phases continues until convergence is achieved, and an image is found. Each optimization process shown in figure 4 is also an iterative algorithm detailed in figure 5, and is implemented
Topology optimization applied to EIT
Data Input
I2 φ1
FE Analysis
Φ2
Φ1
Calculation of Objective Function and Constraints
Ω
Ω
φ2 Update of Design Variables
I1
yes End
Linear Programming
Convergence ?
Sensitivity Analysis
no
Figure 6. Two simultaneous current patterns of electrical excitation.
and wj =
Figure 5. Flowchart of topology optimization algorithm.
using C language. The FE model of the domain and each potential φij 0 are supplied to this algorithm as initial data. By the FE model, the electric potentials (φij ) are computed, allowing us to obtain the objective function and constraint values. In the next step, the optimization is done by using the gradients of the objective function and constraints with respect to optimization variables (sensitivity analysis). The procedure of evaluation of gradients is shown in section 6. The optimization algorithm begins with a uniform distribution of material for the whole domain and it supplies a new material distribution (optimization variables), which is updated in the FE analysis. This optimization algorithm is based on sequential linear programming (SLP) (Bendsøe and Sigmund 2003), which allows us to work with a large number of optimization variables, complex objective functions and constraints. The optimization algorithm solves a nonlinear optimization problem considering it as a sequence of linear sub-problems, which can be solved by linear programming (LP) (Haftka et al 1996). The nonlinear optimization problem can be linearized by writing a truncated Taylor series expansion of the objective function, keeping the terms up to the first-order derivatives. For this approach to be valid, it is necessary to limit the variation of the optimization variables in each linear subproblem using moving limits (Haftka et al 1996). In each iteration, SLP finds the optimum values of the optimization variables which will be used in the subsequent iterations as data input. This process continues successively, until the convergence of the objective function is achieved. 5.4. Implementation of a spatial filtering technique A spatial filtering scheme based on a smooth distribution of the nodal optimization variables in the whole domain is applied to control the spatial variation of these variables (Swan and Kosaka 1997). This filter changes the SLP moving limits in the following way: ρi Vi + w nv j =1 ρj Vj nv , (15) ρi = Vi + w j =1 Vj with
nv w=
j =1
nv
wj
(16)
Rmax − Rij , Rmax
(17)
where ρi is the nodal optimization variable, Vi is a volume interpolation of all elements that contain a node i, nv is the number of adjacent nodes j located around the node i, Rij is the distance between nodes i and j and Rmax is a radius that encompasses all adjacent nodes j . In this work, the image is obtained considering a Rmax value that encompasses at least five adjacent nodes around a central node i.
6. Sensitivity analysis The gradients (or derivatives) of objective function and constraints in relation to optimization variables can be obtained analytically using the adjoint method (Haftka et al 1996). Applying the chain rule, the partial derivative of the objective function with respect to an optimization variable (ρ) can be written in the following form, ∂φij ∂F ∂φij ∂F (φij − φij 0 ) = = , ∂ρ ∂φij ∂ρ ∂ρ j =1 i=1 ne
np
(18)
∂φ
where the term ∂ρij can be written in terms of the conductivity matrix. Consider a domain () with two simultaneous current patterns of electrical excitation, as shown in figure 6. Due to the superposition principle, Maxwell’s reciprocity theorem (Cook and Young 1985), and the linearity of current and electrical potential, we can say that IT1 Φ2 = IT2 Φ1 ,
(19)
where the indices 1, 2 indicate two distinct cases of applied current and the superscript T indicates the transposition of the vector. Initially, we admit a fictitious current pattern I2 as an electrical current vector whose only non-zero component is a unit current applied to one point of the body. Using equations (6) and (19), we obtain ΦT1 KΦ2 = φ1 ,
(20)
where φ1 is a component of the vector Φ1 and K is a symmetric global matrix, then KT = K and (K−1 )T = K−1 . From differentiation of the two sides of (20) with respect to an optimization variable, we obtain ∂ ΦT1 KΦ2 ∂ (φ1 ) = . (21) ∂ρ ∂ρ 2851
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(φ5j − φ5j0 )
(φij − φij0 ) Ψij
Φj
(φ4j − φ4j0 )
i
Ψ1j
Φj
Ω
Ω
Ω
Ij
The left-hand side of (21) is derived in the following way: ∂ ΦT1 KΦ2 ∂K ∂ Φ2 ∂ ΦT1 Φ2 + ΦT1 K = KΦ2 + ΦT1 . (22) ∂ρ ∂ρ ∂ρ ∂ρ Now, differentiating the equilibrium equation KΦ1 = I1 (first current pattern) with respect to an optimization variable ∂ Φ1 ∂K ∂I1 Φ1 + K = , (23) ∂ρ ∂ρ ∂ρ and knowing that electrical current vector I1 does not depend on an optimization variable, we have ∂K ∂K ∂ Φ1 ∂ Φ1 Φ1 + K Φ1 . =0⇒ = −K−1 ∂ρ ∂ρ ∂ρ ∂ρ Thus, transposing equation (24), we obtain
(φ2j − φ2j0 ) (φ1j − φ1j0 )
I1j
Figure 7. Current patterns applied to the ith measurement point for the gradient calculation.
(24)
∂ ΦT1 ∂KT −1 T ∂KT −1 = −ΦT1 (K ) = −ΦT1 K . (25) ∂ρ ∂ρ ∂ρ Similarly, differentiating the equilibrium equation KΦ2 = I2 , we obtain ∂K ∂ Φ2 Φ2 . (26) = −K−1 ∂ρ ∂ρ Substituting equations (25) and (26) into (22) and simplifying, we have ∂ ΦT1 K2 ∂K Φ2 . (27) = −ΦT1 ∂ρ ∂ρ Thus, equation (21) can be rewritten as ∂K ∂ (φ1 ) Φ2 = . (28) ∂ρ ∂ρ Equation (28) is then applied to determine the derivative of equation (18). In EIT, measurements are made around the body, at different points (i = 1 to np) and for all applied current patterns ( j = 1 to ne). For clarity, consider the configuration shown in figure 7 for the ith measurement point: Now, the fictitious current of equation (19), I2 , is considered to be a vector whose components are 0 .. . (φij − φij 0 ) (29) Iij = 0 .. . 0
Figure 8. Current patterns applied to all measurement points for gradient calculation.
vector Iij corresponds to the first measurement point (or node). In the domain (), the electrical current vector Iij produces a potential field Ψij . Since the electrical current (φij − φij 0 ) (ith element of vector Iij ) is constant during the current pattern, applying equation (28) we have −ΦTj
and the corresponding electrical potential Φ2 in equation (19) is denoted by Ψij . It is assumed that the first component of the
∂Kj ∂φij Ψij = (φij − φij 0 ) . ∂ρ ∂ρ
(30)
Then, if this procedure is repeated analogously for the remaining measurement points until completing np measurements and the resulting terms are added considering all the ne current patterns, we have ne ∂Kj ∂Kj ∂F ΦTj Ψ1j + ΦTj Ψ2j =− ∂ρ ∂ρ ∂ρ j =1 ∂Kj (31) + · · · + ΦTj Ψ(np)j . ∂ρ Now, grouping the similar terms of (31), we obtain ne ∂Kj ∂F =− (Ψ1j + Ψ2j + · · · + Ψ(np)j ) . ΦTj ∂ρ ∂ρ j =1
(32)
Ψj
As the FE formulation is linear, we can obtain the summation of potentials Ψj through the following equilibrium equation system: Kj Ψj = Ij ,
−ΦT1
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(φ3j − φ3j0 )
Ω
j
where
Ij = I1j + I2j + · · · + I(np)j
(φ1j − φ1j0 ) (φ2j − φ2j0 ) (φ3j − φ3j0 ) . = .. . (φ(np)j − φ(np)j0 )
(33)
(34)
This procedure is illustrated in figure 8. Therefore, considering ne applied current patterns and np measurement points, the expression to calculate the derivative of the objective function is given by ne ∂Kj ∂F ΦTj Ψj . =− (35) ∂ρ ∂ρ j =1 Equation (35) is, therefore, obtained through the adjoint method, which is more advantageous than the so-called direct method, once only 2ne FE problems must be solved instead of N (ne + 1) for the last method (Haftka et al 1996). The matrix
Topology optimization applied to EIT
electrodes
section of body electric current Figure 9. Electrical excitation in EIT. Figure 11. Numerical phantom (3072 elements) and image to be reconstructed.
to 230 mm. Note that near the boundary there is a higher refinement of the mesh (higher number of elements) to model with more accuracy the electrical potential gradients that happen near the electrodes.
(a)
(b)
Figure 10. Different FE meshes to obtain images: (a) elliptical domain with 1120 elements; (b) circular domain with 576 elements. ∂K ∂ρi
is computed directly from the element local matrices and is given by M ∂K ∂Ck ∂ρk = BT B de ∂ρi ∂ρk ∂ρi e=1 e =
nf e=1
e
BT
∂Ck Ni (x) B de , ∂ρk
(36)
where M is the number of finite elements, B is a function of the derivative of shape functions and nf is the number of elements connected to node i.
7. Results In this section, some examples with numerical and experimental data will be presented to illustrate image reconstruction, using the implemented software to evaluate its performance. In the examples, the electrical current amplitude is equal to 1 mA, which is applied to a pair of electrodes following the adjacent current pattern of electrical excitation (Cheney et al 1999), as illustrated in figure 9. In this work, 32 electrodes are uniformly attached along the boundary of the image domain. One electrode is chosen as the reference electrode. In this current injection process the pair of electrodes is successively changed (Tang et al 2002). 7.1. Using numerical data The following FE models are used for the image reconstruction of some high resistivity regions inside a low resistivity domain. Both of them use four node quadrilateral elements. The first one (see figure 10(a)) is an elliptical domain (1120 elements), whose major semi-axis is 400 mm and the minor semi-axis is 260 mm. The second one (see figure 10(b)) is a circular domain (576 elements), whose diameter is equal
7.1.1. Results obtained from the elliptical domain. The sought image is shown in figure 11 (numerical phantom), where dark regions represent a material with high resistivity (106 m). The high resistivity inclusion is meant to represent a pneumothorax in a clinical scenario. To represent air inside a thorax the value of 106 m was chosen following a previous work (Brown et al 2002). The remaining domain (clear region) is a material with low resistivity (17 m). In practice, it is equivalent to a phantom with some regions with the presence of air in a saline solution. The numerical phantom shown in figure 11, whose domain is uniformly discretized in 3072 four node quadrilateral elements (with thickness equal to 35 mm), is employed to generate the ‘measured’ electrical potentials (φij 0 ). Observe that a more refined mesh (3072 elements) is assigned to generate ‘measured’ electrical potentials and a less refined mesh (1120 elements, see figure 10(a)) is employed for image reconstruction. This saves computational time since the number of optimization variables (related to the number of mesh elements) is smaller and it avoids the so-called ‘inverse crime’ (Lionheart et al 2004) in the solution of an inverse problem. Moreover, if the software finds the sought image in a less refined mesh by using information of a more refined mesh, it is an indication that the topology optimization algorithm is able to deal with modelling errors. Electrode parameters (see section 2) must also be obtained through the implemented algorithm. An electrode parameter value equal to 0.02 m2 for all electrode elements is assigned to obtain the ‘measured’ potentials (φij 0 ) on electrodes in the numerical phantom. Using these potentials and considering the electrode parameters as optimization variables, the topology optimization algorithm obtains optimized electrode parameter values in the less refined mesh shown in figure 10(a). The optimized electrode parameter values are different from the value (0.02 m2 ) of the phantom. However, they do allow us to obtain better conductivity distribution values. For instance, the optimized electrode parameter values corresponding to results presented in the next section range from 0.012 36 to 0.035 56 m2 , with an average error of 15.7% around the value of the phantom (0.02 m2 ). The authors believe that these discrepancies are mainly due to 2853
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700
1000
.001272 .007662 .014051 .020441 .026831 .03322 .03961 .046 .052389 .058779
600
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400
300
200
200 0.2 0 −0.2
(a)
− 0.1
0 0
0.1
0.2
100
0
− 0. 2
(b)
Figure 12. Results obtained with the WDI based constraint, CAMD, spatial filtering, p = 1 and ρk0 = 0.85: (a) obtained image; (b) absolute resistivity values ( m). 1400
Objective Function
1200 1000 800 600 400 200 0 0
20
40
60
80
100
120
Iteration
Figure 13. Convergence curve of an objective function.
the differences between the numerical phantom and image reconstruction meshes. Initially, the WDI based constraint, equation (10), is applied to obtain an image, and then the tuning control, equation (9), is applied. To obtain the topology optimization results, a penalization coefficient value (power p) is set to 1, and a uniform initial value ρk0 is set to 0.85 for all nodal optimization variables at the beginning of the SLP. Using the WDI based constraint with a power q equal to 2 in equation (10), and also using the CAMD approach and the spatial filtering scheme described in sections 5.1 and 5.4, respectively, the image (conductivity distribution) shown in figure 12(a) is obtained for the elliptical domain. According to these results, the implemented algorithm is able to detect the presence of some low conductivity regions inside a high conductivity domain. Absolute resistivity values ( m) of dark region elements are shown in figure 12(b). Observe that the resistivity (or conductivity) values of elements in the dark region are not close to the expected absolute value of the numerical phantom (106 m), although the image is well defined. In all results presented here, the value of the objective function in the optimization process diminishes quickly (20 iterations approximately), as can be seen in figure 13, and then it continues with a very small oscillation until the best image is found. The tuning control is considered in the optimization problem, equation (14), as an alternative to improve the 2854
estimated resistivity (or conductivity) values. It is known that the dark region in figure 11 represents 18% (V ∗ ) of the total volume of the elliptical domain. The next result is obtained considering the same topology optimization parameters applied in figure 12 (p = 1 and ρk0 = 0.85). Then, using the tuning control, the CAMD approach and the spatial filtering scheme, the following conductivity distribution (shown in figure 14) is obtained. According to these results, the implemented algorithm is able to detect the presence of some low conductivity regions inside a high conductivity domain. Absolute resistivity values ( m) of dark region elements are shown in figure 14(b). Observe that the resistivity (or conductivity) values of some elements in dark region are found closer to the expected absolute value of the numerical phantom (106 m). Thus, the tuning contributes to improve the obtained image. However, previous information about the size or volume of the dark regions is needed. 7.1.2. Results obtained from a circular domain. In this example, the sought images are shown in figure 15, where dark regions represent a material with high resistivity (106 m) and the remaining domain represents a material with low resistivity (17 m). The objective of these examples is to evaluate the performance of the implemented algorithm in two cases. In the first case, the domain has only one circular perturbation whose volume (V ∗ ) represents 3.8% of the total volume and the centre of mass is located at x = 0 and y = 53 mm, as shown in figure 15(a). In the second case, a circular perturbation (V ∗ = 3.8%) is in the centre of the domain, as shown in figure 15(b). As numerical data are used, a numerical phantom (FE model) for each domain shown in figure 15 is constructed to model the ‘measured’ electrical potentials (φij 0 ). The domain of these numerical phantoms is uniformly discretized in 4056 four node quadrilateral elements (with thickness equal to 35 mm), as shown in figure 16. In this case, an electrode parameter value equal to 0.02 m2 for all electrode elements is assigned to obtain the ‘measured’ potentials (φij 0 ) on electrodes. For this example, the WDI-based constraint is applied to obtain the images. A power q equal to 2 is used for distance
Topology optimization applied to EIT resistivity
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.100E-05 .006534 .013067 .019601 .026134 .032667 .039216 .045734 .052267 .058824
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Figure 15. Images to be reconstructed: (a) one circular region (V ∗ = 3.8%) at x = 0 and y = 53 mm in the domain; (b) one circular region (V ∗ = 3.8%) at the centre of the domain (x = 0 and y = 0).
(a)
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Figure 14. Results obtained with tuning control, CAMD, spatial filtering, p = 1 and values ( m).
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Figure 16. Numerical phantom with 4056 quadrilateral elements for each domain shown in figure 15: (a) one circular region at y = 53 mm; (b) one circular region at the centre of the domain.
d(x) in equation (10), except for the model of one circular region at the centre of the domain, in which a power q equal to 4 was assigned to balance conductivity values in the central region of the domain. However, the sensitivity in the central region of the domain is poor (Bacrie et al 1997). Thus, a higher power q is required to make the constraint stronger. Additionally, for each case shown in figure 15, the topology optimization algorithm is tested by varying the penalization coefficient p from 1 to 6. The optimization begins using p = 1 and during the iteration process the value of p is changed consecutively until it reaches 6. This strategy (known as the continuation method) is applied to avoid multiple local minima of the problem (Rozvany et al 1995).
ρk0
= 0.85: (a) obtained image; (b) absolute resistivity
In the examples, 0.85 is used as uniform initial value ρk0 for the nodal optimization variables at the beginning of the optimization process. Our goal is to compare the volume (size) and cartesian coordinate of the centre of the object in relation to the phantom shown in figure 15. Using the estimated electrode parameters, the spatial filtering technique, equation (15), the CAMD (see section 5.1), and the mesh shown in figure 10(b), the images shown in figures 17 and 18 were obtained. The size (V ∗ ) and location (x, y) of the dark region in the obtained images were computed taking into account the value of the resistivity of each node. Note that in each image the location (centre of mass coordinate values of all elements in dark region) is given in millimetres. According to these results, the implemented algorithm is able to detect the presence of high resistivity with reduced size and location errors. The obtained size for the dark region shown in figure 17(a) is 15.8% less than the expected value, considering only nodal optimization variable values larger than 97% of maximum expected value (ρ(x)k = 1). Its location in the domain (x = 3.3 mm, y = 45.8 mm) was found with an error of about 7% in relation to the expected value for the vertical location (y = 53 mm). The obtained size of the dark region shown in figure 18(a) is only 5.2% larger than the expected value and its location was found with a minor error in relation to the x direction (x = 6.7 mm). The convergence curves relative to the results shown in figures 17 and 18 are shown in figure 19, which are similar to the results presented in section 7.1.1, i.e., the objective function diminishes quickly to a minimum value (10 iterations approximately) and continues with a very small oscillation until the best image is found. Moreover, the absolute resistivity values of most elements of dark and clear regions were found to be close to the expected value of the numerical phantom, as shown in figures 17(b) and 18(b). Considering only the nodal optimization variables equal to maximum value (ρ(x)k = 1), which indicates the presence of a perturbation, the resistivity values are equal to the expected value (106 m). It can be observed that the application of the spatial filtering reduces abrupt changes of optimization variables on regions next to the perturbation yielding resistivity values for dark regions less than the expected value. The spatial filtering is turned off five iterations before the end of 2855
C R de Lima et al resistivity
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.100E-05 .006537 .013073 .019609 .026145 .03268 .039216 .045752 .052288 .058824
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Figure 17. Results obtained with a WDI-based constraint, p value varying from 1 to 6, spatial filtering and CAMD: (a) obtained image (V ∗ = 3.2%; x = 3.3; y = 45.8); (b) absolute resistivity values ( m).
resistivity
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Figure 18. Results obtained considering a WDI-based constraint, p value varying from 1 to 6, spatial filtering and CAMD: (a) obtained image (V ∗ = 4.0%; x = 6.7; y = 0); (b) absolute resistivity values ( m).
objective function
1500
1000
500
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iteration Figure 19. Convergence curve of the objective function.
the optimization process to free the algorithm of this constraint after a minimum is found. 7.2. Using experimental data The performance of the implemented algorithm is evaluated to identify the position of a glass object, using data obtained from 2856
an experimental phantom. In this example, the experimental phantom is a cylindrical container, whose diameter is equal to 230 mm and where 32 electrodes (10 mm width) are uniformly attached along its boundary, as shown in figure 20. The container of this phantom was filled up to 35 mm with a saline solution of 0.3 g l−1 of NaCl, whose resistivity is equal to 17 m. The diameter of the glass object is equal to 60 mm, the volume represents 6.8% of the total volume of the domain, and the maximum resistivity inside the domain, which is considered here as prior information for the EIT inverse problem, is equal to 106 m. Figure 20 illustrates this experimental phantom, in which the glass object (circular region in the domain) is positioned between electrodes 5 and 9. The 2D mesh shown in figure 10(b), with quadrilateral elements, is used for image reconstruction. The algorithm must be able to estimate the unknown conductivity distribution inside the phantom, and the electrode parameters, which range from 0.0001 to 0.0042 m2 in this case. Figure 21 shows the results obtained using the WDI based constraint with a power q in equation (10) equal to 2, a penalization coefficient value p varying from 1 to 6 consecutively (continuation method),
Topology optimization applied to EIT
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electrodes 1
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glass object
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Figure 20. (a) Experimental phantom illustration; (b) experimental phantom scheme with a glass object near electrodes 5 and 9.
resistivity
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.100E-05 .006537 .013073 .019609 .026145 .03268 .039216 .045752 .052288 .058824
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Figure 21. Results obtained using experimental data, considering WDI based constraint, p varying from 1 to 6, spatial filtering and CAMD: (a) obtained image; (b) resistivity distribution ( m).
the spatial filtering and the CAMD. In this example, 0.85 was applied as the initial value ρk0 for nodal optimization variables at the beginning of the optimization process. The objective function follows the standard behaviour detected in all results shown in the numerical examples (section 7.1); it diminishes quickly to a minimum value in about 12 iterations. According to the result, shown in figure 21, the implemented algorithm is able to detect the glass object (dark region) inside the phantom. The location of the glass object has been found between electrodes 5 and 9, as expected. The algorithm indicates that its coordinate values are x = 26 mm and y = 62 mm. The size (volume) of the image of the glass object was 2.2% (V ∗ ), smaller than the size of the glass object. The absolute conductivity (or resistivity) values of most elements in the dark and clear regions (that represent the glass object and saline medium) are close to the expected value. The resistivity values of some elements in the dark region were found equal to the expected value (106 m), as shown in figure 21(b), which demonstrates that the implemented algorithm has the potential to reconstruct images of domains with large resistivity amplitude.
8. Conclusion A computational algorithm that applies the topology optimization method to obtain images in electrical impedance
tomography (EIT) was proposed. A new formulation to compute the derivatives of the objective function was presented, taking advantage of Maxwell’s reciprocity theorem. This formulation is more advantageous than the direct method (Haftka et al 1996), as mentioned in the text. The implemented software was tested using numerical and experimental data, and it is able to obtain unknown resistivities of two materials with sharp spatial gradients and high differences between the maximum and minimum resistivity values in the domain. According to the results, the algorithm is able to detect the position within 7% accuracy of one or two perturbations inside an elliptical or cylindrical domain, such as glass objects in a container of saline solution. In the results relative to a domain that has two high resistivity regions (elliptical domain), the images show the robustness of the implemented algorithm to detect complex geometries inside the domain. Although the size of the image of the object (dark region) was smaller than expected, it was improved by the inclusion of some constraints in the topology optimization problem. To demonstrate this, constraints based on tuning control, WDI, spatial filtering and CAMD were applied and yielded better results than without these constraints. In the near future, the implemented algorithm will be evaluated to obtain thorax images through an EIT device for monitoring mechanical ventilation of lungs. 2857
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Acknowledgments The first author thanks the Research Foundation of S˜ao Paulo State (FAPESP) for supporting this research through a doctoral fellowship (no 2002/01625-0). All authors are grateful for the research project support from FAPESP, which was granted through a Thematic Project (no 2001/05303-4).
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