Chapter 7

Electrical Impedance Tomography

Electrical impedance tomography (EIT) produces cross-sectional images of an admittivity distribution inside an electrically conducting object. It has a wide range of applications in biomedicine, geophysics, non-destructive testing and so on. Considering the fact that structural imaging modalities such as X-ray CT and MRI provide images with a superior spatial resolution to EIT, the primary goal of biomedical EIT is to supply functional diagnostic information of organs with a high temporal resolution. It may provide diagnostic information on functional and pathological conditions of biological tissues and organs. Following a brief introduction to EIT, we summarize bioimpedance measurement methods, on which an EIT system is based, to acquire data for image reconstruction. Its forward problem is introduced in the context of a practically feasible measurement setting. Modeling of the forward problem and sensitivity analysis will be the key to understanding and designing an inversion method. Three kinds of EIT inverse problems, including static imaging, time-difference imaging and frequency-difference imaging, will be described.

7.1 Introduction

The material properties of electrical conductivity and permittivity may produce image contrast in EIT. The conductivity (σ) and permittivity (images) values of a biological tissue are determined by its ion concentrations in extra- and intracellular fluids, cellular structure and density, molecular composition, membrane characteristics and other factors. In the frequency range of a few hertz to megahertz, numerous experimental findings indicate that different biological tissues have different electrical properties, and their values are influenced by physiological and pathological conditions (Gabriel et al. 1996a,b; Geddes and Baker 1967; Grimnes and Martinsen 2008). In biomedical applications of EIT, we deal with the admittivity images, where the angular frequency images is in rad s−1 with the frequency f in Hz. For most biological tissues, we may assume that γ ≈ σ at low frequencies below 10 kHz. With abundant membraneous structures in an organism, the images term is not negligible beyond 10 kHz and we should deal with the admittivity images in general at high frequencies.

We consider an electrically conducting object such as the human body with its internal admittivity distribution γ(r) as a function of position r = (x, y, z). To probe the object with the intention of non-invasively sensing γ, we inject current through electrodes attached on its surface. This induces internal current density and voltage distributions that are determined by the admittivity distribution, object geometry and electrode configuration. In the frequency range up to a few MHz, we may adopt the elliptic partial differential equation (PDE) introduced in Chapter 3 to describe the interrelations among the injection current, current density and voltage. By measuring induced voltages on the surface subject to multiple injection currents, an EIT system produces images of the internal admittivity distribution using an inversion method.

Mathematical theory has been developed to support such an EIT system especially for the unique identification of the conductivity σ from knowledge of all possible boundary current-to-voltage data at low frequencies where we can assume γ ≈ σ (Astala and Paivarinta 2006a,b; Brown and Uhlmann 1997; Calderón 1980; Kenig et al. 2007; Kohn and Vogelius 1984; Nachman 1988; Nachman 1996; Sylvester and Uhlmann 1986, 1987, 1988). After the early attempt to build an EIT system (Barber and Brown 1984), numerous studies have accumulated knowledge and experience, summarized in the fairly recently published book on EIT (Holder 2005). The nonlinear inverse problem in EIT suffers from its ill-posedness, related to lack of enough measurable information and insensitivity of measured data to a local change of an internal admittivity value. Though there exist numerous image reconstruction algorithms (Barber and Brown 1984; Berenstein et al. 1991; Brown et al. 1985; Cheney et al. 1990; Fuks et al. 1991; Gisser et al. 1988, 1990; Isaacson and Cheney 1991; Isaacson et al. 1989, 1996; Lionheart et al. 2005; Newell et al. 1988; Santosa and Vogelius 1990; Somersalo et al. 1992; Wexler et al. 1985; Yorkey 1987), it is difficult to reconstruct accurate admittivity images with a high spatial resolution in a practical setting, where modeling and measurement errors are unavoidable. In this chapter, we focus on robust image reconstructions that may overcome the technical difficulties of the ill-posedness.

7.2 Measurement Method and Data

7.2.1 Conductivity and Resistance

We consider a cylinder filled with saline. The saline contains mobile charged ions, and their migration under an external electric field characterizes its conductivity σ (siemens per meter, S m−1). Attaching two electrodes on the top and bottom surfaces, we measure its resistance R (ohms, Ω). Neglecting interfacial phenomena between each electrode and the saline, the resistance R is denoted as

7.1 7.1

where L and A are the length (m) and cross-sectional area (m2) of the cylinder, respectively. If we inject DC current I (amperes, A), the induced DC voltage V (volts, V) follows Ohm's law as

7.2 7.2

Injecting a known DC current I and measuring the induced DC voltage, we may find the resistance R, as is done in an electrical multimeter. If we have geometrical information about L and A, we can find the conductivity σ. For materials such as biological tissues, we denote the conductivity as images to emphasize its frequency dependence. We may measure images by injecting a sinusoidal current images to measure the induced AC voltage images, where t is the time (seconds, s). Assuming a linear component, the resistance R at images also follows Ohm's law as

7.3 7.3

Note that the current and voltage are in phase. Repeating this measurement for multiple frequencies, we may get a conductivity spectrum, which plots conductivity images as a function of frequency images.

7.2.2 Permittivity and Capacitance

We consider a dielectric sandwiched between two parallel conducting plates. When we apply a DC voltage V between the plates, it induces an electric field inside the dielectric. The dielectric contains immobile charges, and their polarization or rotation in the electric field produces surface charges Q and − Q (coulombs, C). The induced charge is proportional to the applied voltage as

7.4 7.4

where the proportionality constant C is called the capacitance (coulombs per volt, C V−1; or farad, F). The capacitance C between the two plates is given by

7.5 7.5

where images is the permittivity (F m−1), A the surface area and d the gap between the plates. The permittivity is a material property determined by the polarization of the dielectric under an external electric field. For most dielectrics, including biological tissues, the permittivity changes with frequency, and we denote it as images.

If we assume a perfect dielectric, there is no mobile charge and its conductivity σ is zero. Applying DC voltage V to the dielectric, we get zero DC current through it. If we apply a sinusoidal voltage images, there occurs an AC displacement current through the dielectric due to time-varying polarizations with frequency images:

7.6 7.6

Note that the current and voltage are out of phase by 90° or the voltage is in quadrature with the current. Assuming that there is no polarization initially, we can express the induced voltage v(t) subject to an injection current i(t) as

7.7 7.7

With known images and I, we may find the capacitance C in farads (F), which equals A s V−1 or s Ω−1. If we have geometrical information on A and d, we can find the permittivity images (F m−1). Repeating this measurement for multiple frequencies, we may get a permittivity spectrum, which plots permittivity images as a function of frequency images.

7.2.3 Phasor and Impedance

Given an electrically conducting object with both mobile and immobile charges, we may view it as a mixture of resistors and capacitors. In this section, we adopt a circuit model using lumped elements since this provides intuitive understanding about the continuum model. Let us consider the series RC circuit in Figure 7.1(a). Injecting a sinusoidal current images, we can express the induced voltage v(t) across the series connection of R and C as

7.8 7.8

where

images

Figure 7.1 (a) Series, (b) parallel and (c) series–parallel RC circuits

7.1

Figure 7.2 shows current i(t) and voltage v(t). Noting that there is no change in frequency between current and voltage for all linear components, we adopt the phasor notation. The current and voltage phasors are defined as complex numbers I = I∠0 and V = V∠θ, so that we can recover time functions i(t) and v(t) from

7.9 7.9

respectively. Using the phasor notation, we can handle the relation between time functions i(t) and v(t) as an algebraic equation instead of an integrodifferential equation of time.

Figure 7.2 Current and voltage waveforms

7.2

We define the impedance Z (Ω) as the ratio of the voltage phasor V to the current phasor I, and it is a measure of the total opposition to current flow through a component or a collection of components. For the case of the series RC circuit, the total impedance is

7.10 7.10

The real part of Z is the resistance (Ω) and its imaginary part is the reactance (also Ω). Note that, for a single resistor R, its impedance ZR = R. For a single capacitor C, images.


Example 7.2.1
For the series RC circuit in Figure 7.1(a) with images and C = 0.1 μF, compute its impedance Z at a frequency of 10 kHz. For images, find v(t) and plot both i(t) and v(t) for images.

 


Example 7.2.2
For the parallel RC circuit in Figure 7.1(b) with images and C = 0.1 μF, compute its impedance Z at a frequency of 10 kHz. For images, find v(t) and plot both i(t) and v(t) for images.

 


Example 7.2.3
For the series–parallel RC circuit in Figure 7.1(c) with images, images and images, compute its impedance Z at a frequency of images. For images, find v(t) and plot both i(t) and v(t) for images.

 


Example 7.2.4
For the series RC circuit in Figure 7.1(a) with images and images, plot the magnitude and phase of its impedance Z = Z∠θ in the frequency range of 1 Hz to 1 MHz.

 


Example 7.2.5
For the parallel RC circuit in Figure 7.1(b) with images and images, plot the magnitude and phase of its impedance Z = Z∠θ in the frequency range of 1 Hz to 1 MHz.

 


Example 7.2.6
For the series–parallel RC circuit in Figure 7.1(c) with images, images and images, plot the magnitude and phase of its impedance Z = Z∠θ in the frequency range of 1 Hz to 1 MHz.

7.2.4 Admittivity and Trans-Impedance

When we consider a material including both mobile and immobile charges, its electrical property is expressed as the admittivity γ (S m−1). To express its frequency dependence, we denote it as images. Note that images and images have the same unit (S m−1). We now assume a cylinder filled with a biological tissue whose admittivity is images. The impedance Z between the top and bottom surfaces is

7.11 7.11

where L and A are the length and cross-sectional area of the cylinder, respectively. If images, that is, images, then images and the material is resistive. If images, that is, images, then images and the material is reactive or capacitive.

Most biological tissues are resistive at low frequencies of less than 10 kHz, for example. Since the capacitive term is not negligible beyond 10 kHz, we will denote the admittivity of a biological tissue at position r as images. We assume an electrically conducting domain Ω with its admittivity distribution images, as illustrated in Figure 7.3. Attaching E electrodes images, we inject current images through a pair of electrodes images and images. Between another pair of electrodes images and images, we measure the induced voltage images. We define the trans-impedance from the jth port to the kth port as

7.12 7.12

In section 7.4, we will show that the admittivity distribution images, domain geometry and electrode configuration affect the trans-impedance Zj, k. The reciprocity principle explained in section 7.4 indicates that Zj, k = Zk, j.

Figure 7.3 Measurement of trans-impedance: (a) Z8, 15 and (b) Z15, 8. From the reciprocity principle described in section 7.4, we have Z8, 15 = Z15, 8

7.3

7.2.5 Electrode Contact Impedance

To inject current and measure voltage, we use electrodes. An electrode is made of a highly conductive material such as copper, silver, platinum and others. Carbon is also used to make a flexible electrode, though its conductivity is not as large as for metallic conductors. When the electrode makes contact with an electrolyte or the skin of an organic object, the interface can be modeled as a contact impedance and a contact potential in series. The contact impedance includes both resistive and reactive terms, and its typical circuit model is the series–parallel RC circuit in Figure 7.1(c). As long as the interface is mechanically stable, the contact potential is stable and less than 1 V for most electrode materials.

We consider a method to measure the impedance Z of a cylinder with homogeneous admittivity images. Attaching a pair of electrodes at the top and bottom surfaces, we inject current I at images from the top to the bottom electrode. Denoting the contact impedances of the top and bottom electrodes as images and images, respectively, the induced voltage will be expressed as

7.13 7.13

assuming that no current flows into the ideal voltmeter. We can ignore the DC contact potential since we measure only the induced voltage at frequency images. Using this two-electrode or bipolar method shown in Figure 7.4(a), it is not possible to extract only Z since two contact impedances are in series with Z.

Figure 7.4 Impedance measurements using (a) two-electrode or bipolar method and (b) four-electrode or tetrapolar method. No current flows through the ideal voltmeter

7.4

By attaching another pair of electrodes around the cylinder near its top and bottom, we inject current through the first pair and measure the induced voltage between the second pair as shown in Figure 7.4(b). Using a well-designed voltmeter, we may safely assume that there is no current flowing through the second pair of voltage-sensing electrodes. This means that the voltmeter sees only the voltage drop across the impedance of the cylinder Z between the second pair of electrodes as

7.14 7.14

This four-electrode or tetrapolar method allows us remove the effects of contact impedances in bioimpedance measurements.

7.2.6 EIT System

We consider an imaging domain Ω with its admittivity distribution images. We attach E electrodes images on its boundary ∂Ω. We use an EIT system equipped with current sources and voltmeters to measure trans-impedances or equivalent current–voltage data sets. We may do this for multiple frequencies at different times. A typical EIT system comprises one or multiple current sources, one or multiple voltmeters, optional switching networks, a computer system and a DC power supply. The computer controls current sources, voltmeters and switches to acquire current–voltage data sets. It produces images of images and/or images by applying an image reconstruction algorithm to the data sets.

There are several EIT systems with different design concepts and technical details in their implementations. The number of electrodes used in available EIT systems ranges from eight to 256. The human interface gets complicated with a large number of electrodes and lead wires. With a large number of electrodes, the induced voltage between a pair of electrodes tends to become small, since the gap between them gets smaller. In chest imaging, eight or 16 electrodes are commonly used, while more electrodes are used in head or breast imaging.

We may classify recent EIT systems into two types. The first is characterized as one current source with switching networks. In this case, current is sequentially injected between a chosen pair of electrodes and there always exists only one active current source. The second type uses multiple current sources without any switching for current injection. With this type, one may inject a pattern of current through multiple electrodes using multiple active current sources. The sum of currents from all active current sources must be zero. In most EIT systems belonging to both types, voltages between many electrode pairs are simultaneously measured using multiple voltmeters. Typical examples of the first and second types are Mk3.5 from Sheffield (Wilson et al. 2001) and ACT3 from RPI (Cook et al. 1994), respectively. Boone et al. (1997) and Saulnier (2005) summarized numerous techniques in the development of EIT systems. Figure 7.5 shows examples of EIT systems and their use for chest imaging (Oh et al. 2007a,b, 2008).

Figure 7.5 EIT systems: (a) and (b) are KHU Mark 1 16- and 32-channel multi-frequency EIT systems, respectively, and (c) is a set-up for chest imaging

7.5

The range of the trans-impedance is from a few milliohms (mΩ) to tens of ohms depending on the imaging object, number of electrodes and their configuration. Assuming injection currents of 1 mArms, for example, induced voltages are in the range of a few microvolts to tens of millivolts. Allowing a noise level of 1% of the smallest voltage, we should restrict the level below 0.1 μV and this requires state-of-the-art electronic instrumentation technology. Modern EIT systems usually acquire a complete set of current–voltage data within 10 ms for frequencies higher than 10 kHz. Temporal resolutions could be higher than 20 frames per second using a fast image reconstruction algorithm.

7.2.7 Data Collection Protocol and Data Set

A data collection protocol defines a series of injection currents and corresponding voltage measurements. In this section, we introduce only the neighboring protocol. One may find numerous data collection protocols in Holder (2005) and other literature on EIT. We assume an EIT system with E electrodes. Injecting the jth current between an adjacent pair of electrodes images and images, we measure induced boundary voltages between all neighboring pairs of electrodes images and images for k = 1, 2, …, E. Any index number must be understood as a modulus of the maximal value of the index number. We define this data set as a projection, a term that has its origin in the X-ray CT area. Repeating this for all pairs of current injection electrodes with j = 1, 2, …, E, we can obtain a full set of data from E projections. The kth boundary voltage phasor in the jth injection current or the jth projection is denoted as

images

for j, k = 1, 2, …, E. Since the number of injection currents or projections is E and the number of boundary voltage phasors per projection is also E, the full data set includes E2 boundary voltage phasors.

From the reciprocity theorem introduced in section 7.4 and Kirchhoff's voltage law, only E × (E − 1)/2 boundary voltage data are independent. This is the maximal amount of measurable information using E electrodes regardless of the adopted data collection protocol. This imposes a fundamental limit on the achievable spatial resolution in EIT using E electrodes regardless of the inversion method.

For each injection current between a chosen pair of neighboring electrodes, boundary voltage data between three adjacent pairs of electrodes are involved with at least one current injection electrode. These three voltage data contain the effects of unknown contact impedances between the electrodes and the skin. We may discard or include these data depending on the way in which contact impedances are treated in the chosen inversion method and electrode model, as discussed in section 7.4.

Figure 7.6 shows examples of the neighboring protocol assuming a 16-channel EIT system. For each projection, 13 boundary voltage phasors between adjacent pairs of electrodes are measured to adopt the four-electrode method. In this example, the number of projections is 16 and the total number of measured boundary voltage phasors is 16 × 13 = 208. Among them, only 104 boundary voltage phasors carry independent information. This indicates that the best spatial resolution of a reconstructed admittivity image will be about 10% of the size of the imaging object using a 16-channel EIT system with the neighboring protocol. Using a 32-channel system, we may improve it to 5%.

Figure 7.6 Neighboring data collection protocol of a 16-channel EIT system: (a) first projection with the injection current I1 between images and images; and (b) second projection with the injection current I2 between images and images

7.6

We can collect boundary voltage data at multiple frequencies for a certain period of time. Assuming that we collected E2 number of boundary voltage data at each sampling time t and frequency images, we can express the boundary voltage data set in matrix form as

7.15 7.15

Alternatively, we may adopt a column vector representation as

7.16 7.16

where the superscript T means the transpose. This column vector representation will be used in sections where we study image reconstruction algorithms. For images, we may collect F data vectors or matrices for each sampling time t = t1, t2, …, tN of total N times.

7.2.8 Linearity between Current and Voltage

Before we move on to mathematical topics in EIT, we note the linear relationship between injection currents and boundary voltages. We assume that the internal admittivity distribution images, domain geometry and electrode configuration are all fixed. For an injection current Ij or the jth projection, we measure E boundary voltage phasors Vj, k for k = 1, 2, …, E to form the jth projection data vector images as

7.17 7.17

We now inject current I as

7.18 7.18

with some real constants αj for j = 1, 2, …, E. The corresponding projection data vector images is expressed as

7.19 7.19

This stems from the linearity between injection currents and induced voltages when we view the imaging object as a mixture of linear resistors and capacitors.

7.3 Representation of Physical Phenomena

We assume an imaging object occupying a domain Ω with its boundary ∂Ω and an internal admittivity distribution γ(r). Using an E-channel EIT system, we attach E surface electrodes images for j = 1, …, E on ∂Ω and inject current images through an adjacent pair of electrodes as shown in Figure 7.6. We assume that the current source and sink are connected to electrodes images and images, respectively. The injection current produces internal current density and magnetic flux density distributions, which are dictated by Maxwell's equations, as in Table 7.1. Table 7.2 summarizes the variables used in Maxwell's equations.

Table 7.1 Maxwell's equations for time-varying and time-harmonic fields

Name Time-varying field Time-harmonic field
Gauss's law images images
Gauss's law for magnetism images images
Faraday' law of induction images images
Ampère's circuit law images images

Table 7.2 Variables to describe time-harmonic and time-varying electromagnetic fields

NumberTable

In the frequency range of a few hertz to megahertz, we adopt the elliptic PDE studied in Chapter 3 to describe the forward problem in EIT. From Maxwell's equations, we derive the elliptic PDE and its boundary conditions. After analyzing the PDE in terms of its min–max property, we formulate the EIT forward problem and its model.

7.3.1 Derivation of Elliptic PDE

To simplify mathematical derivations, we assume that the admittivity images in Ω is isotropic, σ > 0 and images < ∞. For some biological tissues, such as muscles and neural tissues, the isotropy assumption is not valid, especially at low frequencies. We assume that the magnetic permeability μ of the imaging object is μ0, the magnetic permeability of free space.

In the frequency range of a few hertz to megahertz, we neglect the Faraday induction to get

images

Since E is approximately irrotational, it follows from Stokes's theorem that we can define a potential u between any two points r1 and r2 as

images

where images is a curve in Ω joining the starting point r1 to the ending point r2. The complex potential u satisfies

images

From images, we have the following relation:

images

Since images, the complex potential u satisfies the following elliptic PDE with a complex parameter γ:

7.20 7.20

Note that the complex potential u is equivalent to the voltage phasor introduced in section 7.2. In the rest of this chapter, we denote u as the voltage phasor or time-harmonic voltage.

7.3.2 Elliptic PDE for Four-Electrode Method

Using the four-electrode method, we can neglect the contact impedance introduced in section 7.2. Investigating the boundary ∂Ω of the imaging object Ω with attached electrodes images with k = 1, 2, …, E, we can observe the following.

Current injection electrodes. Since the total injection current spreads over each current injection electrode,

images

where n is the unit outward normal vector and ds the surface element on ∂Ω.
Boundary without any electrode. Since the air is an insulator,

images

Voltage-sensing electrodes. Since there is no current flowing into a voltmeter,

images

All electrodes. Since u is approximately constant on each electrode with a very high conductivity,

images

From these observations, we can derive the following boundary conditions for the time-harmonic potential u in (7.20).
BC 1: images.
BC 2: images on images.
BC 3: images for images.
BC 4: images on images for k = 1, 2, …, E.

We define g as

7.21 7.21

and call it the Neumann data of u. In practice, it is difficult to specify the Neumann data g in a pointwise sense because only the total injection current I is known. Note that the Neumann boundary data g have a singularity along the edge of each electrode and gL2(∂Ω). Fortunately, we can prove that gH−1/2(∂Ω) by the standard regularity theory in PDE. The total injection current through the electrode images is images. The condition images ensures that images is approximately a constant for each electrode since images is normal to its level surface.

Expressing the boundary conditions by g, the time-harmonic voltage u is governed by

7.22 7.22

Since g is the magnitude of the current density on ∂Ω due to the injection current, g = 0 on images and images. Setting a reference voltage u(r0) = 0 for a fixed point r0 ∈ Ω, we can obtain a unique solution u of (7.22) from γ and g. Note that u depends on γ, g and the geometry of Ω. When γ changes with images, so does u.


Example 7.3.1
Assume that the electrodes are perfect conductors and images. The potential images satisfies

7.23 7.97

The above non-standard boundary value problem is well-posed and has a unique solution within H1(Ω) up to a constant. Figure 7.7 illustrates a numerical example.

Figure 7.7 (a) An example of an electrically conducting domain with a given conductivity distribution. Numbers inside ellipsoids are conductivity values (S m−1). (b) Voltage and current density distributions induced by the injection current. Black and white lines are equipotential and current density streamlines, respectively

7.7

 


Example 7.3.2
Assume that u is a solution of (7.23). Then, images is singular at the edge of a current injection electrode. To estimate this singularity, we consider the simplified model images and images. Let w be the H1(Ω)-solution of the following mixed boundary value problem:

7.24 7.98

Let images. Prove that g satisfies the integral equation:

7.25 7.99

Find a representation formula for w and find the behavior of g near the circular edge of the electrode images.
Solution. Assume that g satisfies (7.25). Define

images

where images is the Neumann function

images

It is easy to see that UH1(Ω) satisfies

images

It follows from the uniqueness theory that the solution w of (7.24) must be U = w in Ω. Since g is radial, images satisfies

images

where

images

Noting that

images

for 0 < t < 1, one can show that images.

7.3.3 Elliptic PDE for Two-Electrode Method

When we adopt the two-electrode method where we measure voltages on current injection electrodes, we must take into account of the contact impedance. We introduce the complete electrode model (Cheng et al. 1989; Somersalo et al. 1992; Vauhkonen et al. 1996), where the complex potential u satisfies

7.26 7.23

where zk is the contact impedance of the kth electrode images and Uk is the voltage on images. Setting a reference voltage having images, we can obtain a unique solution u of (7.26).

In this case, measured boundary voltages are

images

Using an E-channel EIT system, we may inject E number of currents through adjacent pairs of electrodes and measure the following voltage data set:

images

The voltage data are influenced by contact impedances whose values are unknown. Since the reciprocity principle Vk, j = Vj, k in section 7.4 still holds, images contains at most E(E − 1)/2 number of independent data.

7.3.4 Min–Max Property of Complex Potential

The variational form of the problem (7.22) with the Neumann boundary condition is

7.27 7.24

According to the Lax–Milgram theorem in Chapter 4, for a given gH−1/2(∂Ω) with ∫∂Ωg ds = 0, there exists a unique solution uH1(Ω) with ∫∂Ωu ds = 0 satisfying (7.27). When images, we can figure out the global structure of images using its weighted mean value property, maximum principle and minimization property of the corresponding energy functional:

7.28 7.25

When images, the potential images does not have the minimization property (7.28), mean value property and maximum principle. Denoting images and images, u = v + ih satisfies the following coupled system:

7.29 7.26

The complex potential u has the min-max property (Cherkaeva and Cherkaev 1995) in the sense that

7.30 7.27

and

7.31 7.28

7.4 Forward Problem and Model

We describe the forward problem of EIT using the Neumann-to-Dirichlet (NtD) data, which depend on the admittivity γ. After introducing the continuous NtD data and some theoretical issues, we formulate the discrete NtD data of an E-channel EIT system.

7.4.1 Continuous Neumann-to-Dirichlet Data

We define the continuous NtD data set Λγ as

7.32 7.29

7.33 7.30

where images is the unique solution of the Neumann boundary value problem

7.34 7.31

This NtD data Λγ include all possible Cauchy data. With this full data set, the forward problem of EIT is modeled as the map

7.35 7.32

and the inverse problem is to invert the map in (7.35).

There are two major theoretical questions regarding the map.

Uniqueness: Is the map γ → Λγ injective?
Stability: Find the estimate of the form:

images

where || · ||* is an appropriate norm for the admittivity, images is a continuously increasing function with Ψ(0) = 0, and images is the operator norm on images.

The NtD data Λγ are closely related with the Neumann function restricted on ∂Ω. The Neumann function images is the solution of the following Neumann problem: for each r,

images

where δ is the Dirac delta function. With the use of the Neumann function images, we can represent images in terms of the singular integral:

images

Since Λγ is the restriction of images to the boundary ∂Ω, we can represent it as

7.36 7.33

The kernel images with r, r′ ∈ ∂Ω can be viewed as an expression of the NtD data Λγ. Note that Λγ is sensitive to a change in the geometry of the surface ∂Ω since images is singular at r = r′.

For the uniqueness in a three-dimensional problem, Kohn and Vogelius (1985) showed the injectivity of γ → Λγ if γ is piecewise analytic. Sylvester and Uhlmann (1987) showed the injectivity if images. The smoothness condition on γ and ∂Ω has been relaxed by several researchers (Astala and Paivarinta 2006b; Brown and Uhlmann 1997; Isakov 1991; Nachman 1988, 1996).

For a two-dimensional problem, Nachman (1996) proved the uniqueness under some smoothness conditions on γ and provided a constructive way of recovering γ. Based on Nachman's proof on two-dimensional global uniqueness, Siltanen et al. (2000) developed the d-bar algorithm, which solves the full nonlinear EIT problem without iteration.

To reconstruct γ by inverting the map (7.35), it would be ideal if the full continuous NtD data Λγ are available. In practice, it is not possible to get them due to a limited number of electrodes with a finite size. It is also difficult to capture the correct geometry of ∂Ω at a reasonable cost. The map in (7.35) is highly nonlinear and insensitive to a local change of γ, as explained in section 7.4.3. All of these hinder a stable reconstruction of γ with a high spatial resolution.

7.4.2 Discrete Neumann-to-Dirichlet Data

We assume an EIT system using E electrodes images for j = 1, 2, …, E. The isotropic admittivity distribution in Ω is denoted as γ. The complex potential u in (7.22) subject to the jth injection current between images and images is denoted as uj and it approximately satisfies the following Neumann boundary value problem:

7.37 7.34

where images and the Neumann data gj are zero on the boundary regions not contacting with the current injection electrodes. Setting a reference voltage at r0 ∈ Ω as uj(r0) = 0, we can obtain a unique solution uj from γ and gj.

We assume the neighboring data collection protocol in section 7.2 to measure boundary voltages between adjacent pairs of electrodes, images and images for k = 1, 2, …, E. The kth boundary voltage difference subject to the jth injection current is denoted as

7.38 7.35

where images can be understood as the average of uj over images.


Lemma 7.4.1
The kth boundary voltage difference subject to the jth injection current satisfies

7.39 7.36


 


Proof.
Integration by parts yields

images

where the last identity comes from the boundary condition (7.37).

Since Vj, k[γ] is uniquely determined by the distribution of γ, it can be viewed as a function of γ. With E projections and E complex boundary voltage data for each projection, we are provided with E2 complex boundary voltage data, which are expressed in matrix form as

7.40 7.37

where Vj, k = Vj, k[γ] for a given γ.


Theorem 7.4.2 (Reciprocity of NtD data)
For a given γ, Vj, k in (7.38) satisfies the reciprocity property:

7.41 7.38


 


Proof.
The reciprocity follows from the identity:

images


 


Observation 7.4.3
Assume that γ is constant or homogeneous in Ω. Then,

7.42 7.39

where wj is the solution of (7.37) with γ = 1.

 


Proof.
Since wj = γuj, we have

images


The data matrix images in (7.40) can be viewed as a discrete version of the NtD data since it provides all the measurable current-to-voltage relations using the E-channel EIT system. With this discrete NtD data set, the forward problem of the E-channel EIT is modeled as the map

7.43 7.40

and the inverse problem is to invert the map in (7.43).

The smoothness condition on γ should not be a major issue in a practical EIT image reconstruction. For any discontinuous admittivity γ and an E-channel EIT system, we always find images, which approximates γ in such a way that

images

Taking account of inevitable measurement noise in the discrete NtD data and the ill-posedness of its inversion process, we conclude that γ and images are not distinguishable in practice.

7.4.3 Nonlinearity between Admittivity and Voltage

As defined in (7.43), the forward model is a map from the admittivity to a set of boundary voltage data. From (7.37), we can see that any change in the admittivity influences all voltage values. Unlike the linear relation between currents and voltages, the map in (7.43) is nonlinear. Understanding the map should precede designing a method to invert it.

A voltage value at a point inside the domain can be expressed as a weighted average of its neighboring voltages, where the weights are determined by the admittivity distribution. In this weighted averaging method, information on the admittivity distribution is conveyed to the boundary voltage, as shown in Figure 7.8. The boundary voltage is entangled with the global structure of the admittivity distribution in a highly nonlinear way, and we investigate the relation in this section.

Figure 7.8 Nonlinearity and insensitivity grow exponentially as the matrix size increases

7.8

We assume that the domain Ω is a square in images with its conductivity distribution σ, that is, γ = σ. We divide Ω uniformly into an N × N square mesh. Each square element is denoted as Ωi, j with its center at (xi, yj) for i, j = 1, 2, …, N. We assume that the conductivity σ is constant in each element Ωi, j, say σi, j. Let

images

For a given σ ∈ Σ, we can express σ as

images

The solution u of the elliptic PDE in (7.37) with σ in place of γ can be approximated by a vector

images

such that each voltage uk for k = i + jN is determined by the weighted average of four neighboring voltages. To be precise, the conductivity equation

images

can be written as the following discretized form

7.44 7.41

with

7.45 7.42

where kT, kD, kR and kL denote top, down, right and left neighboring points of the kth point, respectively.

The discretized conductivity equation (7.44) with the Neumann boundary condition can be rewritten as a linear system of equations:

images

where g is the injection current vector associated with the Neumann boundary data g. Any change in σk for k = 1, 2, …, N2 spreads its influence to all uk for k = 1, 2, …, N2 through the matrix images. We should note the following implications of the entanglement among σk and uk.

Geometry. The recursive averaging process in (7.44) with (7.45) makes the influence of a change in σk upon ul smaller and smaller as the distance between positions of σk and ul is further increased.
Nonlinearity. The recursive averaging process in (7.44) with (7.45) causes a nonlinearity between σk and ul for all k, l = 1, 2, …, N2.
Interdependence. The recursive averaging process in (7.44) with (7.45) makes the influence of a change in σk upon ul affected by all other σm with images.

7.5 Uniqueness Theory and Direct Reconstruction Method

Before we study practical inversion methods to invert the map in (7.43), we review mathematical theories of uniqueness and a direct reconstruction technique called the d-bar method.

7.5.1 Calderón's Approach

In this section, we will assume a full NtD map Λγ as EIT data. Calderón (1980) made the following observation, which plays a key role in achieving the theoretical development of EIT, especially uniqueness theory. For a quick and easy explanation, we assume the following throughout this section:

  • images with its C2 boundary ∂Ω;
  • γ is real and images with γ = 1 in images;
  • images in Ω and q = 0 in images;
  • γ0 = 1 is the background conductivity.

To prove his observation, Caldéron used a set of special pairs of harmonic functions that is dense in L1(Ω).


Lemma 7.5.1
If images (or images) satisfy

7.46 7.43

then both images and images are harmonic in the entire space images (or images). Moreover,

7.47 7.44

where images, images and

images


 


Proof.
Both v and w are harmonic because

images

Since images we have

images

which is clearly dense in L1(Ω) due to the Fourier representation formula.

 


Theorem 7.5.2 (Calderón's approach)
Let images and images. Denote

images

If, for any images,

images

then

images


 


Proof.
For images, let images be a solution of

7.48 7.45

Taking the derivative of the problem (7.48) with respect to t, we have

7.49 7.46

Here, we use the assumption that δσ|∂Ω = 0. By multiplying (7.49) by images and integrating over Ω, we have

images

At t = 0, this becomes

images

We also have the same identity for images:

images

It follows from the assumption images that

7.50 7.47

Hence, images because

images

and images is dense in L1(Ω) from Lemma 7.5.1.

Let us begin by explaining the scattering transform that transforms the conductivity equation images into the Schrödinger equation images. This transform was first used to prove the uniqueness of EIT for images by Sylvester and Uhlmann (1987). The following lemma explains this scattering transform.


Lemma 7.5.3
Let images and u satisfy

images

Then, images satisfies

7.51 7.48


 


Proof.
The proof follows from the direct computation:

images


 


Remark 7.5.4
From Lemma 7.5.3, images is the solution of

7.52 7.49

This fact has been used to develop a two-dimensional constructive identification method of γ named the images (or d-bar) method (Nachman 1988).

7.5.2 Uniqueness and Three-Dimensional Reconstruction: Infinite Measurements

In this section, we briefly explain some impressive results on the uniqueness question and three-dimensional reconstruction in EIT mainly by Sylvester and Uhlmann (1987) and Nachman (1988). We, however, note that the reconstruction formula suggested in this section may not be appropriate for practical cases.

We define the DtN map images by

images

where uj satisfies

7.53 7.50

The goal is to prove that

images


Lemma 7.5.5
Assume that images. For any u1 and u2 satisfying images in Ω, we have

images


 


Proof.
By the definition, we have

images

for any ϕ ∈ H1(Ω). Hence,

images

Subtracting the above two equations yields

images

It then follows that if images, then

images


 


Lemma 7.5.6
For images satisfying images, there exists a solution u of the equation

images

in the form

images

and ψζ → 0 in L2(Ω) as images.

 


Proof.
If images is a solution of images, then ψζ satisfies

images

Since the symbol of the Fourier transform of the operator images is images, Green's function for images can be expressed as

images

This Green's function was first introduced by Faddeev (1965) and the integral should be understood as an oscillatory integral. Hence, ψζ must satisfy

images

The decay condition ψζ → 0 in L2(Ω) as images follows from the fact that

7.54 7.51

where − 1 < δ < 0 and

images


 


Theorem 7.5.7
The set images is dense in L1(Ω). Hence,

images


 


Proof.
For each images and images, we can choose images (six unknowns) satisfying four equations:

images

Denoting

images

we have images and images (j = 1, 2). Let uj be the solution given in Lemma 7.5.6 corresponding to qj, that is,

images

Then,

images

in L1(Ω)-sense. Therefore, we have

images

Since images is arbitrary, we finally have q1 = q2.

The next observation provides an explicit representation formula for q from the knowledge of the NtD map.


Observation 7.5.8 (Nachman's reconstruction)
Let images in Lemma 7.5.6. Then,

7.55 7.52


 


Proof.
Since images as images, we have

7.56 7.53

On the other hand, since images, it follows from the divergence theorem that

images

Thus, we have

7.57 7.54


7.5.3 Nachmann's D-bar Method in Two Dimensions

Siltanen et al. (2000) first implemented the d-bar algorithm based on Nachmann's two-dimensional global uniqueness proof of EIT. This d-bar method solves the full nonlinear EIT problem without iteration (Mueller and Siltanen 2003; Murphy and Mueller 2009).

The d-bar method is based on the fact (Lemma 7.5.3) that: images is a solution of

7.58 7.55

where images is the standard solution of the conductivity equation. We know that, for each k = k1 + ik2, there exists a unique solution ψ( ·, k) of

images

The scattering transform of qC0(Ω) can be expressed as

images

where images is a Dirichlet-to-Neumann (DtN) map given by

images

where uf is a solution of images in Ω with the Dirichlet boundary data uf|∂Ω = f.

Using the fact that images and the above property of t(x, k), it is easy to prove that

images

satisfies the d-bar equation:

7.59 7.56

From (7.58), solving the d-bar equation (7.59) for μ(z, k) leads to the reconstruction algorithm for γ:

images

For the reconstruction algorithm, we need the following steps:

Step 1. Compute ψ( ·, k)|∂Ω for each k = k1 + ik2.
Step 2. Compute t(k) using step 1.
Step 3. Solve the d-bar equation (7.59) for μ(z, k).
Step 4. Visualize images, z = (x, y) ∈ Ω.

For a precise explanation of the reconstruction algorithm, let us fix notation and definitions:

  • For a complex variable z = x + iy at a point z = (x, y), define the d-bar operator images by

images

  • For images,

images

Note that gk satisfies images.
  • Define a single-layer operator images for images by

images

where Gk(z) = eik(x+iy)gk(z). Note that images.

The direct method for reconstructing γ without iteration is based on the following theorem.


Theorem 7.5.9
Nachmann's constructive result:
1. For each images, there exists a unique solution ψ( ·, k) ∈ H1/2(∂Ω) satisfying the integral equation

images

where ikz = ik(x + iy) and images denotes the DtN map of the homogeneous conductivity γ = 1.
2. For each z = (x, y), the solution μ of (7.59) satisfies the integral equation

images

3. We reconstruct γ by

images


 


Proof.
For the detailed proof, please see Nachman (1996).

7.6 Back-Projection Algorithm

Barber and Brown (1983) introduced the back-projection algorithm as a fast and practically useful algorithm in EIT. Since it was motivated by the X-ray CT algorithm, we can view it as a generalized Radon transform. However, there exists a clear difference between EIT and CT. In CT, we can obtain projected images in various directions; while, in EIT, we cannot control current pathways since the current flow itself depends on the unknown conductivity distribution to be imaged. Under the assumption that the conductivity is a small perturbation of a constant value, we can approximately apply the back-projection algorithm.

Let us begin by reviewing the well-known Radon transform. In CT, we try to reconstruct a cross-sectional image f from its X-ray projections in several different directions (cosθ, sinθ). The projection of f in direction θ can be defined by

images

Taking the Fourier transform of Pθf leads to

images

The reconstruction algorithm is based on the following expression of f in terms of its projection:

images

where θj = jπ/N. Hence, the image f can be computed from knowledge of its projection images, j = 1, 2, …, N.

To quickly explain the back-projection algorithm in EIT, we assume the following:

  • Ω is the unit disk in images;
  • γ = γ0 + δγ and γ0 = 1;
  • images;
  • Pθ = (cosθ, sinθ) and z = (x, y) (or z = x + iy).

Let u0 and u denote the electric potentials corresponding to γ0 = 1 and γ with the same Neumann dipole boundary data

images

Writing u = u0 + δu, δu approximately satisfies the equation

7.60 7.57

where we neglect the term images. When images is very small, u0 can be computed approximately as

images

where images.

Next, we introduce a holomorphic function in Ω whose real part is − u0:

images

Then, Ψθ maps from the unit disk onto the upper half-plane as shown in Figure 7.9:

images

Define

images

Viewing ξ = ξ(x, y) and η = η(x, y), we have

images

Hence, the perturbed equation (7.60) implies that images satisfies

7.61 7.58

For the moment, we assume that images is independent of the η variable. With this temporary assumption, images is independent of the η variable and hence

images

Therefore,

images

For a fixed z, denote Ψθ(z) = ξθ + iηθ and images (see Figure 7.10).

Figure 7.9 The Ψθ transformation

7.9

Figure 7.10 Diagram of images

7.10

Using the relation among Ψθ, z and z*, Barber and Brown (1983) derived the reconstruction formula

7.62 7.59

where images denotes the tangential derivative at images.

7.7 Sensitivity and Sensitivity Matrix

Recently developed image reconstruction algorithms are based on sensitivity analysis. We investigate the sensitivity of a boundary voltage Vj, k[γ] to a change in γ. We assume that the discrete NtD data images in (7.40) are available. Since images can be viewed as a function of γ, we denote it by images. In order to explain the sensitivity matrix, we use the vector form images as

7.63 7.60

or

7.64 7.61

for j, k = 1, 2, …, E.

We assume a reference admittivity images, which is a homogeneous admittivity minimizing

7.65 7.62

We may assume that images is a measured data set and images is a computed data set by numerically solving (7.37) with a known γ0 in place of γ.

7.7.1 Perturbation and Sensitivity

We consider γ that is different from a known admittivity γ0. Assume that we inject the same currents into two imaging domains with γ and γ0.


Lemma 7.7.1
The perturbation δγ: = γ − γ0 satisfies

7.66 7.63

where images is the solution of (7.37) with γ0 in place of γ.

 


Proof.
Using integration by parts and the reciprocity theorem, we have

images


The sensitivity expression in (7.66) provides information about how much boundary voltage changes by the admittivity perturbation δγ.

7.7.2 Sensitivity Matrix

The effects of a perturbation δγ depend on the position r of the perturbation. In order to construct an explicit expression, we divide the domain Ω into small subregions and assume that γ, γ0 and δγ are constant in each subregion. With this kind of discretization, we can transform (7.66) into matrix form.


Observation 7.7.2
We discretize the domain Ω into N subregions as images. We assume that γ, γ0 and δγ are constants in each Tn. We can express (7.66) as

7.67 7.64

where

images

and images is the value of δγ in Tn. The E2 × N sensitivity matrix images is given by

images


Note that the sensitivity matrix depends nonlinearly on the admittivity distributions γ and γ0.

7.7.3 Linearization

We let γ0 be a variable and make a link between changes in boundary voltages and a small admittivity perturbation δγ around γ0.


Observation 7.7.3
Assuming the same discretization of the domain Ω as explained in the previous section, the admittivity is an N-dimensional variable. When the perturbation images is small,

7.68 7.65

where images can be viewed as a Fréchet derivative of images with respect to γ at γ = γ0.

 


Proof.
Let p = (j − 1) × E + k. From (7.66),

images

since images. Hence,

images

where images is the characteristic function of Tn. The proof follows from the fact that

images


 


Observation 7.7.4
We let images. The linearized EIT problem is expressed by

7.69 7.66

or

images

images


The matrix images is called the sensitivity matrix or Jacobian of the linearized EIT problem.

7.7.4 Quality of Sensitivity Matrix

Each data collection protocol is associated with its own sensitivity matrix. We may apply the singular value decomposition explained in Chapter 2 to the sensitivity matrix. Performance of the data collection protocol is closely related with the distribution of singular values. Evaluating several sensitivity matrices from chosen data collection protocols, we may choose a best one. One may also adopt the point spreading function and analyze performance indices of a chosen data collection method, including the spatial resolution, amount of artifacts, uniformity of image contrast and others. This may suggest an optimal data collection method for a specific application.

7.8 Inverse Problem of EIT

Providing intuitive understanding about the inverse problem in EIT using RC circuits as examples, we will formulate three EIT inverse problems including static imaging, time-difference imaging and frequency-difference imaging. Based on the observations in section 7.4.3, we study the ill-posedness in those inverse problems.

7.8.1 Inverse Problem of RC Circuit

We consider two simple examples of elementary inverse problems in RC circuits.


Example 7.8.1
Consider the series RC circuit. The injection current and measured voltage are I = I∠0 and V = V∠θ, respectively, in their phasor forms. The inverse problem is to find the resistance R and the capacitance C from the relation between I and V.
UnFigure
Solution. From

images

we find

images

The number of unknowns is two and the number of measurements is also two, including the real and imaginary parts of the impedance Z.

 


Example 7.8.2
Repeat the above example for the parallel RC circuit.

 


Example 7.8.3
Consider the series RC circuit with two resistors and two capacitors. The inverse problem is to find R1, R2, C1 and C2 from the data I = I∠0 and V = V∠θ.
UnFigure
Solution. From

images

we find

images

The number of unknowns is four and the number of measurements is two, including the real and imaginary parts of the impedance Z. This results in infinitely many solutions.

The inverse problem in Example 7.8.3 has no unique solution and is ill-posed in the sense of Hadamard. Note that we may increase the number of measurements by separately measuring two voltages across R1C1 and R2C2 to uniquely determine R1, C1, R2 and C2. One may think of numerous RC circuits with multiple measurements that are either well-posed or ill-posed.

7.8.2 Formulation of EIT Inverse Problem

We assume an EIT system using E electrodes images for j = 1, 2, …, E. The admittivity inside an imaging domain Ω at time t, angular frequency images and position r is denoted as images.

7.8.2.1 Static Imaging

Static imaging in EIT is to produce an image of the admittivity images from the NtD data images in (7.40). The image reconstruction requires inversion of the map

images

for a fixed time t and frequency images. We may display images of images and images separately. In each image, a pixel value is either images or images (S m−1). This kind of image is ideal for all applications since it provides absolute quantitative information. One may conduct multi-frequency static imaging by obtaining multiple NtD data sets at the same time at multiple frequencies. We may call this “spectroscopic imaging”. We may perform a series of static image reconstructions consecutively at multiple times to provide a time series of admittivity images. Since static EIT imaging is technically difficult in practice, we consider difference imaging methods.

7.8.2.2 Time-Difference Imaging

Time-difference imaging produces an image of any difference, images, between two times t1 and t2 from the difference of two NtD data sets, images. For single-frequency time-difference imaging, images is fixed. One may also perform multi-frequency time-difference imaging. Time-difference imaging is desirable for functional imaging to monitor physiological events over time. Though it does not provide absolute values of images and images, it is more feasible in practice for applications where reference NtD data at some time are available.

7.8.2.3 Frequency-Difference Imaging

For applications where a time-referenced NtD data set is not available, we may consider frequency-difference imaging. It produces an image of any difference between images and images using two NtD data sets images and images, which are acquired at the same time. One may perform frequency-difference imaging at multiple frequencies using images. Frequency-difference imaging may classify pathological conditions of tissues without relying on any previous data. Consecutive reconstructions of frequency-difference images at multiple times may provide functional information related to changes over time.

7.8.3 Ill-Posedness of EIT Inverse Problem

Before we study these three inverse problems in detail, we investigate their ill-posed characteristics based on the description in section 7.4.3, where we assumed that γ = σ for simplicity. For an injection current images, we are provided with a limited number of voltage data using a finite number of electrodes. The voltage data vector f corresponds to measured boundary voltages on portions of ∂Ω where voltage-sensing electrodes are attached. The inverse problem is to determine the conductivity vector images or equivalently the matrix images from several measurements of current–voltage pairs (gm, fm) for m = 1, …, P, where P is the number of projections.

The ill-posedness of the EIT inverse problem is related to the fact that the difficulty in reconstructing images from (gm, fm) with m = 1, …, P increases exponentially as the size of images increases. This means that the ill-posedness gets worse as we increase the number of pixels for better spatial resolution. According to (7.44), the voltage at each pixel inside the imaging domain can be expressed as the weighted average of its neighboring voltages, where weights are determined by the conductivity distribution. As explained in section 7.4.3, the measured voltage data vector f is nonlinearly entangled in the global structure of the conductivity distribution. Any internal conductivity value σk has little influence on the boundary measurements f, especially when the position of σk is away from the positions of voltage-sensing electrodes. Figure 7.8 depicts these phenomena, from which the ill-posedness originates.

EIT reveals technical difficulties in producing high-resolution images owing to the inherent insensitivity and nonlinearity. For a given finite number of electrodes, the amount of measurable information is limited. Increasing the size of images for better spatial resolution makes the problem more ill-posed. To supply more measurements, we have to increase the number of electrodes. With reduced gaps among a larger number of electrodes, measured voltage differences will become smaller to deteriorate signal-to-noise ratios. Beyond a certain spatial resolution or the pixel size, all efforts to reduce the pixel size using a larger images result in poorer images, since the severe ill-posedness takes over the benefit of additional information from the increased number of electrodes.

Therefore, we should not expect EIT images to have a high spatial resolution needed for structural imaging. EIT cannot compete with X-ray CT or MRI in terms of spatial resolution. One should find clinical significance of biomedical EIT from the fact that it provides unique new contrast information with a high temporal resolution using a portable machine.

7.9 Static Imaging

7.9.1 Iterative Data Fitting Method

Most static image reconstruction algorithms for an E-channel EIT system can be viewed as a data fitting method, as illustrated in Figure 7.11. We first construct a computer model of an imaging object based on (7.37). With the discretization of the imaging domain into N pixels as explained in section 7.7, we can express γ as an admittivity vector

images

Since we do not know the true admittivity images of the imaging object, we assume an initial admittivity distribution images with m = 0 for the model. When we inject currents into both the object and the model, the corresponding measured and computed boundary voltages are different, since images in general. An image reconstruction algorithm iteratively updates images until it minimizes the difference between measured and computed boundary voltages.

Figure 7.11 Static EIT image reconstruction as a data fitting method

7.11

To illustrate this idea, we define the following minimization problem:

7.70 7.67

where “arg min” is an operator that gives an energy functional minimizer, images is a measured NtD data vector, images is the computed NtD data vector and images is an admissible class for the admittivity. For the solution of (7.70), we may use an iterative nonlinear minimization algorithm such as the Newton–Raphson method (Yorkey and Webster 1987).

In every iteration, we compute the sensitivity matrix or Jacobian images in (7.69) by solving (7.37) with γm in place of γ. Solving the following linear equation

7.71 7.68

for images by

7.72 7.69

we update images as

7.73 7.70

We may stop when

7.74 7.71

where δ is a tolerance.

7.9.2 Static Imaging using Four-Channel EIT System

To understand the algorithm in (7.70) clearly, we consider a simple example using a four-channel EIT system. We inject sinusoidal current images to each electrode pair images and images for j = 1, …, 4 and images. From these four projections, we acquire 16 voltages:

images

Figure 7.12 shows a circular imaging object Ω, images, images, images and images.

Figure 7.12 Current and voltage signals from a four-channel EIT system

7.12

We divide the imaging domain as Ω = T1T2T3T4 in Figure 7.13. Assume that γ is constant on each Tj for j = 2, 3, 4 and γ = 1 on T1. The goal is to recover γ from the NtD data in Table 7.3 using the following iteration process.

1. Let γ0 = 1 be the initial guess.

Figure 7.13 Discretized imaging domain for a four-channel EIT system

7.13

Table 7.3 NtD data from a four-channel EIT system

NumberTable
2. For each images with m = 1, 2, …, solve the forward problem of (7.37) with γ = γm and get images. Figure 7.14 shows the distributions of images for j = 1, 2, 3 and 4.

Figure 7.14 Voltage distributions inside the imaging object

7.14
3. Compute the sensitivity matrix images in (7.69) as

images

and compute

images

4. Calculate images by solving

images

5. Update images
6. Repeat steps 2, 3, 4 and 5 until images is smaller than a predetermined tolerance.

In step 4, we used images to update images. Recall that solving the minimization problem of images with the four-channel EIT is to find a minimizing sequence images such that images approaches its minimum effectively. The reason for this choice is that images in step 4 makes images smallest with a given unit norm of images.

To see this rigorously, assume that the true conductivity is γ* and the measured data are exact so that Vj, k = Vj, k[γ*]. According to (7.66),

images

Computation of the Frechét derivative of the functional Φ(γ) requires one to investigate the linear change δu: = uγ+δγuγ subject to a small conductivity perturbation δγ. Note that images. For simplicity, we assume that δγ = 0 near ∂Ω. The relationship between δγ and the linear change δu can be explained by

images

We have the following approximation:

images

We want to find the direction δγ that makes Φ(γ + δγ) − Φ(γ) smallest with a given unit norm of ||δγ||. The steepest descent direction δγ = (δγ1, δγ2, δγ3)T can be calculated by solving the matrix equation:

images

To understand this, we recall that images and

images

We choose the direction δγ that makes Φ(γ + δγ) − Φ(γ) smallest with a given norm ||δγ|| as

images

Owing to

images

the steepest descent direction δγ must satisfy

images

7.9.3 Regularization

Since the Jacobian matrix in (7.71) is ill-conditioned, as explained in section 7.7, we often use a regularization method. Using the Tikhonov type regularization, we set

7.75 7.72

where λ is a regularization parameter and images is a function measuring a regularity of images. This results in the following update equation for the mth iteration:

7.76 7.73

where images is a regularization matrix.

This kind of method was first introduced in EIT by Yorkey and Webster (1987), followed by numerous variations and improvements (Cheney et al. 1990, 1999; Cohen-Bacrie et al. 1997; Edic et al. 1998; Hyaric and Pidcock 2001; Lionheart et al. 2005; Vauhkonen et al. 1998; Woo et al. 1993). These include utilization of a priori information, statistical information, various forms of regularity conditions, adaptive mesh refinement and so on. Though this iterative approach is widely adopted for static imaging, it requires a large amount of computation time and produce static images with a low spatial resolution and poor accuracy for the reasons discussed in the next section. Beyond this classical technique in static imaging, new ideas are in demand for better image quality.

7.9.4 Technical Difficulty of Static Imaging

In a static EIT imaging method, we construct a forward model of the imaging object with a presumed admittivity distribution. Injecting the same currents into the model as the ones used in measurements, boundary voltages are computed to numerically simulate measured data. Since the initially guessed admittivity distribution is in general different from the unknown admittivity distribution of the object, there exist some differences between measured and computed voltages. Most static EIT imaging methods are based on a minimization technique, where a sum of these voltage differences is minimized by adjusting the admittivity distribution of the model (Adler and Lionheart 2006; Cheney et al. 1990; Lionheart et al. 2005; Woo et al. 1993; Yorkey and Webster 1987). Other methods may include layer stripping (Somersalo et al. 1991) and d-bar (Siltanen et al. 2000) algorithms.

For a static EIT image reconstruction algorithm to be reliable, we should be able to construct a forward model that mimics every aspect of the imaging object except the internal admittivity distribution. This requires knowledge of the boundary geometry, electrode positions and other sources of systematic artifacts in measured data. In practice, it is very difficult to obtain such information within a reasonable accuracy and cost, and most static EIT image reconstruction algorithms are very sensitive to these errors.

When we inject current through a pair of electrodes images and images, the induced voltage images is dictated by the applied Neumann data gj of the injection current, the geometry of the domain Ω and γ. That is, images satisfies approximately

7.77 7.74

where gj represents the Neumann data in (7.37).

Taking account of the nonlinearity and ill-posedness in EIT, most image reconstruction methods for EIT use the assumption that γ is a perturbation of a known reference distribution γ0 so that we can linearize the nonlinear problem. The inverse problem is to find δγ: = γ − γ0 from the integral equation

7.78 7.75

where images and dS is the surface element (Cheney et al. 1990, 1999, Lionheart et al. 2005). In practice, the value of the right-hand side of (7.78) is the potential difference uj between electrodes images and images.

If the change δγ is small, we can approximate

images

and (7.78) becomes

7.79 7.76

where images and bγ, Ω are L × L vectors with (j − 1)L + k component

images

respectively. We may view images as a linear operator acting on δγ and its discretized version in terms of the admittivity distribution is called the sensitivity matrix.

To solve the inverse problem (7.79), we construct a forward model of the imaging object with a presumed reference admittivity images:

7.80 7.77

where images is a computational domain mimicking the geometry of the imaging subject, images is the Neumann data mimicking the applied current gj and images is the internal potential induced by the current corresponding to the Neumann data images.

The forward model (7.80) is used to compute the reference boundary voltage images, which is expected to be substituted for images in (7.79). If we have the exact forward modeling images and images, we may obtain reasonably accurate images of δγ by inverting the discretized version of the linear operator images with the use of regularization. Knowing that we cannot avoid forward modeling errors, a major drawback of static imaging stems from the fact that the reconstruction problem (7.79) is very sensitive to geometric modeling errors in the computed reference data images, including boundary geometry errors on Ωc and electrode positioning errors on images (Barber and Brown 1988; Kolehmainen et al. 2005; Nissinen et al. 2008). It would be very difficult to get accurate data images at a reasonable cost in a practical environment.

To deal with undesirable effects of modeling errors, we investigate two difference imaging methods in the following sections. We expect that time or frequency derivatives of the NtD data Λσ, Ω may cancel out the effects of geometry errors on ∂Ω.

7.10 Time-Difference Imaging

In time-difference EIT (tdEIT), measured data at two different times are subtracted to produce images of changes in the admittivity distribution with respect to time. Since the data subtraction can effectively cancel out common errors, tdEIT has shown its potential as a functional imaging modality in several clinical application areas. In this section, we consider multi-frequency time-difference EIT (mftdEIT) imaging. After formulating the mftdEIT imaging problem, we study the mftdEIT image reconstruction algorithm.

7.10.1 Data Sets for Time-Difference Imaging

We assume an imaging object Ω bounded by its surface ∂Ω. The isotropic admittivity in Ω at time t, angular frequency images and position r = (x, y, z) is denoted images. Attaching surface electrodes images for j = 1, 2, …, E on ∂Ω, we inject a sinusoidal current images between a chosen pair of electrodes. A distribution of voltage in Ω is produced and we can express it as images.

Assuming an EIT system using E electrodes, we inject the jth current between an adjacent pair of electrodes denoted as images and images for j = 1, 2, …, E. The time-harmonic voltage subject to the jth injection current is denoted as images, which is a solution of (7.37) with g replaced by gj. We assume that the EIT system is equipped with E voltmeters and each of them measures a boundary voltage between an adjacent pair of electrodes, images and images for k = 1, 2, …, E.

Using an mftdEIT system, we collect complex boundary voltage data at multiple frequencies for a certain period of time. Assuming that we collected E2 number of complex boundary voltage data at each sampling time t and frequency images, we can express a complex boundary voltage data vector as (7.16). We rewrite it using a column vector representation as

7.81 7.78

For t = t1, t2, …, tN and images, we are provided with N data vectors for each one of F frequencies. To perform tdEIT imaging, we need a complex boundary voltage data vector at a reference time t0:

7.82 7.79

for images. The mftdEIT imaging problem is to produce time series of difference images using images for t = t1, t2, …, tN at each one of images.

7.10.2 Equivalent Homogeneous Admittivity

For a given admittivity distribution images, we define the equivalent homogeneous admittivity images as a complex number that minimizes

images

where images is the voltage satisfying (7.37) with images in place of images and images is a weighting constant. We assume that images is a small perturbation of images.

We set a reference frequency images as well as the reference time t0. We assume that the complex boundary voltage vector images is available at t = t0 and images. Defining

images

it measures the quantity images roughly because

7.83 7.80

where vj and vk are solutions of (7.37) with images for the jth and kth injection currents, respectively.

We now relate a time change of the complex boundary voltage with a time change of the internal admittivity. For p = (k − 1) × E + j with j, k = 1, 2, …, E,

7.84 7.81

where ep is the unit vector in the E2 dimension having 1 at its pth component. Note that we have utilized the reciprocity theorem in section 7.4. Since we assumed that images and images are small perturbations of images and images, respectively, we have the following approximation:

images

Hence, for all p = (k − 1) × E + j, we have

7.85 7.82

7.10.3 Linear Time-Difference Algorithm using Sensitivity Matrix

We construct a computer model of the imaging object Ω. Assume that the domain of the model is Λ with its boundary ∂Λ. Discretizing the model into Q elements or pixels as images, we define the time-difference image images at time t and frequency images as

7.86 7.83

with

images

where images and images for q = 1, 2, …, Q are the admittivity values of the imaging object at times t and t0, respectively, inside a local region corresponding to the qth pixel Λq of the model Λ.

The model is assumed to be homogeneous, with images in Λ. Using E electrodes, we inject current between the jth adjacent pair of electrodes to induce voltage vj in Λ. We numerically solve (7.37) for vj by using the finite element method. We can formulate the sensitivity matrix images in section 7.7 as

7.87 7.84

for j, k = 1, 2, …, E and q = 1, 2, …, Q. The maximal size of images is E2 × Q and all of its elements are real numbers. Using the discretization and linearization, the expression (7.85) becomes

7.88 7.85

Computing the truncated singular value decomposition (TSVD) of images, we find PQ singular values that are not negligible. We can compute a pseudo-inverse matrix of images after truncating its (QP) negligible singular values. Denoting this inverse matrix as images, we have

7.89 7.86

Note that images is a real matrix whose maximal size is Q × E2. Since we do not know images in (7.86), we replace (7.86) by the following equation:

7.90 7.87

where images and images are the real and imaginary parts of a reconstructed complex tdEIT image images, respectively.

We may reconstruct a time series of mftdEIT images images for f = 1, 2, …, F at n = 1, 2, …, N. Choosing images at a low frequency below 1 kHz, we may assume that images since we can neglect the effects of the permittivity at low frequencies. In such a case, (7.90) becomes

7.91 7.88

Note that images in (7.91) has the same phase angle as images.

7.10.4 Interpretation of Time-Difference Image

The mftdEIT image reconstruction algorithm based on (7.91) produces both real- and imaginary-part tdEIT images at multiple frequencies. It provides a theoretical basis for proper interpretation of a reconstructed image using the equivalent homogeneous complex conductivity. From (7.91), we can see that the real- and imaginary-part images represent images and images, respectively. We can interpret them as fractional changes of σ and images between times t and t0 with respect to the square of the equivalent homogeneous conductivity images at time t0 at a low frequency images.

We should note several precautions in using the mftdEIT image reconstruction algorithm of (7.91). First, since (7.84) is based on the reciprocity theorem, the EIT system must have a smallest possible reciprocity error. Second, the true admittivity distribution images inside the imaging object at time t and images should be a small perturbation of its equivalent homogeneous admittivity images in order for the approximations in (7.84) and (7.85) to be valid. This is the inherent limitation of the difference imaging method using the linearization. Third, the computed voltage v in (7.87) may contain modeling errors. It would be desirable for the model Λ of the imaging object Ω to have correct boundary shape and size. We may improve the model by incorporating a more realistic boundary shape in three dimensions. Fourth, the number of non-negligible singular values of the sensitivity matrix should be maximized by optimizing the electrode configuration and data collection protocol.

7.11 Frequency-Difference Imaging

Since tdEIT requires time-referenced data, it is not applicable to cases where such time-referenced data are not available. Examples may include imaging of tumors (Kulkarni et al. 2008; Soni et al. 2004; Trokhanova et al. 2008) and cerebral stroke (McEwan et al. 2006; Romsauerova et al. 2006a,b). Noting that admittivity spectra of numerous biological tissues show frequency-dependent changes (Gabriel et al. 1996b; Geddes and Baker 1967; Grimnes and Martinsen 2008; Oh et al. 2008), frequency-difference EIT (fdEIT) has been proposed to produce images of changes in the admittivity distribution with respect to frequency.

In early fdEIT methods, frequency-difference images were formed by back-projecting the logarithm of the ratio of two voltages at two frequencies (Fitzgerald et al. 1999; Griffiths 1987; Griffiths and Ahmed 1987a,b; Griffiths and Zhang 1989; Schlappa et al. 2000). More recent studies adopted the sensitivity matrix with a voltage difference at two frequencies (Bujnowski and Wtorek 2007; Romsauerova et al. 2006a,b; Yerworth et al. 2003). All of these methods are basically utilizing a simple voltage difference at two frequencies and a linearized image reconstruction algorithm. Alternatively, we may consider separately producing two static (absolute) images at two frequencies and then subtract one from the other. This approach, however, will suffer from the technical difficulties in static EIT imaging.

In this section, we describe an fdEIT method using a weighted voltage difference at two frequencies (Seo et al. 2008). Since the admittivity spectra of most biological tissues change with frequency, we will assume an imaging object with a frequency-dependent background admittivity in the development of fdEIT theory. We may consider two different contrast mechanisms in a reconstructed frequency-difference image. First, there exists a contrast in admittivity values between an anomaly and background. Second, the admittivity distribution itself changes with frequency.

7.11.1 Data Sets for Frequency-Difference Imaging

We assume the same setting as in section 7.10.1. Using an E-channel EIT system, we may inject E number of currents through adjacent pairs of electrodes and measure the following voltage data set:

7.92 7.89

For t = t1, t2, …, tN and images, we are provided with N data vectors for each one of F frequencies. Let us assume that we inject currents at two frequencies of images and images to obtain corresponding voltage data sets images and images, respectively. The goal is to visualize changes of the admittivity distribution between images and images by using these two voltage data sets.

In tumor imaging or stroke detection using EIT, we are primarily interested in visualizing an anomaly. This implies that we should reconstruct a local admittivity contrast. For a given injection current, however, the boundary voltage images is significantly affected by the background admittivity, boundary geometry and electrode positions, while the influence of a local admittivity contrast due to an anomaly is much smaller. Since we utilize two sets of boundary voltage data, images and images in fdEIT, we need to evaluate their capability to perceive the local admittivity contrast. As in tdEIT, the rationale is to eliminate numerous common errors by subtracting the background component of images from images, while preserving the local admittivity contrast component.

7.11.2 Simple Difference images

The simple voltage difference images may work well for an imaging object whose background admittivity does not change with frequency. A typical example is a saline phantom. For realistic cases where background admittivity distributions change with frequency, it will produce artifacts in reconstructed fdEIT images. To understand this, let us consider a very simple case where the imaging object has a homogeneous admittivity distribution, that is, images is independent of position. In such a homogeneous object, induced voltages images and images satisfy the Laplace equation with the same boundary data, and the two corresponding voltage data vectors images and images are parallel in such a way that

images

When there exists a small anomaly inside the imaging object, we may assume that the induced voltages are close to the voltages without any anomaly. In other words, the voltage difference images in the presence of a small anomaly can be expressed as

images

for a complex constant images. This means that the simple difference images significantly depends on the boundary geometry and electrode positions except for the special case where images. This is the main reason why the use of the simple difference images cannot deal with common modeling errors even for a homogeneous imaging object.

7.11.3 Weighted Difference images

An imaging object including an anomaly has an inhomogeneous admittivity distribution images. We define a weighted difference of the admittivity at two different frequencies images and images at time t as

7.93 7.90

where α is a complex number. We assume the following two conditions:

1. In the background region, especially near the boundary, images.
2. In the anomaly, images is significantly different from 0.

In order to extract the anomaly from the background, we investigate the relationship between images and images. We should find a way to eliminate the background influence while maintaining the information of the admittivity contrast across the anomaly. We decompose images into a projection part onto images and the remaining part:

7.94 7.91

where 〈 ·, · 〉 is the standard inner product of two vectors. Note that images is orthogonal to images.

In the absence of the anomaly, we may set images and this results in images. The projection term images mostly contains the background information, while the orthogonal term images holds the anomaly information. To be precise, images provides the same information as images, which includes influences of the background admittivity, boundary geometry and electrode positions. The orthogonal term images contains the core information about a nonlinear change due to the admittivity contrast across the anomaly. This explains why the weighted difference images must be used in fdEIT.

7.11.4 Linear Frequency-Difference Algorithm using Sensitivity Matrix

In this section, we drop the time index t to simplify the notation. Applying the linear approximation in section 7.7, we get the following relation:

7.95 7.92

Given α, we can reconstruct an image of images using the weighted difference images. Since α is not known in practice, we need to estimate it from images and images using (7.94).

We discretize the imaging object Ω as images, where Ωi is the ith pixel. Let images be the characteristic function of the ith element Ωi, that is, images in Ωi and zero otherwise. Let ξ1, …, ξN be complex numbers such that images approximates

images

By approximating images, where Uj is the solution of (7.26) with images, it follows from (7.95) that

7.96 7.93

The reconstruction method using the approximation (7.95) is reduced to reconstructing the images that minimizes the following:

7.97 7.94

where α is the complex number described in section 7.11.3. In order to find ξ = (ξ1, …, ξN), we use the sensitivity matrix images in (7.87). We can compute ξ = (ξ1, …, ξN) by solving the following linear system through the truncated singular value decomposition (TSVD):

images

It remains to compute the fdEIT image images from knowledge of ξ. We need to estimate the equivalent homogeneous (constant) admittivity images corresponding to images to use the following approximation

images

From the divergence theorem, we obtain the following relation:

images

For an E-channel mfEIT system, we may choose

7.98 7.95

where we identify E + j = j and − j = Ej for j = 1, 2, 3. We reconstruct an fdEIT image images by

7.99 7.96

where images is a pseudo-inverse of images.

7.11.5 Interpretation of Frequency-Difference Image

In (7.99), images can be estimated from (7.98) using another low-frequency measurement images. If we choose images low enough, images may have a negligibly small imaginary part. In such a case, we may set images as a reconstructed fdEIT image, which is equivalent to the complex image images divided by an unknown real constant. In practice, it would be desirable to set images smaller than 1 kHz, for example 100 Hz.

This scaling will be acceptable for applications where we are mainly looking for a contrast change within an fdEIT image. These may include detections of tumors and strokes. In order to interpret absolute pixel values of an fdEIT image quantitatively, we must estimate the value of images, which requires knowledge of the object size, boundary shape and electrode positions. Alternatively, we may estimate values of images and images in (7.99) without using the third frequency images. This will again need geometrical information about the imaging object and electrode positions.

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