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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. X, NO. X, MONTH 20121

Impact of model shape mismatch on reconstruction

quality in Electrical Impedance Tomography

Bartłomiej Grychtol, William RB Lionheart, Marc Bodenstein, Gerhard K Wolf and Andy Adler

Abstract—Electrical Impedance Tomography (EIT) is a low-

cost, non-invasive and radiation free medical imaging modality

for monitoring ventilation distribution in the lung. Although

such information could be invaluable in preventing ventilator-

induced lung injury in mechanically ventilated patients, clinical

application of EIT is hindered by difficulties in interpreting the

resulting images. One source of this difficulty is the frequent use

of simple shapes which do not correspond to the anatomy to

reconstruct EIT images. The mismatch between the true body

shape and the one used for reconstruction is known to introduce

errors, which to date have not been properly characterized. In the

present study we therefore seek to 1) characterize and quantify

the errors resulting from a reconstruction shape mismatch for

a number of popular EIT reconstruction algorithms and 2)

develop recommendations on the tolerated amount of mismatch

for each algorithm. Using real and simulated data, we analyze the

performance of 4 EIT reconstruction algorithms under different

degrees of shape mismatch. Results suggest that while slight shape

mismatch is well tolerated by all algorithms, using a circular

shape severely degrades their performance.

Index Terms—EIT, model, shape, mechanical ventilation, ALI,

ARDS

I. INTRODUCTION

E

distribution in the lung. In thoracic EIT, imperceptible current

injection and voltage measurement through surface electrodes

around the thorax are used to reconstruct a conductivity map

across a transverse slice of the body. EIT is low-cost, non-

invasive, radiation free and available at the bedside. One of

the most promising applications of EIT is for monitoring

and/or guiding mechanical ventilation therapy. The ability

of EIT to measure regional distribution of ventilation has

been validated against single photon emission computed to-

mography (SPECT) [1], X-ray computed tomography (CT)

[2], [3] and positron emission tomography (PET) [4]. No

LECTRICAL Impedance Tomography (EIT) is a promis-

ing medical imaging modality for monitoring ventilation

B Grychtol is with the German Cancer Research Centre (DKFZ), De-

partment of Medical Physics in Radiology, 69120 Heidelberg, Germany

b.grychtol@dkfz.de

WRB Lionheart is with School of Mathematics, University of Manchester,

Manchester M13 9PL, England

M Bodenstein is with Department of Anesthesiology, Johannes Gutenberg-

University Mainz, Mainz, Germany

GK Wolf is with the Division of Critical Care Medicine, Department of

Anesthesiology, Children’s Hospital Boston, Harvard Medical School, Boston,

MA 02115, USA

A Adler is with Systems and Computer Engineering, Carleton University,

Ottawa, ON K1S 5B6, Canada

Manuscript received March 22, 2012; revised May 14, 2012

Copyright c ?2012 IEEE. Personal use of this material is permitted. How-

ever, permission to use this material for any other purposes must be obtained

from the IEEE by sending a request to pubs-permissions@ieee.org.

other currently available technology can provide real-time long

term monitoring of the regional functional state of the lungs.

Although such information could be invaluable in preventing

ventilator-induced lung injury (VILI), clinical application of

EIT is hindered by difficulties in interpreting the resulting

images.

Such difficulties are often a result of errors in the for-

ward modeling of the electrical fields, a necessary step in

reconstructing the conductivity distribution. In particular, no

two-dimensional model can fit EIT data obtained from a

three-dimensional domain (body) [5] and, even when a three-

dimensional model of a domain is used, it is generally im-

possible to accurately fit data from an isotropic conductivity

distribution if the boundary shape is wrong [6].

Because in clinical practice the boundary shape is generally

unknown and changes with breathing and posture, the problem

is often reduced to reconstructing the changes rather than ab-

solute conductivity, which is less sensitive to shape mismatch

and easier to solve. A circular shape has traditionally been used

to represent a cross-section of the subject’s body [7]. This lack

of correspondence to the anatomy imposes several limitations

on the analysis of EIT images. Because expected organ shape

and position on circular images is unknown, it is difficult

to distinguish some artifacts from correct images. Images of

different patients cannot be directly compared. Moreover, the

mismatch between the true body shape and the shape used

for reconstruction is known to produce image errors [8], [6],

which to date have not been properly characterized.

In a preliminary study of one reconstruction algorithm [9],

we showed that using the correct body shape obtained from

a CT scan produces reconstructions qualitatively superior to

those produced with a circular shape. However, for practical

reasons, EIT reconstruction cannot depend on the availability

of a CT scan of each individual subject. Patient shape could

instead be obtained by means of, for example, wearable sen-

sors or through optical 3D surface reconstruction (from images

obtained with a multi-camera system). However, we believe

that developing a set of pre-defined shapes to choose from

for each patient based on easy to measure parameters (weight,

height, etc.) is the most practical and least expensive approach.

In order to develop such a set, a deeper understanding of the

errors and tolerances of different EIT algorithms with respect

to shape mismatch is required.

In the present study we therefore seek to 1) characterize and

quantify the errors resulting from reconstruction shape mis-

match for a number of popular EIT reconstruction algorithms

and 2) develop recommendations on the tolerated amount of

mismatch for each algorithm.

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2IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. X, NO. X, MONTH 2012

II. METHODS

Shortly, the external boundary shape of a human and a swine

were obtained from sample CT images in the electrode plane.

From each so obtained true shape a number of progressively

more circular contours was derived. For each contour a 3D

finite element model (FEM) was built by extrusion along the

long axis of the body. The models were used to reconstruct

simulated and real data using four EIT reconstruction algo-

rithms. The results were evaluated in terms of the performance

measures agreed by a representative group of researchers and

practitioners in the field [10]. The procedure is presented

schematically in Fig. 1.

All calculations have been carried out with Matlab (Math-

warks, Natick, USA) using the EIDORS1toolbox [11], to

which all relevant tools developed as part of this project have

been contributed and were included in the latest release (3.5).

A. Model shape

To investigate the impact of mismatch between the actual

body shape and that of the FEM used for EIT image recon-

struction, a number of progressively more inaccurate shapes

were obtained as follows. First, the true shape was obtained

from a single CT image of the thorax at the electrode plane by

manual delineation with a number of points (37 for the human

shape and 41 for the swine). The original pixel coordinates

of the resulting points were rigidly transformed such that

the entire shape fitted in a square with side 2 centered at

the origin. All subsequents unit-less quantities are reported in

this coordinate system. Second, the points were interpreted in

the complex plane (with origin coinciding with the Cartesian

coordinates just defined) and a parametric description of the

shape was obtained by taking the discrete Fourier transform

(DFT) of the resulting complex vector. Third, progressively

smoother shapes were obtained by truncating the Fourier

series, two components at a time, down to a length of 5 terms

(at 3 the resulting shape was an almost perfect circle but off-

center with respect to the other shapes). At each series length

a new shape of 41–45 equidistant points, the exact number

adjusted to allow the creation of a finite element model with

Netgen [12] as described below, was obtained by padding the

truncated Fourier series with zeros and taking the inverse DFT.

All shapes were scaled to have the same area as the original

(π). The last shape was a circle with radius 1. Representative

shapes for both animal and human geometries are presented

in Fig. 2.

The mismatch ∆S between a smooth shape and the original

was quantified as the area of symmetric difference (non-

overlapping area) between the two shapes (divided by π), as

illustrated in Fig 2c. Fig. 2d presents ∆S as a function of the

number of Fourier coefficients retained (assuming 1 for the

circular model).

In order to create 3D models from thus defined contours,

we extended the EIDORS interface to Netgen [12] to support

extrusion. For each shape we then created a 3D model by

extruding the 2D outline to a height of 1. Sixteen circular

1http://eidors3d.sourceforge.net/

electrodes were placed equidistantly around the perimeter

of the model at a height of 0.5. The mesh was refined

locally around the electrodes. Occasionally the locations of

the electrodes and the outline points interacted in ways that

prevented Netgen from successfully meshing the geometry.

In such cases, the number of points describing a shape was

increased, as mentioned earlier. Sample meshes are presented

in Fig. 1.

B. Reconstruction Algorithms

The reconstruction of conductivity values inside a body

based on surface voltage measurements is a severely ill-

posed non-linear inverse problem. However, because of the

large uncertainties about measurement noise, domain shape

and electrode impedance present in clinical and experimental

EIT data alike, various linearized approximations to solving

difference data have proven useful. In difference EIT, a vector

of conductivity change x = σ − σr between the current

conductivity σ and the reference σr is reconstructed from

measurements y = v − vr of the corresponding change in

recorded voltage. Often, both differences are element-wise

normalized (such that yi= (vi− vri)/vri), as is also the case

in the present study.

For sufficiently small changes, the relationship between x

and y can be approximated by the linear relationship

y = Jx + n

(1)

where J is the Jacobian or sensitivity matrix calculated for

each element of the FEM as Jij =

the measurement noise (assumed to be uncorrelated white

Gaussian). Because the number of conductivity elements is

much greater than the number of measurements, x is longer

than y, and J is not square and therefore does not have an

inverse. Instead, a linear reconstruction algorithm calculates

an estimate of x

ˆ x = Ry

∂yi

∂xjand n represents

(2)

using a reconstruction matrix R. Many algorithms to derive

R have been proposed, four of which are used in this

study: TSVD (truncated singular value decomposition [13],

[14]), GREIT (Graz consensus Reconstruction algorithm for

EIT [10]), and two variants of the one-step Gauss-Newton

(GN) method.

In the TSVD algorithm, R is the truncated pseudoinverse

J+

t= VD+

tU∗

(3)

of J, where J = UDV∗is the singular value decomposition

of J and D+

?

0

As the threshold t is increased less components of D are

retained, which means that only the more significant singular

values are used, thus increasing the amount of regularization.

The GREIT reconstruction matrix is calculated from simu-

lated measurements Y and the corresponding desired solutions

˜X as

R =˜XY(JΣxJT+ λΣn)−1,

tis obtained as

D+

t[i,i] =

D[i,i]−1

if ?D[i,i]? ≥ t

otherwise.

(4)

(5)

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GRYCHTOL et al.: IMPACT OF MODEL SHAPE MISMATCH ON RECONSTRUCTION QUALITY IN EIT3

............

...

...

... ...

...

...

... ...

Evaluate each image

Reconstruct

Simulate

targets

Amplitude response map for each model shape

-1

0

1

Fig. 1.

electrode positions. An FEM with a lung contrast conforming to the CT slice is created and used to simulate a number of targets covering the whole

body. Several homogeneous models with distorted shape are created and used for reconstruction. Based on the individual target reconstructions, a map

representing performance metric as a function of position is constructed for each model shape.

Overview of the methods. A CT slice in the electrode plane is segmented to obtain the boundary shape, the contour of the lungs and the

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4IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. X, NO. X, MONTH 2012

41

13

9

5

1

(a)

38

13

9

5

1

(b)

(c)

05 10 1520 253035

Number of Fourier coeficients retained

(d)

40

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

Area of symmetric difference / π (∆S)

Porcine thorax

Human thorax

Fig. 2. Representative swine (a) and human (b) thorax model shapes. Legends

indicate the number of retained Fourier coefficients. (c) Area of symmetric

difference between the original pig thorax shape and a smooth one. (d) Area of

symmetric difference for both human and porcine model shapes as a function

of the number of Fourier coefficients retained.

where˜X =

matrices obtained by horizontal concatenation of n desired

solution or simulated measurement vectors, respectively, while

Σn and Σx represent the noise and image covariance ma-

trices [10]. The trade-off between the different performance

measures is embedded in the desired solutions while the

hyperparameter λ controls the amount of regularization. We

extended the original implementation of the GREIT algorithm,

previously only defined for cylinders, to arbitrary shapes [9].

For the Gauss-Newton algorithms, R can be expressed as

?

where P is a regularization prior matrix and λ is again a

hyperparameter controlling the amount of regularization. We

test the NOSER [15] and discrete Laplace filter [16] priors.

We use the normalized difference imaging approach, whereby

conductivity differences are not reconstructed as absolute

values but as unit-less ratios to the reference background

conductivity, defined as an expiration or target-less simulation

measurement for clinical and simulation data, respectively.

All tested algorithms employ the dual model approach

whereby the Jacobian J is calculated on a 3D forward model

obtained as detailed above, but values are only reconstructed

on a 2D rectangular grid in the electrode plane. After [10],

we adjusted the hyperparameter value for each model and

algorithm such as to achieve noise amplification (as defined by

the Noise Figure parameter in [17]) of 0.5 in the center of the

image. This method of choosing the regularization parameter is

configuration-independent and has been shown to consistently

produce good reconstructions [18].

1

n(˜ x(1)... ˜ x(n)) and Y =

1

n(y(1)...y(n)) are

R =JTJ + λP

?−1

JT

(6)

Amplitude

∝∑k[ x]k

Set ¼-Max

Amplitude ( )

rt

Simulation

Target

Radial Position

Error

Shape

Deformation

Aoutside/Aq

Resolution

Aq/A0

Ringing

Inverted image

Reconstructed

Image ( ) x

rq

rt−rq

Aq

xq

Fig. 3.

and actual reconstructed images. Adapted from [10].

Algorithm evaluation criteria calculated by comparing the desired

C. Evaluation criteria

For each shape, each reconstruction algorithm was evaluated

using the performance figures of merit defined in [10]. Briefly,

these are: amplitude response (AR), resolution (RES), shape

deformation (SD), position error (PE), ringing (RNG), and

position error (PE). Each figure of merit is measured empiri-

cally on a reconstructed image of a small simulated target, as

illustrated in Fig. 3.

To evaluate the spatial variability of each algorithm with

respect to the performance measures, calculations were carried

out for a large set of regularly spaced small targets in the

electrode plane. Thus, for each shape and figure of merit,

the performance of the algorithm is represented as an image

with each pixel corresponding to one simulated target and

its value reflecting the respective figure of merit obtained by

reconstructing that single simulated target. The mean of values

within each such image and the ratio of standard deviation to

the mean are analyzed as a function of the shape deformation

∆S.

D. Simulation

Simulated measurements were obtained through the FEM

method using a mesh with the true thorax shape (obtained

as described above) and a conductivity contrast in the lung

region segmented from the corresponding CT image. The lung

to other tissue conductivity ratio was 0.1875 (as the average

of the expiration and inspiration values assumed in [8] and in

agreement with the ranges observed by Gabriel et al. [19] for

100 kHz current frequency). This simulation setup represents

well the practical use of EIT where measurements obtained on

a heterogeneous body are reconstructed using a homogeneous

model. The meshes of the human and pig chest contained 31×

103and 33×103first order tetrahedral elements, respectively.

E. Data

The animal data used in this study were obtained at

the University of Mainz, Germany, under appropriate eth-

ical approval (license no. 1.5 177-07/041-75, Landesunter-

suchungsamt Rheinland-Pfalz, 56028 Koblenz, Germany). CT

data were acquired during a period of apnea in a healthy 23 kg

swine. EIT data were recorded during conventional mechanical

ventilation in the same animal.

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GRYCHTOL et al.: IMPACT OF MODEL SHAPE MISMATCH ON RECONSTRUCTION QUALITY IN EIT5

41 11951

GN

Noser

GN

Laplace

TSVD

GREIT

Fig. 4.

Images show difference between inspiration and expiration in one breath cycle.

The number of Fourier coefficients used to define each shape is indicated at

the top.

Reconstructions on various model shapes of data from a healthy pig.

The human CT used in this study originates from a diagnos-

tic scan of a male volunteer (54 y.o., BMI 25.4, healthy lung

and heart) taken to investigate a non-thoracic condition and

donated by the subject to the EIDORS project for scientific

purposes.

III. RESULTS

A. Animal experiment data

Sample reconstructed images of animal data using all four

algorithms are presented in Fig. 4. For all algorithms, the more

circular the model shape is, the more distorted the lung shape

appears. Features along the longer vertical axis are pushed

together and lost, particularly at the ventral side. Qualitatively,

images reconstructed with the GREIT algorithm exhibit the

least artifacts but also the smoothest boundaries.

B. Performance measure analysis

Fig. 5 depicts for both models the mean and standard

deviation (normalized to the mean) across the map of each

performance measure as a function of the number of Fourier

coefficients retained to describe the model shape, while sample

performance measure maps of GREIT reconstructions on

selected porcine thorax model shapes are presented in Fig. 6

With few exceptions, the performance measures visibly worsen

as shapes become smoother, but not until the descriptor is

truncated to below 13 coefficients for the porcine thorax

(∆S = 2.77%) and 7 coefficients for the human chest shape

(∆S = 3.91%), cf. Fig. 2d. This corresponds to approximately

4% difference in model shape ∆S (for the porcine shape

described by 11 coefficient ∆S = 4.32%). None of the studied

algorithms are immune to the effect.

The performance of the two Gauss-Newton solvers is very

similar. The GREIT algorithm stands out for its higher (i.e.

worse) but more uniform resolution (c.f Fig. 7 and 8), lower

ringing and shape deformation. It also has higher position error

020 40

0.3

0.35

0.4

0.45

0.5

AR

mean

020 40

0.8

1

1.2

1.4

1.6

std/mean

0 2040

0

0.05

0.1

0.15

0.2

abs(PE)

0 2040

1

2

3

4

5

0 20 40

0.2

0.25

0.3

0.35

0.4

RES

02040

0

0.2

0.4

0 20 40

0.1

0.15

0.2

0.25

SD

02040

0.7

0.9

1.1

1.3

02040

0

1

2

3

4

RNG

0 2040

0

5

10

GREIT

(a) porcine thorax models

NOSERTSVD Laplace

020 40

0.5

1

1.5

2

AR

mean

020

0.75

0.8

0.85

0.9

0.95

std/mean

0 2040

0

0.05

0.1

0.15

0.2

abs(PE)

0 20

0.4

0.6

0.8

1

02040

0.2

0.25

0.3

0.35

0.4

RES

020

0

0.2

0.4

0 2040

0.1

0.15

0.2

0.25

SD

020

0.7

0.9

1.1

1.3

0 2040

0

1

2

3

4

RNG

020

0

1

2

3

4

5

GREIT

(b) human thorax models

NOSERTSVD Laplace

Fig. 5.

Fourier coefficients used to describe the model shape.

Dependence of algorithm performance measures on the number of

close to the boundary than other algorithms, which means that

changes close to the boundary are reconstructed more centrally

than they ought to. The TSVD algorithm exhibits low average

amplitude response, high shape deformation, high ringing as

well as lowest and most variable resolution – low (i.e. good)

at the boundary and high in the center.

IV. DISCUSSION

A. Figures of merit

In [10] a group of domain experts defined the desired

characteristics of an EIT reconstruction algorithm as (in order

of importance): 1) uniform amplitude response (AR), 2) small

and uniform position error (PE), 3) low and uniform ringing

(RNG), 4) uniform resolution (RES), 5) small shape deforma-

tion (SD), and 6) small RES. These are discussed below in

turn.

The behavior of amplitude response is similar for all tested

algorithms. In general, AR was not uniform across the image.

Higher values were recorded within the lung as compared

with the surrounding tissue, as shown on Figures

and 8. The results suggest that conductivity changes in less

conductive tissues (e.g. inflated lung) are overestimated. This

effect has potentially far reaching consequences for ventilation

monitoring in patients with atelectasis (lung collapse), where

EIT could underestimate ventilation in the more conductive

collapsed regions of the lung. We also observed that AR is

1, 7