The Mainz Electric Impedance Tomograph

EIT: Imaging of resistance or impedance densities in the interior of an objectfrom measurements of currents and potentials on its surface

� Karl Schilcher, Hubert Spiesberger, Karl-Heinz Georgi, Christian Hähnlein(Johannes Gutenberg Universität Mainz )

� Cristiana Sebu (Oxford Brookes)

Physics Kolloquium, UCT, March 2011

Industrial applications:

pressure ltration, polymerization, solid-state fermentation, product ageing, u-idized beds, water or air bubbles in oil pipelines

Geophysics

Medical applications:

detection of pulmonary emboli, monitoring of apnoea,

monitoring heart function and blood ow;

breast cancer detection and identication

.

Tissue Conductivity (mS/cm)Cerebrospinal uid 15.4Blood 6.7Sceletal muscle 8.0 (long), 0.6 (trans.)Cardiac muscle 6.3 (long), 2.3 (trans.)Neural Tissue 1.7Lung (expiration-inspiration) 1.0-0.4Fat 0.36Bone 0.06Breast (normal tissue) 0.3-0.4Breast (malignant tissue) 2.0-8.0

Well posed (direct) problem:

a unique solution exists for all data,

the solution depends continuously on the data

Ill-posed (inverse) problem: (Hadamard criteria)

(a): Existence: solution does not exist for arbitrary data

(b): Uniqueness: solutions are not unique

(c): Stability: solution does not depend continuously on the data,

leads to unstable solutions for error-prone measurements

The last criterion creates the most trouble in EIT

How to cure ill-posedness

A physically acceptable solution need not be the exact solution to the prob-lem but only an approximate solution, i.e., it reproduces the data within theexperimental errors.

The set of approximate solutions is extremely large.

Search for approximate solutions satisfying additional constraints coming fromthe physics of the problem

Regularization: choose the solution which is "closest" to the exact solution andhas the smallest norm

Examples: Tikhonov regularization, truncated SVD

Mathematical Setting of EIT

Smooth domain � Rn, n � 2, boundary @, conductivity �(~x), 0 < c �� � C.

A potential �(~x) in is described by the Poisson equation (a low-frequencyapproximation of Maxwells equations)

~r �h�(~x)~r�(~x)

i= 0 in

At 10-50 kHz the magnetic eld can be neglected (quasi-static situation), onlythe ohmic component of the impedance is taken.

Measurements: (on the boundary)

Voltage (Dirichlet data): f = �j@

Current ux (Neumann data): j = �

@�

@n

����@

Neumann-Dirichlet map:

�� : j 7�! f

The D-N map includes all possible (innitely many) boundary measurementstaken with innite precision.

Direct problem: Given �(~x), nd ��. This is a non-linear but well-posed(nice) problem.

Inverse problem: Concerns the inversion of the mapping � 7�! ��, i.e. given��, nd �(~x).

Theorem: Under certain smoothness assumptions, �� determines �(~x) uniquely.

Example: Suppose = D � R2 is a unit disc in two dimensions with aconcentric circular anomaly in the conductivity

�(x) =

(�1; � < jxj < 1�2 < jxj � �

dene � =�1 � �2�1 + �2

The direct problem can be solved by separation of variables with the result

��

"cosn�sinn�

#= �n

"cosn�sinn�

#with

�n =1

n� ��n and ��n =

2

n

��2n

1 + ��2n(� < 1)

��n !n!1 0 exponentially fast

Therefore cosn� and sinn� are the eigenfunctions of the Neumann-Dirichletmapping with eigenvalues �n, while 1=n are the eigenvalues of the N-D map-ping for � = const:(� = 0). If we want to disentangle deviations from� = const: we have to know the ��n.

Inverse problem: Given �� or equivalently �n nd �.

Arbitrary Neumann boundary data can be expanded as

j(�) =Xnan cosn� + bn sinn�

Then the Fourier coe¢ cients of the boundary potential data are�1

n� ��n

�an and

�1

n� ��n

�bn with ��n ! 0 exponentially fast

and hence in practice only few eigenvalues are experimentally available. Fora given level of measurement precision, we can construct a circular anomalyundetectable at that precision. Hadamards 3rd condition is violated.

EIT is a non-linear, exponentially ill-posed problem.

Integral Equation

The basic equation of EIT

~r �h�(~x)~r�(~x)

i= 0

can be written as

��(~x) = ��~r ln�(~x)

�� ~r�(~x)): (1)

Formal solution (Greens second identity):

�(~x) = h�iS+Z@

GN(~x; ~x0)@�

@n

����@

da0 +

Z

GN(~x; ~x0)r ln�(~x0)�r�(~x0)d3x0 :

(2)where GN(~x; ~x

0) is the Neumann-Greens function of the Laplace equation forthe given boundary condition (current on the boundary).

Assuming that the conductivity is constant near the boundary,

�(~x)j@ = �0 = 1 ;

the solution (2) becomes

�(~x) = �0(~x) + h�iS +Z

GN(~x; ~x0)r ln�(~x0) � r�(~x0)d3x0; (3)

where �0(~x) is the (known) solution of our boundary value problem for � =�0 = 1, i.e.

�0(~x) =Z@

GN(~x; ~x0)j(~x0)da0

We can impose the normalization

h�iS =I@

�(~x)da = 0 .

To linearize the problem, we approximate �(~x0) = �0(~x0) on the r.h.s. ofeq.(3) and carry out a partial integration to get

��(~x) =Z

r0GN(~x; ~x0) �r�0(~x0)d3x0 ln�(~x0) (4)

where

��(~x) = �(~x)� �0(~x)

with �(~x) measured and (�0(~x) calculated).

If ��(~x) is known on @ (standard EIT), Eq.(4) represents an integral equationfor ln�.

(the same holds if ��(~x) is given at points inside )

Standard EIT for a circular domainThe Neumann Greens function of the Laplace equation for a circular disc ofradius R reads

GN(r; r0; �; �0) = � lnf(r2+r02�2rr0 cos(���0))�(R2+ r

2

R2r02�2rr0 cos(���0))g

(5)We apply a special set of "current congurations", the eigenfunctions of theNeumann-Dirichlet mapping for � = 1.

jk = uk(�) �"cos k�sin k�

#The associated "free" potentials �k0(~x) are determined by the same Greensfunction

�k0(~x) =I@

GN(~x; ~x0)uk(~x0)da0 =

1

k

�r

R

�kuk(�):

From eq.(4). we then obtain a system of integral equationsD��; ui

E@

=Z

Ki(~x0) ln�(~x)d3x0 (6)

whereD��; uj

E@

=I@

��(~x)ui(~x)da and Ki(~x) = �r�i0(~x)r�0(~x):

Eq.(6) can be inverted to obtain the conductivity �(~x).

EIT of the lungs

A ring of outer electrodes on which currents can be injected and voltagesmeasured.

The inversion of the measured Neumann-Dirichlet mapping allows, in principle,to calculate the conductivity in the interior.

Distinguishability is maximal when the voltages (or currents) are modulatedwith the eigenfunctions of the Neumann-Dirichlet mapping corresponding tothe largest eigenvalue. In the case of a circular two-dimensional boundary,these are the sinus and cosinus functions

The Problem of Contact Impedance

EIT requires skinelectrode contact and typically uses ECG or EEG-type elec-trodes, which are placed on the skin with a conducting gel to improve contact.

Contact impedance: Total electrode-gel-skin impedance,

1. an electrochemical component, arising from the conversion ofelectron current to ionic current at the interface betweenthe metal electrode and ionic gel,

2. the skin impedance of the layers of the stratum corneum.

Sources of error:

� Changes in sweat ducts over time and variations in electrode size and areaof e¤ective contact may cause variations in contact impedance of up to20% on the same subject.

� This is a serious concern for EIT as the contact-impedance may be ordersof magnitude higher than the impedance of interest.

Example: The resistance between two electrodes of 1cm diameter placed onthe forearm should be about 2 if the conductivities in the table are used. Anactual measurement, however, gives at about 1k at 20kHz.

The errors due to contact impedance uncertainties are often larger than thee¤ect of inhomogenities in the interior.

Active electrodes: Currents can be measured accurately, potentials measure-ments are useless

Passive electrodes: Potentials are measured with no current drawn. The po-tential on the skin surface is equal to the potential in the adjacent inner tissue.

The Mainz Electric Impedance Mammograph (EIM)

Alternative EIT device: Reconstruct the electrical conductivity from boundarymeasurements of currents only, plus interior measurements of the potential atisolated points

Active electrodes: 12 circular outer electrodes, arranged regularly on a circle,currents up to 50mA and frequencies between 20 to 40kHz, measured with anaccuracy of about 0:1%. Their amplitudes are distributed over the electrodesin the form of sinusoidal patterns.

Passive electrodes: 54 small spring mounted high impedance inner electrodes

Voltages can be measured with 16 bit accuracy in the interior on a grid con-sisting of regularly distributed hexagons.

Two reconstruction algorithms: one is based on discrete resistance networkmodel, the other one, for smooth conductivities, on an integral equation.

Alternative EIT setup which is not based on the Neumann-Dirichlet mapping.

Combination of applied currents on the boundary with potential measurementsat isolated points in the interior.

Separating current and voltage electrodes circumvents the issue of the contactimpedance.

Data: so far only synthetic data and data taken on a salt water tank withvarious objects immersed.

Absolute data reconstruction without use of a reference conductivity

Resistance network

Solution of the Inverse Problem

We introduce a compound index nfk; lg to enumerate the N links connectingknot k; with knot l via

n = n(fk; lg) = 1 : : : N ; k > l ;an N -dimensional vector of conductivities Sn(fi;jg) assigned to the links.and a K �N dimensional matrix Uik of potential di¤erences,

Uin =

(�i � �k with n = n(i; k) and k is a neighbor of i

0 otherwise

Kirchho¤s rst law (current conservation) combined with Ohms law give0BBBBBB@U11 U12 : : : : : : U1NU21 � : : : : : : �� � : : : : : : �� � : : : : : : �UK1 � : : : : : : UkN

1CCCCCCA �0BBBBBB@S1:::SN

1CCCCCCA =0BBBBBB@c1:::cK

1CCCCCCA ;

For a single measurement, this linear system of equations for the unknownconductivities Si is, in general, under-determined .

If the measurements are repeated enough equations are generated to allow asolution.

This system is then usually over-determined and can be solved conveniently bygeneralized inversion and the singular value decomposition.

Integral equation method

This is formally the same as in the standard EIT approach described above,except that the input boundary potentials are replaced by internal potentials

Types of current patterns:

There are Sext = 12 active external electrodes, Sext � 1 linearly independentcurrent patterns.

a) 10 patterns of the varying-frequencyform I(m)(�j) � sinm�j and I(m)(�j) �cosm�j, where �j =

2�Sext

is the angular position of the external electrodej, m = 1; : : : ; Sext � 1 and j = 1; : : : ; Sext. (eigenfunctions of theNeumann-Dirichlet map for a constant conductivity).

b) 12 patterns of the shifted-phasetype I(m)(�j) = sin(�j +m2�12)

(lead to parallel current lines). Only 2 patterns are linearly independent,for example sin � and cos �.

Procedure:

a) We reconstruct the resistances using the 54(36) internal potentials and the18(24) currents entering the hexagonal net.

b) We use our knowledge of how the 12 currents applied to the tomographsplit up at the rst layer in the case of constant conductivities.

c) We use from 2 to 10 measurements in the reconstruction algorithm.

d) We add errors to the inputs of currents and potentials.

Example:

Reconstruct the resistances using the 36 internal potentials and the 24 currentsentering the hexagonal net

Use all 10 current patterns of the varying-frequencytype for

r = 4mm; x0 = �10mm; y0 = 0; w = 0:4mm;�0 = 0:3; �1 = 1 (arbitrary units):

Original zero errors

100% errors oncurrents

1% errors onvoltages

The reconstruction based on the Neumann mapping is remarkably insensitivewith respect to errors in the injected currents!

Real Tank Data:

Images of a metallic cylinder of diameter 15mm and height 15mm

pos_1_2mm.png pos_1_5mm.png pos_1_10mm.png

pos_2_2mm.pngv

pos_2_5mm.png pos_2_10mm.png

pos_3_2mm.png pos_3_5mm.png

pos_3_10mm.png

Conclusions:

� Astonishing insensitivity to errors on the injected currents

� Relatively high sensitivity to errors of potentials

� 2 measurements su¢ ce for a unique solution, more measurements improvethe quality of the image.

� The ill-posed nature of the problem has been resolved by the additionalinternal potential data.

� Promising results with two independent reconstruction algorithms

Next Steps:

a) Tests under realistic medical environment

b) Optimize imaging

c) Improved accuracy of voltage measurements (di¤erential ampliers)

d) 3D extension (halfspace)

Additional Material

Breast Cancer

Breast cancer is the most common malignant neoplasm in women:

A¤ects every 11 th woman in her lifetime

constitutes 40% of newly diagnosed cancers (35-59 j.)

causes 30% of cancer deaths

(Robert Koch Institute, 2005)

Breast cancer is presently routinely detected by palpation, by X-ray mammog-raphy and by ultrasound imaging. These classical diagnostic methods, however,yield rather unspecic results. Only one in ve biopsies of suspicious lesionsleads to a malignant histological diagnosis. In addition, it is estimated, that10-25% of breast carcinomas are not detected by classical screening. Researchis therefore aimed at developing alternatives imaging techniques to diagnosebreast cancer more accurately and earlier. One approach is to make use ofelectrical impedance tomography (EIT). As the specic conductivity of maligntissues is up to four times higher than that of healthy tissues imaging the elec-trical properties of breast tissues will yield important information which canimprove the specicity of the diagnosis.

Example of an inverse problem: The Cauchy problem for the Laplace equa-tion in two variables

@2u

@x2+@2u

@y2= 0

Boundary condition (data)

u(x; 0) =1

ncos(nx);

@u

@y(x; 0) = 0

) unique solution

u(x; y) =1

ncos(nx) cosh(ny)

for n!1 the Cauchy data tends to zero while the solution tends to innityany nite value of y (cosh(ny) = 12e

ny + 12e�ny). Small errors in the data

lead to large errors in the solution.

Synthetic data

�(x; y) = 1000 �"�1

1� exp (d� r)

w

!+ �0

#�(R2 � x2 � y2)

with d =q(x� x0)2 + (y � y0)2 :

We vary the following parameters:

the height of the object: h = �1=�0, with both h > 0 and h < 0,

the size of the object: r,

the position of the object: x0; y0,

the smoothness of the object at its boundary: w.

Neumann-Dirichlet mapping for � = const: = 1:

�� fj(~x)j@g = �(~x)j@ :

The current density on the boundary ( � = 1; r = 1) can be expanded in aFourier series:

j(�) = �@�(r; �)

@rj@ =

1Xk=1

ckeik�

This leads to

�(r; �) =1

k

1Xk=0

ckrkeikr

and on the boundary

�(r = 1; �) =1Xk=0

ckeik� :

Discretization (Standard EIT)

��i =

NpXk

Kik ln�k

with��i = h��; uii@

Kik = �~r�i0(~xk)~r�0(~xk)�V (~xk)ln�k = ln�(~xk)

where �V (~xk) is the volume element.

Resistance Network Details:

� 54 Knots (at the electrodes where potentials are measured)

knot number: k; k = 1; ::::K with potentials �k

� 72 Links fk; lg (k 6= l) connecting knot k to knot l

with resistances R(fk;lg

� 18 currents ck are the currents at the knots. For the outer layer of knots ckare the applied (calculated splitting ratios) currents. For the inner knots,ck = 0. Current conservation:

Pkck = 0.

For a single measurement, this linear system of equations for the unknownconductivities Si is, in general, under-determined .

The measurements are repeated:

C(a) = (c(a)1 ; c

(a)2 ; : : : ; c

(a)K )

T independent current patterns

U(a)ik corresponding potential di¤erence matrices

� = (S1; S2; : : : SN)T the unknown conductance vector

Enough equations are generated to allow a unique solution:

U (1) � � = C(1)U (2) � � = C(2)

��

�!

0BBBB@U (1)

U (2)

��

1CCCCA � � =0BBBB@C(1)

C(1)

��

1CCCCA : (7)This system is usually over-determined and can be solved conveniently by gen-eralized inversion and the singular value decomposition.