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«Incorporating  a­priori  anatomical   information   into   image  reconstruction in electrical impedance tomography ...»

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Incorporating  a­priori  anatomical   information   into   image 

reconstruction in electrical impedance tomography

H Dehghani†, D C Barber‡ and I Basarab­Horwath†*

† School   of   Engineering,   Sheffield   Hallam   University,   City   Campus,   Pond 

 

Street, Sheffield, S1 1WB, UK

‡  Department   of   Medical   Physics   and   Clinical   Engineering,   Sheffield 

University and  Health  Authority, Royal Hallamshire  Hospital, Sheffield, S10  2JF, UK *Corresponding author Abstract. Image reconstruction in electrical impedance tomography using the  sensitivity   theorem   is   generally   based   on   the   assumption   that   the   initial  conductivity distribution of the body being imaged is uniform. The technique of  image   reconstruction   using   this   method   is   described   and   reconstructed  images   are   presented.   Improvements   in   image   quality   and   accuracy   are  demonstrated when accurate a­priori ‘anatomical’ information, in the form of a  model of the distribution of conductivity within the region to be imaged, is used  to construct the sensitivity matrix. In practice correct a­priori information is not  available, for example, the conductivity values within the various anatomical  regions will not be known.   An iterative algorithm is presented which allows  the   conductivity   parameters   of   the  a­priori  model   to   be   determined   during  reconstruction.

1. Introduction The methodology of two­dimensional (2D) electrical impedance tomography  (EIT) has been described in detail elsewhere (Barber and Brown 1984, 1990,  Brown   and   Seagar   1987).   Electrodes   are   positioned   with   equal   spacing  around   the   body   to   be   imaged   thus   defining   a   plane   through   the   object.  Voltage   profiles   are   collected   for   all   drive   and   receive   electrode­pair  combinations and images are reconstructed as though the data were from a  2D   object.   In   this   work   16   electrodes   have   been   placed   around   a   circular  object.   Current   is   driven   into   the   object   through   two   adjacent   (drive)  electrodes. For each of the 16 drive­electrode pairs, 13 voltage differences  between the remaining adjacent non­current carrying (receive) electrodes are  recorded. In practice objects are three­dimensional (3D); current cannot be  confined   to   one   plane.   However,   the   reconstruction   algorithm   usually  assumes   that   the   object   is   2D.   This   paper   will   only   deal   with   proper   2D  reconstruction.

Previous investigations of such an approach includes the incorporation of a­ priori  information   into   the   Sheffield   filtered   backprojection   image  reconstruction algorithm, which showed some success (Avis et al 1995). Also,  Zadehkoochak et al (1991) presented a reconstruction algorithm based on the  inversion of the sensitivity matrix associated with a non­uniform conductivity  distribution using the singular value decomposition (SVD) method. They did  not present any reconstructed images. Zadehkoochak et al (1993) have also  investigated the use of a­priori information associated with imaging the thorax  and reported artefacts in the resulting images. The use of a universal model  (a standard model of internal conductivity distribution within the human thorax)  has been speculated by Zadehkoochak et al (1993) to be invalid; however, it  will   been   shown   that   providing   the   assumed   anatomical   model   is   closely  matched to the patient's anatomy an improvement in the reconstructed image  is achieved.

The conductivity of a region, discretised into small areas or elements, can be  written as a column matrix or vector. When this conductivity changes from a  reference   conductivity   distribution  σ ref  (where   each   element   has   units   of  ­1 conductivity (Sm )) to a conductivity  σ dat  (where each element has units of  ­1 Sm ) it represents a change in conductivity ∆ σ  = σ dat ­ σ ref, where ∆ σ  is a  vector   of   the   same   size   as   both  σ dat  and  σ ref.   Images   of   this   change   in  internal   impedance   distribution   can   be   reconstructed   from   the   resulting  change in differential boundary voltages measured in volts (∆ v = vdat  ­ vref)  from a uniform conductivity distribution, where vdat , vref  and  ∆ v are column  vectors, usually of a different size from the conductivity vectors; the actual  size depending on the number of independent differential boundary voltage  measurements. These images of conductivity change can be obtained using a  relationship   described   by   a   sensitivity   matrix   (S)   which   is   derived   using   a 

theorem by Geselowitz (1971). This relation is given by:





∆v = S∆σ (1) where  S  is   a   sensitivity   matrix     (usually   non­square),  ∆ σ   is   a   vector  containing   the   changes   in   conductivity   and  ∆ v   is   a   vector   containing   the  corresponding   changes   in  voltage   profiles.  S  relates   the   small   conductivity  change   within   each   individual   pixel   of   the   image   to   the   corresponding  differential boundary voltage change for each and every pixel and each and  every drive­receive electrode combination. The relationship between ∆ v and  ∆ σ   is often assumed to be linear (Barber and Brown 1990) and thus the  elements   of  S  are   independent   of   conductivity.  S  is   usually   calculated   by  assuming that the initial conductivity distribution is uniform, i.e. all elements  within a region to be imaged have the same conductivity value (Barber and  Brown 1990) and then calculating the resulting boundary voltage changes due  to small perturbations in the conductivity values for each and every element.  The   sensitivity   matrix   derived   in   this   manner   is,   in   this   paper,   called   the  uniform sensitivity matrix, denoted by Sunif. However, the relationship between  the conductivity changes and the corresponding differential boundary voltages  is   not   linear   and   virtually   no   interrogated   region   is   uniform.   The   current  distribution and therefore the voltage distribution within a body depends on  the conductivity distribution within the body and as a consequence this is also  true of the sensitivity matrix. For example, a small change in conductivity next  to   a   physically   large   conductive   area   will   have   a   smaller   effect   on   the  boundary   voltages   than   the   same   small   change   in   conductivity   next   to   a  relatively small area. Thus the coefficients of the true sensitivity matrix vary  with the conductivity distribution and this matrix is referred to in this paper as  the ideal sensitivity matrix, Sideal. In general therefore, although the sensitivity  matrix  S  changes   with   the   conductivity   distribution   and   equation   (1)   is  therefore non­linear, it has been argued that for small changes in conductivity  this non­linearity can be ignored (Barber and Brown, 1990). Shaw et al (1993)  have also reported a similar result, namely that for a small region, the linear  approximation   is   valid   for   the   conductivity   changes   of   up   to   300%   of   the  background.

We have determined that the assumption of linearity by Barber and Brown  (1990) is violated in conditions where there is a wide range of conductivity  values; this is in accordance with findings by Seagar (1983). Given accurate  spatial and conductivity information about a region to be imaged, an accurate  image of the impedance distribution can be reconstructed from the boundary  voltage data. Although this may be self­evident, it shows whether or not the  relationship described by equation (1) is valid. It also shows the extent of the  blurring of the image due to the algorithm and computational process. Also, it  serves as a benchmark for images obtained using less  a­priori  information  and as such is the "best" image that can be obtained. Recognising that in  general complete anatomical information will not be available, this paper will  address the degree of accuracy needed in the a­priori information in order to  produce a useful reconstructed image. If the method is reasonably robust it  may be possible to use anatomical information taken from a database rather  than   from   the   subject   being   imaged,   contrary   to   previous   speculations  (Zadehkoochak et al (1993)). However it will be shown in the work presented  in   the   paper   that   providing   the   model   is   closely   matched   to   that   of   the  interrogated area an improvement in image quality is achieved. It also will be  demonstrated   how   the   present   widely   used   sensitivity   algorithm   can   be  combined   with   an   approximate   knowledge   of   the   spatial   distribution   of   the  tissues to provide a better estimate of the conductivity values.

2. Method

In order to generate the sensitivity matrix S the area of interest, a 2D circular  tank, is split into 1920 triangular and brick elements as shown in figure 1. The  elements   representing   the   modelled   electrodes   are   not   shown   in   the  reconstructed images and the central elements are not displayed well due to  their small size.

The sensitivity matrix S is a matrix of 208 by 1920 coefficients. The 208 rows  of the S matrix relate to the 208 different differential voltage readings and the  1920   columns   to   the   sensitivity   coefficient   for   each   independent   element.  There are 13 voltage measurements for each of the sixteen projections, giving  208   voltage   readings.   Each   of   the   1920   elements   in   the   model   has   its  conductivity   value   perturbed   from   a   uniform   value   and   this   small   change  generates   208   differential   boundary   voltage   changes.

  The   sensitivity  coefficient for each element i was calculated from Geselowitz’s lead theory  (Geselowitz 1971) and is given by:

S( j,i ) = − ∫ ∇Φ m ⋅ ∇Ψn ⋅ du (2) u where j = j(m,n), Φ m is the potential distribution generated in an object when  unit   current   is   passed   through   the   electrode   pair   m   before   a   change   in  conductivity (σ ref) and Ψ n is the potential distribution produced if unit current  had been injected through electrode pair n after the change in conductivity to  σ dat  has occurred. j is the drive and receive electrode combination and the  integration   is   over   the   area   of   the   element   u,   i.e.   multiplication   of   the   dot  product with the area of the element u.

The  electric  fields  ∇Φ   and  ∇Ψ   are  calculated   for each   element  using  an  available finite element package. The two vector components of the electric  field (Exi and Eyi) are calculated at the centre of each element for every drive­ electrode combination.

As Φ m and Ψ n are derived for different conductivity distributions equation (1)  is   non­linear   in   terms   of   conductivity.   However,   for   small   changes   in  conductivity   about  σ ref,  Ψ n  can   be   replaced   by   the   equivalent   potential  calculated   for  σ ref  and   this   linearises   the   problem.   For   large   changes   the  assumption of linearity is no longer correct: a problem this paper addresses.

The   linearised   sensitivity   matrix  Sunif  is   calculated   assuming   that   the  conductivity distribution is uniform before and after a change has occurred  and this is the matrix which has been used to date even when the reference  conductivity is not uniform (Barber and Brown 1990, Kotre 1989).

The   sensitivity   coefficient   for   each   element   of  Sunif  was   calculated   using  equation (3), where for the electric field (Exm, Eym) in element i is due to the  current injected through electrode pair m and the electric field (Ex n, Eyn) is  that produced when the same current is injected through electrode pair n.

–  –  –

The integration is again over the area of the element u.

In principle the calculated sensitivity matrix can now be used to reconstruct an  image of the change in conductivity distribution calculated from the boundary  potential differences.  This relationship  can  be  derived  from equation  (1) to 

give:

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