WO1989009564A1

WO1989009564A1 - Tomography

Info

Publication number: WO1989009564A1
Application number: PCT/GB1989/000363
Authority: WO
Inventors: Brian Hilton Brown; David Charles Barber
Original assignee: The University Of Sheffield
Priority date: 1988-04-15
Filing date: 1989-04-10
Publication date: 1989-10-19
Also published as: GB9023905D0; GB2236397B; AU3446589A; IT1229015B; GB2236397A; FR2630235A1; GB8808969D0; FR2630235B1; IT8920139A0; AU628334B2

Abstract

A method of determining the external shape of a body comprises placing a plurality of electrodes adjacent the surface at spaced intervals round the body, causing currents to flow in the body by applying an electrical potential between pairs of electrodes in turn, measuring potentials between other pairs of electrodes, relating the measured potentials to the distances between the corresponding points of current introduction and of potential measurement, and determining electrode positions consistent with these distances, which electrode positions define the external shape of the body.

Description

TOMOGRAPHY

This invention relates to tomography of the type described in GB-PS 2119520B in which tomographic images of a body are constructed by placing a plurality of surface electrodes at spaced intervals on the body, causing currents to flow in the body, and measuring the potentials between pairs of electrodes, calculating the potential in each case on the assumption that the body consists of one uniform medium, plotting the i sopotenti als corresponding to the calculated results to create a uniform image of the body, obtaining the ratio between the measured potential and the calculated potential in each case, and modifying the image in accordance with the respective ratios by increasing the impedance along an iεopotential in proportion to a ratio greater than unity or decreasing the impedance in proportion to a ratio less than unity.

The calculations of potentials and the obtaining of ratios have been carried out using a computer, and the plotting of the i so potential s have been carried out by a visual display unit (VDU) and/or a print-out run off the computer.

An additional method is provided for indicating a change of state in the body in GB-PS 2160323B, and this method includes the step of determining ratios between initial and subsequent measured actual potentials between electrodes.

Both methods are described in US-PS 4617939 and both methods assume that the cross-sectional shape of a body is circular and thus produce circular images, which may not be satisfactory if the true cross- sectional shape is far from circular, as is frequently the case.

An object of the present invention is the determination of the external shape of a body preparatory to obtaining to ographic images of it.

Another object is to construct a tomographic image that accurately depicts the internal structure of the body.

According to the present invention, a method of determining the external shape of a body comprises placing a plurality of electrodes adjacent the surface at spaced intervals round the body, causing currents to flow in the body by applying an electrical potential between pairs of electrodes in turn, measuring potentials between other pairs of electrodes, relating the measured potentials to the distances between the corresponding points of current introduction and of potential measurement, and determining electrode positions consistent with these distances, which electrode positions define the external shape of the body.

The external shape of the body can be plotted, e.g., on a VDU and or as a print-out, by means of a computer, on the assumption that the body consists of one uniform medium and plotting the image of electrode positions corresponding to the measured values.

The method of determining the external shape of a body will now be described with reference to the diagrammatic

Figures 1 to 3 of the accompanying drawings.

The voltage measured between a pair of electrodes connected to a body surface when a current is applied between a second pair of electrodes applied to the body surface depends on (a) the relative positions of the electrodes on the surface; (b) the shape of the body surface and (c) the internal impedance distribution within the body. (a) is the dominant factor followed by (b) and then (c). However the use of a calculated or measured reference set allows the contributions from (a) and (b) to be largely eliminated from the image reconstruction calculation; the differences in the sets of measurements between the 'reference' set and the •data' set are dependent almost solely on the changes in internal impedance. Because the effect of boundary shape is supressed the shape of the boundary for the reconstructed images is by default circular.

The following describes how a single set of measurements ('data') can be used to calculate the shape of the boundary and generate a computed set of reference measurements.

THEORY

If N electrodes defining a plane are connected to the body surface then a measurement set consists of N(N-3)/2 independent measurements if drive electrodes are adjacent to each other or N(N-4)/2 otherwise. The voltages measured between a pair of electrodes (k,l) when a current I is applied between a second pair of electrodes (i,j) is given by (Figure 1)

^Vijkl = I.F(_ri,_rj,_rk,_rι,b(_r) ,c(_r)) where ri r-i rk_rrl ^{are t}*^ιe positions of the electrodes, b (r_) is the function representing the boundary and c(_r) is the internal impedance distribution. Since the Vijkl ^{are on}--y weakly dependent on c(r) this can be replaced by c, a uniform impedance throughout the object. Similarly since the dependence of V^^ _{on b} .) _{is wea}ker than the dependence on electrode position, these can be combined into a single function F¹ of the electrode position. in the absence of a boundary this function can be obtained from simple geometric considerations alone. In the presence of a boundary this function will deviate from this ideal form in a way that may be modelled empirically. In summary

Vijkl - ICP' irJZjrXkril) ⁽¹⁾

Given the set of measured V and a model of the function F¹ the position vector s r. of the N electrodes may be computed and hence the boundary determined. In the absence of an exact model of F¹ the fact that there are at least N(N-4)/2 independent measurements and only 2 electrode coordinates to compute means that the problem is usefully over-determined leading to a robust solution for the electrode positions. For a similar reason these computed electrode positions are not sensitive to the internal impedance distribution; data from non-uniform and uniform internal impedance distributions will generate almost the same electrode positions. By inserting the electrode positions back into Equation 1 values

°^ ^Vijkl corresponding to a 'reference' distribution may be generated and used to calculate images of absolute impedance in conjunction with the measured 'data¹ set.

REAL ISA ION

If the voltages are always measured between adjacent electrodes, current is also always applied between adjacent electrodes, the spacing between adjacent electrodes is constant and this spacing is small compared to the dimensions of the body then empirically the function F may be represented by

F^» (d) = 1 ikⁿ or, representing

^vi,i+l,k,k+l by Vik

Vik Ic ^dikⁿ where di •**•-**■* ^t*^tιe distance between the midpoint between the electrodes i and i + 1 and the midpoint between electrodes k and k + 1 (Figure 2). A value of n = 3.7 has proved effective.

The starting point for determining the boundary shape is a circular boundary with equally spaced electrodes. Two pairs of electrodes are taken (typically those placed at the ends of the maximum diameter of the body) and the coordinates of these electrode pairs fixed. The distance between them is chosen, (this fixes the scale of the boundary) and the constant C in ^vik = §-₇ (2) d.. ^■3-

1.. is computed from this distance and the measured value of V-^. Changing the scale of this reference d simply changes the value of the constant C. For the remaining pairs of 'drive' and 'receive' electrode pairs the distances between them are

Y calculated from di ~

Consider a 'drive' pair initially situated on the circular boundary and a 'receive' pair also on the boundary. The value of d is computed and this distance is measured out along the line joining the 'drive' and 'receive' pairs from the 'drive' pair. The point at the end of this distance is now taken to be a new estimate of the position of the 'receive' pair. This process is repeated for all combinations of 'drive' and 'receive' electrode pairs. For each electrode pair there are N-3 position estimates relative to all other pairs seen as 'drive* electrode pairs and the position estimates are averaged. These position estimates now define a new boundary, no longer circular. The process of position estimation is now repeated until no further change in the position estimates occurs. Estimation of the electrode position (and boundary shape) is now complete (Figure 3). Faster convergence may be obtained if a starting boundary shape (e.g. an ellipse) closer to that of the required boundary shape than a circle is chosen. Once the boundary shape has been defined an appropriate form of the function F' in equation 1 may be determined from the measured values of V_i;j_kl and the calculated electrode positions and used to calculate reference values.

More accurate estimates of boundary shape may be achieved by (a) a more complex version of F' than that given in Equation 2, such as

1.0 and m = 2.6,

or (b) choice of a simple form for F¹ but restriction of its use to a pre-εelected subset of drive and receive pairs.

A tomographic image that accurately depicts the internal structure of the body can be constructed by calculating the potential between pairs of electrodes on the assumption that the body consists of one uniform medium, plotting the isopotential s corresponding to the calculated results to form an image, obtaining the ratio between the measured potential and the calculated potential in each case, and modifying the image in accordance with the respective ratios by increasing the impedance along an isopotential in proportion to a ratio greater than unity or decreasing the impedance in proportion to a ratio less than unity, as described in GB-PS 2119520B. A typical print-out image is shown in Figure 4 of the accompanying drawings.

Again, as described in GB-PS 2160323B and US-PS 4617939, the method can be adapted for indicating a change of state in the body by determining ratios between initial and subsequent measured actual potentials between electrodes.

Claims

1. A method of determining the external shape of a body comprising placing a plurality of electrodes adjacent the surface at spaced intervals round the body, causing currents to flow in the body by applying an electrical potential between pairs of electrodes in turn, measuring potentials between other pairs of electrodes, relating the measured potentials to the distances between the corresponding points of current introduction and of potential measurement, and determining electrode positions consistent with these distances, which electrode positions define the external shape of the body.

2. A method as in Claim 1, wherein the external shape of the body is plotted on a VDU by means of a computer, on the assumption that the body consists of one uniform medium and plotting the image of electrode positions corresponding to the measured values.

3. A method as in Claim 1 or Claim 2, wherein the external shape of the body is plotted as a print-out by means of a computer, on the assumption that the body consists of one uniform medium and plotting the image of electrode positions corresponding to the measured values.

4. A method of constructing a tomographic image that accurately depicts the internal structure of a body comprising the method of determining the external shape of the body as in Claim 2 or Claim 3 and then calculating the potential between pairs of electrodes on the assumption that the body consists of one uniform medium, plotting the isopotential s corresponding to the calculated results to form an image, obtaining the ratio between the measured potential and the calculated potential in each case, and modifying the image in accordance with the respective ratios by increasing the impedance along an isopotential in proportion to a ratio greater than unity or decreasing the impedance in proportion to a ratio less than unity.

5. A method of indicating a change of state in a body comprising constructing tomographic images by the method of Claim 4 determined by ratios between initial and subsequent measured potentials between electrodes respectively before and after a change in the state of the body.