Iterative approach for applying multiple currents to a body using voltage sources in electrical impedance tomography

• US 20050251062A1
• Filed: 05/06/2005
• Published: 11/10/2005
• Est. Priority Date: 05/10/2004
• Status: Abandoned Application

• Current Assignee: Rensselaer Polytechnic Institute
• Original Assignee: Rensselaer Polytechnic Institute
• Inventors: Isaacson, David, Newell, Jonathan C., Choi, Myoung
• Application Number:
  US11/124,246
• Publication Number:
  US 20050251062A1
• Time in Patent Office: Days
• US Class Current: 600/547
• CPC Class Codes: A61B 5/0536 Impedance imaging, e.g. by tomography

Abstract

Voltage sources produce desired current patterns in an ACT-type Electrical Impedance Tomography (EIT) system. An iterative adaptive algorithm generates the necessary voltage pattern that will result in the desired current pattern. The convergence of the algorithm is shown under the condition that the estimation error of the linear mapping from voltage to current is small. The simulation results are presented along with the implication of the convergence condition.

Claims

1. An electrical impedance tomography method for determining at least one of an electrical conductivity and an electrical permittivity distribution within a body from measurements made at a plurality of electrodes spaced on a surface of the body, the method comprising:

(a) providing a plurality of voltage sources for producing a plurality of voltage patterns that are each calculated using an iterative calculation process;

(b) applying the calculated voltage patterns to the electrodes to create resulting current patterns in the body; and

(c) measuring the resulting current patterns at the electrodes to determine at least one of the conductivity and permittivity distributions within the body;

(d) the calculation process comprising;

(i) selecting a desired current vector and an error tolerance;

(ii) using a first algorithm to compute an orthonormal basis set;

(iii) using a second algorithm with the orthonormal basis set and the desired current vector to compute an estimate of a non-singular linear mapping matrix for converting coordinate vector for voltage vector with respect to the orthonormal basis set to coordinate vector for current vector with respect to the orthonormal basis set and to compute coordinate vector for the desired current vector;

(iv) computing and applying to the electrodes, the voltages of the voltage vector as a function of the estimate of the non-singular linear mapping matrix and the coordinate vector for the desired current vector;

(v) measuring the resulting current vector;

(vi) computing the coordinate vector for the measured resulting current vector with respect to the orthonormal basis set;

(vii) calculating a norm of the actual error between the coordinate vector for the measured resulting current vector and the coordinate vector for the desired current vector; and

(viii) if the norm of the actual error is greater than the selected error tolerance, repeating steps (iv) to (viii), and if the norm of the actual error is less than the selected error tolerance, using the computed voltage vector of step (iv) as one of the calculated voltage patterns to perform step (b).

2. An electrical impedance tomography method according to claim 1, wherein the voltage source comprises a resistor and an operational amplifier as a measuring circuit for measuring a signal I_out which is a measure of current that is fed to said electrodes.

3. An electrical impedance tomography method according to claim 1, wherein the first algorithm includes the steps of:

providing

let T^k;

L×

1 vector, K=1,2, . . . L-1

$T_i^k=\{\begin{array}{cc}1,& i=k\\ -1,& \begin{array}{cc}i=k+1,& i=1,2,\mathrm{\dots <br><br>}<br><br>\text{\hspace{1em}<br><br>},L\end{array}\\ 0,& \mathrm{otherwise}\end{array}$

orthonormalizing the vectors of the matrix; and

generating the orthonormal basis set

${\left\{T^n\right\}}_{n=1}^{L-1}.$

4. An electrical impedance tomography method according to claim 1, wherein the second algorithm includes the steps of:

applying voltage T^k and measuring I^k, k=1, . . . L-1;

computing {circumflex over (B)} based on

$\hat{B}=\left[\begin{array}{cccc}\langle <br><br>T^1,I^1\rangle <br><br>& \langle <br><br>T^1,I^2\rangle <br><br>& \mathrm{\cdots <br><br>}& \langle <br><br>T^1,I^{L-1}\rangle <br><br>\\ \langle <br><br>T^2,I^1\rangle <br><br>& \langle <br><br>T^2,I^2\rangle <br><br>& \mathrm{\cdots <br><br>}& \langle <br><br>T^2,I^{L-1}\rangle <br><br>\\ \mathrm{⋮<br><br>}& \mathrm{⋮<br><br>}& \mathrm{⋮<br><br>}& \mathrm{⋮<br><br>}\\ \langle <br><br>T^{L-1},I^1\rangle <br><br>& \langle <br><br>T^{L-1},I^2\rangle <br><br>& \mathrm{\cdots <br><br>}& \langle <br><br>T^{L-1},I^{L-1}\rangle <br><br>\end{array}\right]$

computing i^d

$i^d=\left[\begin{array}{c}{i}_1^d\\ i_2^d\\ \mathrm{⋮<br><br>}\\ i_{L-1}^d\end{array}\right]=\left[\begin{array}{c}\langle <br><br>I^d,T^1\rangle <br><br>\\ \langle <br><br>I^d,T^2\rangle <br><br>\\ \mathrm{⋮<br><br>}\\ \langle <br><br>I^d,T^{L-1}\rangle <br><br>\end{array}\right];<br><br>$

and

generating {circumflex over (B)} and i^d.

5. An electrical impedance tomography method according to claim 1, wherein the voltages of the voltage vector are computed as a function of the estimate of the non-singular linear mapping matrix and the coordinate vector for the desired current vector by computing ν

^k=ν

^k-1+{circumflex over (B)}<sup>−

1</sup>e_k-1.

6. An electrical impedance tomography method according to claim 1, wherein the voltages of the voltage vector are applied as a function of the estimate of the non-singular linear mapping matrix and the coordinate vector for the desired current vector by applying

$V^k=\sum _{}^{}$

n = 1 L - 1 ⁢

v n k ⁢

 

⁢

T n .

7. An electrical impedance tomography method according to claim 1, wherein the coordinate vector for the measured resulting current vector is computed with respect to the orthonormal basis set by computing

$i^k=[\begin{array}{c}{i}_1^k\\ i_2^k\\ ⋮\end{array}$

i L - 1 k ] = [ 〈

I k , T 1 〉

〈

I k , T 2 〉

⋮

〈

I k , T L - 1 〉

] .

8. An electrical impedance tomography method according to claim 1, wherein a norm of the actual error between the coordinate vector for the measured resulting current vector and the coordinate vector for the desired current vector is calculated by computing e_k=i^d−

i^k.

9. A method for calculating the voltage that will generate a desired electrode current in an EIT system, comprising the steps of:

(a) selecting a desired current vector and an error tolerance;

(b) using a first algorithm to compute an orthonormal basis set;

(c) using a second algorithm with the orthonormal basis set and the desired current vector to compute an estimate of a non-singular linear mapping matrix for converting coordinate vector for voltage vector with respect to the orthonormal basis set to coordinate vector for current vector with respect to the orthonormal basis set and to compute coordinate vector for the desired current vector;

(d) using a third algorithm comprising the steps of;

(i) computing and applying to the electrodes, the voltages of the voltage vector as a function of the estimate of the non-singular linear mapping matrix and the coordinate vector for the desired current vector;

(ii) measuring the resulting current vector;

(iii) computing the coordinate vector for the measured resulting current vector with respect to the orthonormal basis set;

(iv) calculating a norm of the actual error between the coordinate vector for the measured resulting current vector and the coordinate vector for the desired current vector; and

(v) if the norm of the actual error is greater than the selected error tolerance, repeating steps (i) to (v), and if the norm of the actual error is less than the selected error tolerance, using the computed voltage vector of step (i) as a calculated voltage that will generate a desired electrode current.

10. An electrical impedance tomography method according to claim 9, wherein the first algorithm includes the steps of:

providing

let T^k;

L×

1 vector, k=1,2, . . . L-1

$T_i^k=\{\begin{array}{cc}1,& i=k\\ -1,& \begin{array}{cc}i=k+1,& i=1,2,\mathrm{\dots <br><br>}<br><br>\text{\hspace{1em}<br><br>},L\end{array}\\ 0,& \mathrm{otherwise}\end{array}$

orthonormalizing the vectors of the matrix; and

generating the orthonormal basis set

${\left\{T^n\right\}}_{n=1}^{L-1}.$

11. An electrical impedance tomography method according to claim 9, wherein the second algorithm includes the steps of:

applying voltage T^k and measuring I^k, k=1, . . . L-1;

computing {circumflex over (B)} based on

$\hat{B}=\left[\begin{array}{cccc}\langle <br><br>T^1,I^1\rangle <br><br>& \langle <br><br>T^1,I^2\rangle <br><br>& \mathrm{\cdots <br><br>}& \langle <br><br>T^1,I^{L-1}\rangle <br><br>\\ \langle <br><br>T^2,I^1\rangle <br><br>& \langle <br><br>T^2,I^2\rangle <br><br>& \mathrm{\cdots <br><br>}& \langle <br><br>T^2,I^{L-1}\rangle <br><br>\\ \mathrm{⋮<br><br>}& \mathrm{⋮<br><br>}& \mathrm{⋮<br><br>}& \mathrm{⋮<br><br>}\\ \langle <br><br>T^{L-1},I^1\rangle <br><br>& \langle <br><br>T^{L-1},I^2\rangle <br><br>& \mathrm{\cdots <br><br>}& \langle <br><br>T^{L-1},I^{L-1}\rangle <br><br>\end{array}\right]$

computing i^d

$i^d=\left[\begin{array}{c}{i}_1^d\\ i_2^d\\ \mathrm{⋮<br><br>}\\ i_{L-1}^d\end{array}\right]=\left[\begin{array}{c}\langle <br><br>I^d,T^1\rangle <br><br>\\ \langle <br><br>I^d,T^2\rangle <br><br>\\ \mathrm{⋮<br><br>}\\ \langle <br><br>I^d,T^{L-1}\rangle <br><br>\end{array}\right];<br><br>$

generating {circumflex over (B)} and i^d.

12. An electrical impedance tomography method according to claim 9, wherein the voltages of the voltage vector are computed as a function of the estimate of the non-singular linear mapping matrix and the coordinate vector for the desired current vector by computing ν

^k=ν

^k-1+{circumflex over (B)}<sup>−

1</sup>e_k-1.

13. An electrical impedance tomography method according to claim 9, wherein the voltages of the voltage vector are applied as a function of the estimate of the non-singular linear mapping matrix and the coordinate vector for the desired current vector by applying

$V^k=\sum _{}^{}$

n = 1 L - 1 ⁢

v n k ⁢

T n .

14. An electrical impedance tomography method according to claim 9, wherein the coordinate vector for the measured resulting current vector is computed with respect to the orthonormal basis set by computing

$i^k=[\begin{array}{c}{i}_1^k\\ i_2^k\\ ⋮\end{array}$

i L - 1 k ] = [ <

I k , T 1 >

<

I k , T 2 >

⋮

<

I k , T L - 1 >

] .

15. An electrical impedance tomography method according to claim 9, wherein a norm of the actual error between the coordinate vector for the measured resulting current vector and the coordinate vector for the desired current vector is calculated by computing e_k=i^d−

i^k.

16. A method for calculating the voltage that will generate a desired electrode current in an EIT system, comprising the steps of:

(a) selecting a desired current vector and an error tolerance;

(b) using a first algorithm to compute an orthonormal basis set by providing

let T^k;

L×

1 vector, k=1,2, . . . L-1

$i^d<br><br>T_i^k=\{\begin{array}{c}1,i=k\\ -1,i=k+1,i=1,2,\mathrm{\dots <br><br>}<br><br>\text{\hspace{1em}<br><br>},L\\ 0,\mathrm{otherwise}\end{array}$

orthonormalizing the vectors of the matrix and generating the orthonormal basis set {Tⁿ}_n=1^L-1;

(c) using a second algorithm which comprises applying voltage T^k and measuring I^k, k=1, . . . L-1;

computing {circumflex over (B)} based on

$\hat{B}=\left[\begin{array}{cccc}<<br><br>T^1,I^1><br><br>& <<br><br>T^1,I^2><br><br>& \mathrm{\dots <br><br>}& <<br><br>T^1,I^{L-1}><br><br>\\ <<br><br>T^2,I^1><br><br>& <<br><br>T^2,I^2><br><br>& \mathrm{\dots <br><br>}& <<br><br>T^2,I^{L-1}><br><br>\\ \mathrm{⋮<br><br>}& \mathrm{⋮<br><br>}& \mathrm{⋮<br><br>}& \mathrm{⋮<br><br>}\\ <<br><br>T^{L-1},I^1><br><br>& <<br><br>T^{L-1},I^2><br><br>& \mathrm{\dots <br><br>}& <<br><br>T^{L-1},I^{L-1}><br><br>\end{array}\right]$

computing i^d

$i^d=\left[\begin{array}{c}{i}_1^d\\ i_2^d\\ \mathrm{⋮<br><br>}\\ i_{L-1}^d\end{array}\right]=\left[\begin{array}{c}<<br><br>I^d,T^1><br><br>\\ <<br><br>I^d,T^2><br><br>\\ \mathrm{⋮<br><br>}\\ <<br><br>I^d,T^{L-1}><br><br>\end{array}\right]$

and generating {circumflex over (B)} and i^d;

(d) using a third algorithm comprising the steps of;

(i) computing and applying the voltages of the voltage vector by computing ν

^k=ν

^k-1+{circumflex over (B)}<sup>−

1</sup>e_k-1 and applying

$V^k=\underset{n=1}{\overset{L-1}{\sum <br><br>}}<br><br>v_n^k<br><br>T^n;<br><br>$

(ii) measuring the resulting current vector;

(iii) computing the coordinate vector for the measured resulting current vector with respect to the orthonormal basis set by computing

$i^k=\left[\begin{array}{c}{i}_1^k\\ i_2^k\\ \mathrm{⋮<br><br>}\\ i_{L-1}^k\end{array}\right]=\left[\begin{array}{c}<<br><br>I^k,T^1><br><br>\\ <<br><br>I^k,T^2><br><br>\\ \mathrm{⋮<br><br>}\\ <<br><br>I^k,T^{L-1}><br><br>\end{array}\right];<br><br>$

(iv) calculating a norm of the actual error between the coordinate vector for the measured resulting current vector and the coordinate vector for the desired current vector by computing e_k=i^d−

i^k; and

(v) if the norm of the actual error is greater than the selected error tolerance, repeating steps (i) to (v), and if the norm of the actual error is less than the selected error tolerance, using the computed voltage vector of step (i) as a calculated voltage that will generate a desired electrode current.