The current density at each interfacial layer'

1
A Layered Model for Breasts in Electrical
Impedance Tomography
Rujuta Kulkarni, Greg Boverman, David Isaacson,
Gary Saulnier and Jonathan Newell.
Rensselaer Polytechnic Institute, Troy, NY.
Motivation The observed conductivities in
compressed breasts in EIT are smaller than those
seen previously in whole chest imaging. Looking
at the anatomical breast model we could attribute
this to a thin resistive skin layer present in
breasts. To test this hypothesis and try to more
accurately model breasts, we have developed a
layered analytical forward model. Our layered
model has three layers, thin top and bottom
layers representing skin and a thicker middle
layer representing breast tissue.
Why do we go ahead with the Layered Model? The
voltages measured from the patients in clinical
trials were compared with those obtained from the
Layered model and Homogenous model. It is seen
that the voltage from the Layered model fit the
patient data better than the voltages from the
Homogenous model.
The modeled geometry for the breast
Anatomical Model of the breast
This work is supported in part by CenSSIS, the
Center for Subsurface Sensing and Imaging
Systems, under the Engineering Research Centers
Program of the National Science Foundation (Award
Number EEC-9986821) and by NIBIB, the National
Institute of Biomedical Imaging and
Bioengineering under Grant Number R01-EB000456-02.
Homogenous Model
Blue
vs. Current
Patterns Red
vs.
Current Patterns
Forward Voltages in the first, second and third
layers respectively
Admitivitty in the first and third layers
Admitivitty in the second layer
The calculation and fitting of V_ThreeLayer to
the patient data V_Patient however needs
estimation of and . This
is a nonlinear optimization problem of estimating
the uniform conductivity and permittivity within
each layer that best fits the experimental
measurements in the least square sense. The cost
function to be optimized is
Data matrix uses the
voltages calculated using the homogenous model.
The problem is based on the following conditions
being satisfied

• The forward voltage is
  continuous at every point inside the body.

• The current density
  at each
  interfacial layer.

Six Simultaneous equations that define our
forward problem
Interfacial Layers
L number of electrodes K number of current
patterns applied Presently we are using iterative
methods for this nonlinear optimization. We
intend to further explore the possibilities of
obtaining an analytical solution to this problem.
Reconstruction
Generation of Simulated data We generated data
with finite difference simulations. We built a
three layered rectangular body with a target
centered in the upper half of the body (
). We tried the reconstruction
with both the Layered model (
) and the Homogenous model (
) as the forward solver.

• Future Work
• Exploring iterative and analytical methods for
  more accurate estimation of
• and .
• Studying the effect of error in this estimation
  on the reconstructed images.
• Constructing a reconstructor consistent with the
  Layered Model which includes building the
  Jacobian defined as
• Applying the Layered model to the patient data
  and comparing the reconstructions with those
  obtained with the Homogenous model.

Top and Bottom surfaces
In order to make sure the finite difference data
is consistent with the analytical solution, the
finite difference target data was scaled. The
scaling procedure calculated a scaling factor for
each voltage pattern which best fit the finite
difference data for that particular pattern to
the corresponding analytical voltage pattern.
Our earlier work has used the AveGap electrode
model along with a homogenous body, which assumes
a constant admittivity throughout the body. We
expect the high resistance skin layer included in
the new geometry to affect different applied
spatial frequency patterns differently. The low
spatial frequency current patterns result in more
current flow through the interior of the body
than the higher spatial frequency patterns which
result in current flow mostly in the periphery.
As a result the voltages produced by the high
spatial frequency current patterns are expected
to be affected more by the addition of lower
admittivity skin layers. We can check this
expectation by comparing the voltages from the
AveGap Homogenous and Layered Forward Models.
Layered Model
References Publications Acknowledging NSF
Support 1. Ning Liu, Gary J. Saulnier, J.C.
Newell, D. Isaacson and T-J Kao. ACT4 A
High-Precision, Multi-frequency Electrical
Impedance Tomography Conference on
Biomedical Applications of Electrical Impedance
Tomography, University College London,
June 22-24th, 2005. 2 . Choi, M.H., T-J. Kao, D.
Isaacson, G.J. Saulnier and J.C. Newell A
Reconstruction Algorithm for Breast Cancer
Imaging with Electrical Impedance Tomography in
Mammography Geometry IEEE Trans. Biomed. Eng.
54(4) (In Press), 2007.
Data matrix
uses the voltages calculated using the
Layered model.
V_ThreeLayer Forward Voltages calculated
from the Layered Forward Solver V_Homogenous
Forward Voltages calculated from the Homogenous
Forward Solver for the
same applied current pattern set.
Contact Info Gary Saulnier, Ph. D. Professor of
Electrical Engineering E-mail saulng_at_rpi.edu
Rensselaer Polytechnic Institute Web site
http//www.rpi.edu/newelj/eit.html 110 Eighth
St. Troy, NY 12180-3590 Phone [phone]
FAX [phone]
Current Patterns arranged in order of increasing
spatial frequency