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. Anatomical Model of the breast The modeled geometry for 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@rpi.edu Rensselaer Polytechnic Institute Web site: http://www.rpi.edu/~newelj/eit.html 110 Eighth St. Troy, NY 12180-3590 Phone : 518-276-6433 FAX : 518-276-3035 Current Patterns arranged in order of increasing spatial frequency