4.0 Crossover Technologies in Geophysics and Biomedical Imaging Chester J. Weiss

 

4.1  Introduction

 

A common goal to both the medical and geoscience communities is to address the problem of developing non-invasive techniques for peering into the interior of an object.  Other disciplines share this same objective (such as industrial engineering for quality assurance testing) but for brevity we restrict our discussion to those technologies that are well established in the geosciences (specifically geophysics) and find application as a method for biomedical imaging.  Furthermore, we will narrow the expanse of geophysical methods by examining only those that utilize quasi-static electromagnetic fields and electrical impedance estimates for in situ characterization of biological structures within the human body.  While at the surface this may seem overly restrictive, these methods are still very much in the research development phase in the biomedical imaging community and thus represent areas of professional cross-fertilization between geophysicists and medical imaging specialists.

 

In short, the problem of imaging an object interior via some non-invasive method can be broadly lumped into two primary steps:  measurement of some physical quantity (such as voltage, current, or the electromagnetic fields) on the surface of the body; and utilization of an algorithm which infers the distribution of some physical property (such as electrical conductivity) within the body based solely on these surface measurements.  Interpretation of the spatial distribution of physical properties in terms of other meaningful quantities (e.g., Is this area of anomalous electrical conductivity in the image a tumor or benign fatty deposit?) is beyond the scope of this paper but it's worth mentioning that in geophysics and medical imaging alike, the interpretation problem is often based on heuristic arguments which assimilate a variety of complementary datasets and observations.

 

 

4.2  Geophysical Imaging Methods

 

Two imaging methods used by geophysicists for subsurface imaging have recently found utility as methods for imaging the interior of the human body:  electrical impedance tomography (EIT) and magnetic induction tomography (MIT).  The principles of operation for each method are based on the observation that different biological tissues have varying capacities for sustaining a flow of electric current (Schwan and Takashima, 1993; Jossinet and Lavandier, 1998), and is influenced by, for example, the density of vascularization in a given region.  The variations in vascularization can, in some cases, be readily correlated to differences in tissue type such as lung, muscle, and bone tissues in the thorax.  In other cases, the variations in the density of vascularization, and hence electrical conductivity, can be attributed to the presence of tumorous growths within the body.  Thus, medical imaging by mapping electrical conductivity gives rise to two key applications: monitoring of pulmonary activity (e.g. Metherall, 1998), screening for early cancer  detection (Cherpenin et al., 2001; Wang et al., 2001).  Other applications of EIT include imaging of human head (Tidswell et al., 2001) and gastrointestinal and esophogeal function (e.g. Devane, 1993; Erol et al., 1995).

 

 

 

Figure 4.1  Experimental setup (top) for EIT imaging of the upper arm.  EIT image of upper arm (lower-left) and interpretation of EIT image (lower-right).  Graphics from Dartmouth,  Thayer School of Engineering (www-nml.dartmouth.edu/biomedprg/EIS/index.html).

 

The EIT methods utilize an array of electrodes that are placed in direct electrical contact with the skin.  The experiment proceeds as a small amount of electric current is injected into the skin through a pair of electrodes that, in turn, propagates through body and generates a potential difference elsewhere on the skin that can be measured by the other electrodes in the array.  Thus, the distribution of measure voltages on the skin is directly affected by the electric current paths generated within the body, and hence, the distribution of electrical conductivity within the body.  The method is attractive because data acquisition time is relatively brief (on the order of a few minutes), the electrodes are small and easily attached/removed (Figure 4.1), and the acquisition computer is usually small and portable (Figure 4.2), thus providing an imaging tool which can be used on subjects who have limited physical mobility or are bedridden.  Currents are applied with a harmonic time-dependence and typically operate in the frequency range of D.C to 1MHz.  If induction is neglected, spatial variation of the measured voltages are dictated by a complex-valued Laplace equation for the electric potential—an equation that is readily solved in 2D/3D by applied mathematicians and geophysicists alike using the usual suite of discrete numerical methods such as finite elements or finite differences (e.g. Molinari et al., 2001).

 

 

Figure 4.2  Data acquisition computer for Dartmouth EIT imaging system (www-nml.dartmouth.edu/biomedprg/EIS/index.html).

 

An example of an experimental setup for the imaging of breast tissues is provided in Figures 4.3 to 4.5.  The approach used at Dartmouth University illustrates the ability to discriminate between benign and malignant breast tissues using the EIT method.

 

 

Figure 4.3.  Experimental setup for Dartmouth EIT breast cancer imaging system (www-nml.dartmouth.edu/biomedprg/EIS/index.html).

 

 

Figure 4.4  Anatomy of a human breast for reference in the EIT images of Figure 4.5 (www-nml.dartmouth.edu/biomedprg/EIS/index.html).

 

 

Figure 4.5  EIT images a normal human breast (top), breast with benign cyst (middle) and breast with tumor (bottom) measured using the Dartmouth EIT system.  Actual data on is on the left, interpretation of the data is on the right (www-nml.dartmouth.edu/biomedprg/EIS/index.html).

In contrast to the EIT methods, is MIT method (also known as the “induced-EIT” or “touchless” method) works along the same operating principles as the low-frequency induction methods of geophysics whereby an electric current is induced (according to Faraday's Law of Induction) into the electrically conducting body by a nearby antenna loop which is powered by a small time-harmonic electric current.  The electrical eddy currents induced in the conducting body therefore also time-harmonic and generated secondary electromagnetic fields that can be measured directly by electrodes (Gencer et al., 1994) or by a separate induction coil (Gencer, 1999).  In conjunction with the usual EIT method, the MIT method has the potential to greatly improve the resolution of electrical anomalies within the body since the orientation and location of the transmitting antenna is not constrained to the surface of the skin, but rather, can be located almost arbitrarily above the body and therefore subject the body to a greater variety of EM excitation modes.

 

Mathematical solution to the EM induction problem in 3D is significantly more challenging than that the EIT problem.  In short, the 3D induction problem is characterized by finding solutions of a vector function that obey the complex valued Helmholtz equation: a solution that typically is wrenched out a large, ill-conditioned system of equations (e.g. Newman and Alumbaugh, 1995; Everett and Schultz 1995).  For MIT to be practical, the solution to this equation must be available at a minimum of computational cost, a requirement that is presently remedied by so-called `approximate' methods (Gencer et al., 1994).  However, the highly non-linear (with respect to electrical conductivity) nature of the governing partial differential equation makes reliance on the approximation methods a risky coarse of action.  Instead, rapid and resource-efficient solutions to the fully 3D problem recently appearing in the geophysical literature (e.g. Weiss, 2001; Weiss and Newman, 2002) may find application in the MIT problem.

 

Solution to the governing equations for both the EIT and MIT problems is a necessary step in any frequentist-based (in contrast to a Bayesian approach) image reconstruction algorithm whereby the image is computed by minimizing the misfit between the predicted and observed data, usually subject to some stabilization or regularization criterion (e.g. Constable et al., 1987).  Techniques presently used in the EIT reconstruction problem include linearization of the conductivity term (Mueller et al., 1999), one-step approximate Newton method (Le Hyraic and Pidcock, 2001) and multiple-step Newton method (Edic et al., 1998).   Similar strategies are utilized within the geophysics community (e.g. Non-linear conjugate gradients in Newman and Alumbaugh, 2000; Figure 4.6) however the application of other more “exotic” techniques, such as homotopy (Jegen et al., 2001), have yet to appear in the medical imaging literature.

 

Figure 4.6  Front (left) and back (right) image of a resistive oil-bearing formation in the subsurface reconstructed using a non-linear conjugate gradient inversion of synthetic electromagnetic induction data.  Courtesy of G. Newman, Sandia National Laboratories.

 

 

4.3  References

 

Cherepenin V., Karpov, A., Korjenevsky, A., Kornienko, V., Mazaletskaya, A., Mazourov, D., and Meister, D. (2001) A 3D electrical impedance tomography (EIT) system for breast cancer detection.  Physiological Measurement, 22, 9-18.

 

Constable, S. C., Parker, R. L., and Constable, C. G. (1987) Occam's inversion:  A practical algorithm for generating smooth models from electromagnetic sounding data.  Geophysics, 52, 289-300.

 

Devane, S. P. (1993) Application of EIT to gastric emptying in infants:  Validation against residual volume method.  In D. Holder, Ed., Clinical and Physiological Application of Electrical Impedance Tomography.  UCL Press, London, 113-123.

 

Edic, P. M., Isaacson, D., Saulnier, G. L., Jain, H., and Newell, J. C. (1998) A iterative Newton-Raphson method to solve the inverse admittivity problem.  IEEE Transactions, of Biomedical Engineering, 45, 899-908.

 

Erol, R. A., Mangnall, R., Leathard, A. C., Smallwood, R. H., Brown, B. H., Cherian, P., and Bardhan, K. D. (1995) Identifying esophogeal contents using electrical impedance tomography.  Physiological Measurement, 16, 253-261.

 

Everett, M. E. and Schultz A. (1995) Electromagnetic induction in eccentrically nested spheres.  Physics of the Earth and Planetary Interiors, 92, 189-198.

 

Gencer, N. G., Kuzuoglu, M., and Ider, Y. Z. (1994) Electrical impedance tomography using induced currents.  IEEE Transactions on Medical Imaging, 13, 338-350.

 

Gencer, N. G. (1999) Electrical conductivity imaging via contactless measurements.  IEEE Transactions on Medical Imaging, 18, 617-627.

 

Jegen, M. D., Everett, M. E., and Schultz, A. (2001) Using homotopy to invert geophysical data.  Geophysics, 66, 1749-1760.

 

Jossinet, J. and Lavandier, B. (1998) The discrimination of excised cancerous breast tissue samples using impedance spectroscopy.  Bioelectrochemistry and Bioenergetics, 45, 161-167.

 

Le Hyaric, A. and Pidcock, M. K. (2001) A image reconstruction algorithm for three-dimensional electrical impedance tomography.  IEEE Transactions, of Biomedical Engineering, 48, 230-235.

 

Metherall, P, (1998) Three dimensional electrical impedance spectroscopy of the human thorax.  Ph.D. Thesis, University of Sheffield, U.K.

 

Molinari, M., Cox, S. J., Blott, B. H., and Baniell, G. J. (2001) Adaptive mesh refinement techniques for electrical impedance tomography.  Physiological Measurement, 22, 91-96.

 

Mueller, J. L., Isaacson, D., and Newell, J. C. (1999) A reconstruction algorithm for electrical impedance tomography data collected on rectangular arrays.  IEEE Transactions, of Biomedical Engineering, 46, 1379-1386.

 

Newman, G. A. and Alumbaugh, D. L. (1995) Frequency-domain modelling of airborne electromagnetic responses using staggered finite differences.  Geophysical Prospecting, 43, 1021-1042.

 

Newman, G. A. and Alumbaugh, D. L. (2000) Three-dimensional magnetotelluric inversion using non-linear conjugate gradients.  Geophysical Journal International, 140, 410-424.

 

Schwan, H. P. and Takashima, S. (1993) Electrical conduction and dielectric behavior in biological systems.  Encyclopedia of Applied Physics, VHC Publishers, New York, 177-200.

 

Tidswell, A. T., Gibson, A., Bayford, R. H., and Holder, D. S. (2001) Validation of a 3D reconstruction algorithm for EIT of human brain function in a  realistic head-shaped tank.  Physiological Measurement, 22, 177-185.

 

Wang, W., Tang, M., McCormick, M., and Dong, X. (2001) Preliminary results from an EIT breast imaging simulation system.  Physiological Measurement, 22, 39-48.

 

Weiss, C. J. (2001) A matrix--free approach to solving the fully 3D electromagnetic induction problem, 3-D anisotropic Earth.  71st Annual Meeting of the Society of Exploration Geophysicists, San Antonio TX.

 

Weiss, C. J. and Newman, G. A. (2002) A fast preconditioner for the electromagnetic induction problem in 3D anisotropic media.  Geophysics, in press.