Design of a multi-electrode array to measure cardiac conductivities

Accurate determination of cardiac tissue parameters is essential in bidomain models that simulate the electrical activity of the heart and thereby contribute to understanding cardiovascular disease. Recent experimental work indicated the need for six parameters, which measure electrical conductivity in two domains (extracellular and intracellular), along and across the cardiac fibres within a sheet and also between sheets. This is in contrast to the available experimentally determined conductivities, which are sets of four values, where it is assumed that conductivities across the fibres within a sheet and between the fibre sheets are equal. This study presents a mathematical model that incorporates six bidomain conductivities. It also discusses the design of a multi-electrode array and inversion method to retrieve these conductivities (as well as a value for fibre rotation between the sheets). The sensitivity of electrode spacing in the array design is investigated. References R. M. Arthur and D. B. Geselowitz. Effect of inhomogeneities on the apparent location and magnitude of a cardiac current dipole source. IEEE Trans. Biomed. Eng., 17:141--146, 1970. doi:10.1109/TBME.1970.4502713 R. C. Barr and R. Plonsey. Electrode systems for measuring cardiac impedances using optical transmembrane potential sensors and interstitial electrodes--theoretical design. IEEE Transactions on Biomedical Engineering, 50(8):925--934, 2003. doi:10.1109/TBME.2003.814529 Bryan J. Caldwell, Mark L. Trew, Gregory B. Sands, Darren A. Hooks, Ian J. LeGrice, and Bruce H. Smaill. Three distinct directions of intramural activation reveal nonuniform side-to-side electrical coupling of ventricular myocytes. Circulation: Arrhythmia and Electropysiology, 2:433--440, 2009. doi:10.1161/CIRCEP.108.830133 R. H. Clayton, O. Bernus, E. M. Cherry, H. Dierckx, F. H. Fenton, L. Mirabella, A. V. Panfilov, F. B. Sachse, G. Seemann, and H. Zhang. Models of cardiac tissue electrophysiology: Progress, challenges and open questions. Progress in Biophysics and Molecular Biology, 104(1--3):22--48, 2011. doi:10.1016/j.pbiomolbio.2010.05.008 L. Clerc. Directional differences of impulse spread in trabecular muscle from mammalian heart. Journal of Physiology, 255:335--346, 1976. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1309251/ Eliad Gilboa, Patricio S. La Rosa and Arye Nehorai. Estimating electrical conductivity tensors of biological tissues using microelectrode arrays. Annals of Biomedical Engineering, 40(10):2140--2155, 2012. doi:10.1007/s10439-012-0581-9 D. A. Hooks and M. L. Trew. Construction and validation of a plunge electrode array for three-dimensional determination of conductivity in the heart. IEEE Transactions on Biomedical Engineering, 55(2):626--635, 2008. doi:10.1109/TBME.2007.903705 Darren Hooks. Myocardial segment-specific model generation for simulating the electrical action of the heart. BioMedical Engineering OnLine, 6(1):21--21, 2007. doi:10.1186/1475-925X-6-21 Darren A. Hooks, Karl A. Tomlinson, Scott G. Marsden, Ian J. LeGrice, Bruce H. Smaill, Andrew J. Pullan, and Peter J. Hunter. Cardiac microstructure: Implications for electrical propagation and defibrillation in the heart. Circulation Research, 91(4):331--338, 8 2002. doi:10.1161/01.RES.0000031957.70034.89 Darren A. Hooks, Mark L. Trew, Bryan J. Caldwell, Gregory B. Sands, Ian J. LeGrice, and Bruce H. Smaill. Laminar arrangement of ventricular myocytes influences electrical behavior of the heart. Circulation Research, 101(10):e103--112, 2007. doi:10.1161/CIRCRESAHA.107.161075 Barbara M. Johnston and Peter R. Johnston. Possible four-electrode configurations for measuring cardiac tissue fibre rotation. IEEE Transactions on Biomedical Engineering, 54(3):547--550, 2007. doi:10.1109/TBME.2006.890511 Barbara M. Johnston, Peter R. Johnston, and David Kilpatrick. Analysis of electrode configurations for measuring cardiac tissue conductivities and fibre rotation. Annals of Biomedical Engineering, 34(6):986--996, June 2006. doi:10.1007/s10439-006-9098-4 Barbara M. Johnston, Peter R. Johnston, and David Kilpatrick. A new approach to the determinination of cardiac potential distributions: Application to the analysis of electrode configurations. Mathematical Biosciences, 202(2):288--309, 2006. doi:10.1016/j.mbs.2006.04.004 Barbara M. Johnston, Peter R. Johnston, and David Kilpatrick. A solution method for the determination of cardiac potential distributions with an alternating current source. Computer Methods in Biomechanics and Biomedical Engineering, 11(3):223--233, 2008. doi:10.1080/10255840701747594 Peter R. Johnston. Cardiac conductivity values--A challenge for experimentalists? Noninvasive Functional Source Imaging of the Brain and Heart and 2011 8th International Conference on Bioelectromagnetism (NFSI and ICBEM), pages 39--43, 13--16 May 2011. doi:10.1109/NFSI.2011.5936816 P. Le Guyader, P. Savard, R. Guardo, L. Pouliot, F. Trelles, and M. Meunier. Myocardial impedance measurements with a modified four electrode technique. 16th IEEE-EMBS, pages 880--881, 1994. doi:10.1109/IEMBS.1994.415193 P. Le Guyader, F. Trelles, and P. Savard. Extracellular measurement of anisotropic bidomain myocardial conductivities. I. Theoretical analysis. Annals of Biomedical Engineering, 29:862--877, 2001. doi:10.1114/1.1408923 M. C. MacLachlan, J. Sundnes, and G. T. Lines. Simulation of ST segment changes during subendocardial ischemia using a realistic 3-D cardiac geometry. IEEE Transactions on Biomedical Engineering, 52(5):799--807, 2005. doi:10.1109/TBME.2005.844270 R. Plonsey and R. C. Barr. The four-electrode resistivity technique as applied to cardiac muscle. IEEE Transactions on Biomedical Engineering, 29(7):541--546, 1982. doi:10.1109/TBME.1982.324927 A. E. Pollard and R. C. Barr. Cardiac microimpedance measurement in two-dimensional models using multisite interstitial stimulation. American Journal of Physiology-Heart and Circulatory Physiology, 290(5):H1976--H1987, 2006. doi:10.1152/ajpheart.01180.2005 A. E. Pollard, C. D. Ellis, and W. M. Smith. Linear electrode arrays for stimulation and recording within cardiac tissue space constants. Biomedical Engineering, IEEE Transactions on, 55(4):1408--1414, 2008. doi:10.1109/TBME.2007.912401 Andrew E. Pollard and Roger C. Barr. A biophysical model for cardiac microimpedance measurements. American Journal of Physiology-Heart and Circulatory Physiology, 298:H1699--H1709, 2010. doi:10.1152/ajpheart.01131.2009 D. E. Roberts, L. T. Hersh, and A. M. Scher. Influence of cardiac fiber orientation on wavefront voltage, conduction velocity and tissue resistivity in the dog. Circ. Res., 44:701--712, 1979. doi:10.1161/01.RES.44.5.701 D. E. Roberts and A. M. Scher. Effects of tissue anisotropy on extracellular potential fields in canine myocardium in situ. Circ. Res., 50:342--351, 1982. doi:10.1161/01.RES.50.3.342 B. J. Roth. Electrical conductivity values used with the bidomain model of cardiac tissue. IEEE Transactions on Biomedical Engineering, 44(4):326--328, April 1997. doi:10.1109/10.563303 S. Rush, J. A. Abildskov, and R. McFee. Resistivity of body tissues at low frequencies. Circulation Research, 12:40--50, 1963. doi:10.1161/01.RES.12.1.40 R. Sadleir and C. Henriquez. Estimation of cardiac bidomain parameters from extracellular measurement: Two dimensional study. Annals of Biomedical Engineering, 34(8):1289--1303, 2006. doi:10.1007/s10439-006-9128-2 O. H. Schmitt. Biological information processing using the concept of interpenetrating domains. In K. N. Leibovic, editor, Information Processing in the Nervous System, chapter 18, pages 325--331. Springer--Verlag, New York, 1969. Kirsten H. W. J. Ten Tusscher, R. Hren, and A. V. Panfilov. Organization of ventricular fibrillation in the human heart. Circulation Research, 100(12):87--101, 2007. doi:10.1161/CIRCRESAHA.107.150730 Mark L. Trew, Bryan J. Caldwell, Thiranja P. Barbarenda Gamage, Gregory B. Sands, and Bruce H. Smaill. Experiment-specific models of ventricular electrical activation: Construction and application. In 30th Annual International IEEE EMBS Conference, pages 137--140, 2008. doi:10.1109/IEMBS.2008.4649109 L. Tung. A Bi-domain model for describing ischaemic myocardial DC potentials. PhD thesis, Massachusetts Institute of Technology, June 1978. http://hdl.handle.net/1721.1/16177 Mikael Wallman, Nicolas P. Smith, and Blanca Rodriguez. A comparative study of graph-based, eikonal, and monodomain simulations for the estimation of cardiac activation times. IEEE Transactions in Biomedical Engineering, 59(6):1739--1748, 2012. doi:10.1109/TBME.2012.219339