Spectroscopic imaging of a dilute cell suspension

Abstract The paper aims at analytically exhibiting for the first time the fundamental mechanisms underlying the fact that effective biological tissue electrical properties and their frequency dependence reflect the tissue composition and physiology. For doing so, a homogenization theory is derived to describe the effective admittivity of cell suspensions. A new formula is reported for dilute cases that gives the frequency-dependent effective admittivity with respect to the membrane polarization. Different microstructures are shown to be distinguishable via spectroscopic measurements of the overall admittivity using the spectral properties of the membrane polarization. The Debye relaxation times associated with the membrane polarization tensor are shown to be able to give the microscopic structure of the medium. A natural measure of the admittivity anisotropy is introduced and its dependence on the frequency of applied current is derived. A Maxwell–Wagner–Fricke formula is given for concentric circular cells, and the results can be extended to the random cases. A randomly deformed periodic medium is also considered and a new formula is derived for the overall admittivity of a dilute suspension of randomly deformed cells.

[1]  H. Schwan.  Electrical properties of tissue and cell suspensions. , 1957, Advances in biological and medical physics.

[2]  L. Rayleigh,et al.  LVI. On the influence of obstacles arranged in rectangular order upon the properties of a medium , 1892 .

[3]  L. Berlyand,et al.  Generalized Clausius–Mossotti Formula for Random Composite with Circular Fibers , 2001 .

[4]  Jean Plana.  Mémoire sur la Théorie du Magnétisme , 1855 .

[5]  Hugo Fricke.  A MATHEMATICAL TREATMENT OF THE ELECTRICAL CONDUCTIVITY OF COLLOIDS AND CELL SUSPENSIONS , 1924, The Journal of general physiology.

[6]  J. Schwartz,et al.  CONVERGENCE ALMOST EVERYWHERE OF OPERATOR AVERAGES. , 1955, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Rui Nie,et al.  Optimizing measurement of the electrical anisotropy of muscle , 2008, Muscle & nerve.

[8]  W. Jing.  Homogenization of Randomly Deformed Conductivity Resistant Membranes , 2014, 1406.0580.

[9]  D. Cioranescu,et al.  Homogenization in open sets with holes , 1979 .

[10]  Sverre Grimnes,et al.  INTERFACE PHENOMENA AND DIELECTRIC PROPERTIES OF BIOLOGICAL TISSUE , 2002 .

[11]  C. Poignard.  Asymptotics for steady‐state voltage potentials in a bidimensional highly contrasted medium with thin layer , 2008 .

[12]  Damijan Miklavčič,et al.  Time course of transmembrane voltage induced by time-varying electric fields—a method for theoretical analysis and its application , 1998 .

[13]  Jin Keun Seo,et al.  Effective Admittivity of Biological Tissues as a Coefficient of Elliptic PDE , 2013, Comput. Math. Methods Medicine.

[14]  G. Westby,et al.  Time domain processing of electric organ discharge waveforms by pulse-type electric fish. , 1986, Brain, behavior and evolution.

[15]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[16]  John William Strutt,et al.  Scientific Papers: On the Influence of Obstacles arranged in Rectangular Order upon the Properties of a Medium , 2009 .

[17]  Pierre-Louis Lions,et al.  Stochastic homogenization and random lattices , 2007 .

[18]  H. Ammari,et al.  Effective parameters of elastic composites , 2006 .

[19]  P. Souganidis,et al.  Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments , 2011, 1103.2814.

[20]  G. Nguetseng.  A general convergence result for a functional related to the theory of homogenization , 1989 .

[21]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[22]  Liju Yang,et al.  Electrical impedance spectroscopy for detection of bacterial cells in suspensions using interdigitated microelectrodes. , 2008, Talanta.

[23]  Habib Zribi,et al.  Asymptotic expansions for the voltage potentials with two‐dimensional and three‐dimensional thin interfaces , 2011 .

[24]  A. Angersbach,et al.  Effects of pulsed electric fields on cell membranes in real food systems , 2000 .

[25]  Christopher L. Davey,et al.  The dielectric properties of biological cells at radiofrequencies : Applications in biotechnology , 1999 .

[26]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[27]  Y. Almog.  Averaging of Dilute Random Media: A Rigorous Proof of the Clausius–Mossotti Formula , 2013 .

[28]  Seward B Rutkove,et al.  Discriminating neurogenic from myopathic disease via measurement of muscle anisotropy , 2009, Muscle & nerve.

[29]  S. Kozlov,et al.  The method of averaging and walks in inhomogeneous environments , 1985 .

[30]  D. Rubens,et al.  Shear wave dispersion measures liver steatosis. , 2012, Ultrasound in medicine & biology.

[31]  S. Kozlov.  AVERAGING OF RANDOM OPERATORS , 1980 .

[32]  C. Poignard.  About the transmembrane voltage potential of a biological cell in time-harmonic regime. , 2009 .

[33]  H. Schwan.  Mechanisms responsible for electrical properties of tissues and cell suspensions. , 1993, Medical progress through technology.

[34]  Grégoire Allaire,et al.  Homogenization of a Conductive and Radiative Heat Transfer Problem , 2008, Multiscale Model. Simul..

[35]  Sara Monsurrò,et al.  Homogenization of a two-component composite with interfacial thermal barrier , 2003 .

[36]  Claude Le Bris,et al.  Une variante de la thorie de l'homognisation stochastique des oprateurs elliptiques , 2006 .

[37]  A. Einstein.  Eine neue Bestimmung der Moleküldimensionen , 1905 .

[38]  H. Fricke,et al.  A Mathematical Treatment of the Electric Conductivity and Capacity of Disperse Systems I. The Electric Conductivity of a Suspension of Homogeneous Spheroids , 1924 .

[39]  V. Zadorozhnaya,et al.  Mathematical modeling of membrane polarization occurring in rocks due to applied electrical field , 2009 .

[40]  Habib Ammari,et al.  Effective viscosity properties of dilute suspensions of arbitrarily shaped particles , 2012, Asymptot. Anal..

[41]  L. Slater,et al.  Resistivity and induced polarization monitoring of salt transport under natural hydraulic gradients , 2000 .

[42]  J. Nédélec.  Acoustic and Electromagnetic Equations : Integral Representations for Harmonic Problems , 2001 .

[43]  H. Fricke,et al.  A Mathematical Treatment of the Electric Conductivity and Capacity of Disperse Systems ii. The Capacity of a Suspension of Conducting Spheroids Surrounded by a Non-Conducting Membrane for a Current of Low Frequency , 1925 .

[44]  B H Brown,et al.  The use of electrical impedance spectroscopy in the detection of cervical intraepithelial neoplasia , 2006, International Journal of Gynecologic Cancer.

[45]  Habib Ammari,et al.  Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory , 2010 .

[46]  Y. Feldman,et al.  Time domain dielectric spectroscopy study of human cells. II. Normal and malignant white blood cells. , 1999, Biochimica et biophysica acta.

[47]  Graeme W. Milton,et al.  Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics , 2003 .

[48]  B. Brown,et al.  Relation between tissue structure and imposed electrical current flow in cervical neoplasia , 2000, The Lancet.

[49]  G. Milton.  The Theory of Composites , 2002 .

[50]  Koji Asami,et al.  Characterization of biological cells by dielectric spectroscopy , 2002 .

[51]  Koji Asami,et al.  Characterization of heterogeneous systems by dielectric spectroscopy , 2002 .

[52]  Thomas Deffieux,et al.  Shear Wave Spectroscopy for In Vivo Quantification of Human Soft Tissues Visco-Elasticity , 2009, IEEE Transactions on Medical Imaging.

[53]  Grégoire Allaire,et al.  Homogenization of a Conductive, Convective, and Radiative Heat Transfer Problem in a Heterogeneous Domain , 2013, SIAM J. Math. Anal..

[54]  H. Shapiro,et al.  Poincaré’s Variational Problem in Potential Theory , 2007 .

[55]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[56]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[57]  Habib Ammari,et al.  Boundary layer techniques for deriving the effective properties of composite materials , 2005 .

[58]  Michael C. McAlpine,et al.  Electrical detection of pathogenic bacteria via immobilized antimicrobial peptides , 2010, Proceedings of the National Academy of Sciences.

[59]  H. Fricke,et al.  The Maxwell-Wagner Dispersion in a Suspension of Ellipsoids , 1953 .

[60]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .